A LEAST-SQUARES FINITE ELEMENT METHOD FOR …...A LEAST-SQUARES FINITE ELEMENT METHOD FOR THE STOKES AND NAVIER-STOKES EQUATIONS A thesis submitted to The University of Manchester

A LEAST-SQUARES FINITE ELEMENT METHOD

FOR THE STOKES AND NAVIER-STOKES

EQUATIONS

A thesis submitted to The University of Manchester Institute of

Science and Technology

for the degree of Doctor of Philosophy

By

Paul Bolton

Department of Mathematics

December 2002

Abstract

THE UNIVERSITY OF MANCHESTER INSTITUTE

OF SCIENCE AND TECHNOLOGY

ABSTRACT OF THESIS submitted by Paul Bolton for the Degree of Doctor

of Philosophy and entitled A Least-Squares Finite Element Method for the

Stokes and Navier-Stokes Equations.

Month and Year of Submission: December 2002

In this thesis the least-squares finite element method for first-order systems is set

out. We present a number of established first-order reformulations of the planar Stokes

equations. In particular, we discuss in detail a reformulation which involves the gradi-

ents of the stress and stream functions. Difficulties in modelling fluid flow common to

all of these methods are highlighted with a set of test problems and possible methods

of ameliorating these deficiencies discussed. We introduce a new recasting of the planar

Navier-Stokes equations as a first-order system based on stress and stream functions

and give results obtained with this formulation. Formulations based on stress func-

tions which are equivalent to the Stokes and Navier-Stokes systems of equations in

three dimensions are developed.

ii

Declaration

No portion of the work referred to in this thesis has been submitted in support of an

application for another degree or qualification of this or any other university or other

institution of learning.

iii

Acknowledgements

The guidance of my supervisor Ronald Thatcher has been invaluable. I would also

like to thank the Engineering and Physical Sciences Research Council for the financial

assistance they gave me. The advice, companionship and encouragement of my fellow

students and other researchers has also been much appreciated. Special mention is

due to Ozgur Akman, Thebe Basebi, Richard Booth, Natasha Kenny, Sean Norburn,

Catherine Powell and Akeel Shah. My family has also been very supportive over the

entire course of my studies.

iv

Contents

Abstract ii

Declaration iii

Acknowledgements iv

List of Figures xii

List of Tables xvi

1 The Least-Squares Finite Element Method 1

1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The Least-Squares Finite Element Method . . . . . . . . . . . . . . . . . 6

1.2.1 Other Variational Formulations . . . . . . . . . . . . . . . . . . . 7

1.2.2 The Least-Squares Variational Method . . . . . . . . . . . . . . . 7

1.3 Implementing the Finite Element Method . . . . . . . . . . . . . . . . . 13

1.3.1 Linear Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

v

Contents vi

1.3.2 An Example Finite Element Implementation of the Galerkin For-

mulation of Poisson’s Equation . . . . . . . . . . . . . . . . . . . 15

1.3.3 An Example Least-Squares Finite Element Method . . . . . . . 19

2 First-Order Reformulations of the Stokes System of Equations 22

2.1 The Stokes Equations for Incompressible Flow in the Plane . . . . . . . 22

2.2 The Mixed Finite Element Method . . . . . . . . . . . . . . . . . . . . . 23

2.3 The Stress and Stream Function Reformulation . . . . . . . . . . . . . . 26

2.4 Other First-Order Reformulations of the Stokes Equations . . . . . . . . 30

3 Experimental Comparison of First-Order Stokes Systems 38

3.1 Poiseuille Flow in a Square Region . . . . . . . . . . . . . . . . . . . . . 39

3.1.1 The Finite Element Grid Used in the Solution of Poiseuille Flow 39

3.1.2 Error Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.3 Results for the S Formulation . . . . . . . . . . . . . . . . . . . . 41

3.1.4 Results for the J Formulation . . . . . . . . . . . . . . . . . . . . 47

3.1.5 Results for the G Formulations . . . . . . . . . . . . . . . . . . . 50

3.1.6 Summary of Results in the Square Region . . . . . . . . . . . . . 54

3.2 Poiseuille Flow in a Long Channel . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Exact solution in the S Formulation . . . . . . . . . . . . . . . . 56

3.2.2 Results for the S Formulation . . . . . . . . . . . . . . . . . . . . 56

Contents

Contents vii

3.2.3 Normal Velocities and Tangential Stresses . . . . . . . . . . . . . 64

3.2.4 Tangential Velocities and Normal Stresses . . . . . . . . . . . . . 67

3.2.5 Summary of Results Obtained by the S Formulation in the Long

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.6 Results for the J Formulation . . . . . . . . . . . . . . . . . . . . 71

3.2.7 Results for the G3 Formulation . . . . . . . . . . . . . . . . . . . 74

3.2.8 Summary of Results in the Long Channel for the Three Formu-

lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3 Backward Facing Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.1 Boundary Conditions and Results for the S Formulation . . . . . 81

3.3.2 Boundary Conditions and Results for the J Formulation . . . . . 88

3.3.3 Boundary Conditions and Results for the G Formulations . . . . 90

3.3.4 Effect of Further Refinement near the Re-entrant Corner . . . . . 92

3.3.5 Summary of Results on Grid with Refinement Near to the Re-

entrant Corner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.3.6 Summary of Results for Stokes Problems in Backward Facing

Step Region Obtained with Linear Triangles . . . . . . . . . . . . 94

3.4 Flow over a Backward Facing Step with a Long Outflow Region Modelled

Using Quadratic Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.4.1 Results in the S Formulation . . . . . . . . . . . . . . . . . . . . 95

3.4.2 Results in the J Formulation . . . . . . . . . . . . . . . . . . . . 97

Contents

Contents viii

3.4.3 Results in the G3 Formulation . . . . . . . . . . . . . . . . . . . 98

3.5 Flow around a Cylindrical Obstruction . . . . . . . . . . . . . . . . . . . 99

3.5.1 Solution of S Formulation for Cylinder Moving through Fluid at

Rest at Infinity in Symmetric Half Region . . . . . . . . . . . . . 103

3.6 Poiseuille Flow over a Semicylindrical Restriction . . . . . . . . . . . . . 109

3.7 Other Means of Overcoming the Lack of Mass Conservation . . . . . . . 115

3.8 The Null Matrix Least-Squares Finite Element Method . . . . . . . . . 116

3.9 Solutions of the S Formulation by the Null Space Method . . . . . . . . 124

3.9.1 Results obtained with Enclosed Flow Boundary Conditions . . . 125

3.9.2 Results obtained for Long Channel with Downstream Stress Bound-

ary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.9.3 Commentary on the Null Matrix Least-Squares Finite Element

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

3.10 Summary of Results for Planar Stokes Flow . . . . . . . . . . . . . . . . 127

4 A First-Order Reformulation of the Navier-Stokes Equations for Steady

Flow Using Stress and Stream Functions 129

4.1 The Planar Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . 129

4.2 The Planar Navier-Stokes Equations in Terms of Velocity, Vorticity and

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.3 The Planar Navier-Stokes Equations in Terms of Velocity, Vorticity and

Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Contents

Contents ix

4.4 Backward Facing Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.4.1 Enclosed Flow Boundary Conditions in the SN Formulation for

Backward Facing Step Geometry . . . . . . . . . . . . . . . . . . 135

4.4.2 Downstream Stress Boundary Conditions in the SN Formulation

for Backward Facing Step Geometry . . . . . . . . . . . . . . . . 139

4.4.3 Enclosed Flow Boundary Conditions in the JN Formulation for

Backward Facing Step Geometry . . . . . . . . . . . . . . . . . . 143

4.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.5 Flow over a Semicylindrical Restriction . . . . . . . . . . . . . . . . . . 146

4.5.1 Results in the SN formulation . . . . . . . . . . . . . . . . . . . . 147

4.6 Navier-Stokes Flow around a Cylindrical Obstruction . . . . . . . . . . . 149

4.6.1 Navier-Stokes Flow around a Cylindrical Obstruction Modelled

Using the SN Formulation . . . . . . . . . . . . . . . . . . . . . . 149

4.6.2 Uniform Navier-Stokes Flow around a Cylindrical Obstruction

Modelled Using the JN Formulation . . . . . . . . . . . . . . . . 153

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5 First-Order Reformulations of the Stokes and Navier-Stokes Equa-

tions in Three Dimensions 157

5.1 Vector Calculus in Three Dimensions . . . . . . . . . . . . . . . . . . . . 157

5.2 The Stokes and Navier-Stokes Equations in a Three-Dimensional Carte-

sian Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Contents

Contents x

5.3 A Reformulation of the Stokes Equations in Three Dimensions in Terms

of Stress Functions and Velocities . . . . . . . . . . . . . . . . . . . . . . 158

5.3.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 163

5.3.2 A Reformulation of the Navier-Stokes Equations in Three Di-

mensions in Terms of Stress Functions and Velocities . . . . . . . 164

5.4 The Three-Dimensional Velocity-Vorticity-Pressure Formulation . . . . . 165

5.4.1 The Navier-Stokes Velocity-Vorticity-Pressure Formulation in Three

Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.4.2 The Navier-Stokes Velocity-Vorticity-Head Formulation in Three

Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.5 The Three-Dimensional Velocity-Velocity Gradient-Pressure Formula-

tion of the Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.6 Navier-Stokes Equations in Velocity-Velocity Gradient-Pressure Form . 172

6 Concluding Remarks 175

A ADN Ellipticity Analysis of the Stress and Stream System 177

A.1 ADN Ellipticity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

A.2 Conditions on an ADN Elliptic System of Equations . . . . . . . . . . . 180

A.3 The Boundary Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 181

A.3.1 The Complementing Condition . . . . . . . . . . . . . . . . . . . 181

A.4 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Contents

Contents xi

A.4.1 The Primitive Second-Order Stokes Equations . . . . . . . . . . 183

A.4.2 The System of Equations of the S Formulation . . . . . . . . . . 186

A.5 The Application of ADN Theory to the Development of Mesh Dependent

Weights in Least-Squares Functionals . . . . . . . . . . . . . . . . . . . . 190

B Preconditioning Matrices for Least-Squares Solutions of the Stokes

Equations 193

B.1 Enclosed Flow Boundary Conditions . . . . . . . . . . . . . . . . . . . . 195

B.2 Normal Velocities and Tangential Stresses . . . . . . . . . . . . . . . . . 201

B.3 Conditioning of Linearised Systems arising from the SN Formulation . . 203

C A Stress and Stream Formulation of the Unsteady Planar Stokes

Equations 204

References 207

Contents

List of Figures

1.1 Reference triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 A macro-element K ∈ Mh . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1 Union Jack grid of size 8× 4 . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Planar backward facing step grid at ny = 2 . . . . . . . . . . . . . . . . 79

3.3 Velocity field with equal weights at ny = 8 . . . . . . . . . . . . . . . . . 82

3.4 Velocity field with equal weights at ny = 16 . . . . . . . . . . . . . . . . 83

3.5 Velocity field with weights of 1, 1, 103, 103 at ny = 8 . . . . . . . . . 83

3.6 Velocity field with weights of 1, 1, 103, 103 at ny = 16 . . . . . . . . 84



3.9 Velocity field with weights of 1, 1, 103, 103 at ny = 8 . . . . . . . . . 87


xii

List of Figures xiii

3.11 Planar backward facing step grid with further refinement close to the

re-entrant corner at ny = 2 . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.12 Geometry of region for flow around solid circular cylinder . . . . . . . . 100

3.13 Grid generated by MATLAB PDE Toolbox . . . . . . . . . . . . . . . . 100

3.14 Mesh for region around a cylindrical obstruction at ng = 1 . . . . . . . . 101

3.15 First refinement at ng = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.16 Second refinement at ng = 4 . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.17 Geometry of symmetric half cylinder problem . . . . . . . . . . . . . . 104

3.18 Unweighted solution at ng = 4 . . . . . . . . . . . . . . . . . . . . . . . 106

3.19 Weighted solution at ng = 4 . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.20 Plot of ux on line PQ at ng = 1 . . . . . . . . . . . . . . . . . . . . . . 108

3.21 Plot of ux on line PQ at ng = 16 . . . . . . . . . . . . . . . . . . . . . . 109

3.22 Plot of ux in solutions on line PQ at ng = 1 . . . . . . . . . . . . . . . 113

3.23 Plot of ux in solutions on line PQ at ng = 16 . . . . . . . . . . . . . . . 113





4.5 Velocity field with weights of 1, 1, 103, 103 at ny = 16 close to E . . 138

List of Figures

List of Figures xiv





4.10 Velocity field with weights of 1, 1, 103, 103 at ny = 16 close to E . . 142

4.11 Velocity field for the enclosed flow solution of the JN formulation with

equal weights at ny = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

4.12 Velocity field for the enclosed flow solution of the JN formulation with

equal weights at ny = 16 . . . . . . . . . . . . . . . . . . . . . . . . . . 144

4.13 Velocity field in the enclosed flow solution of the JN formulation with

weight of 103 on mass conservation term at ny = 16 . . . . . . . . . . . 145

4.14 Velocity field with weight of 103 on mass conservation term at ny = 16

close to corner E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.15 Velocity field in solution with equal weights at ng = 4 . . . . . . . . . . 148

4.16 Velocity field in solution with weights of 1, 1, 103, 103 at ng = 4 . . 148

4.17 Recirculation in solution with weights of 1, 1, 103, 103 at ng = 8 . . 148

4.18 Plot of ux on the line PQ at ng = 1 . . . . . . . . . . . . . . . . . . . . . 150


4.20 Velocity field in the solution of the SN formulation with equal weights

at ng = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

List of Figures

List of Figures xv

4.21 Velocity field in the solution of the SN formulation with weights of

1, 1, 103, 103 at ng = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 152



4.24 Velocity field in the solution of the JN formulation with equal weights

at ng = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.25 Velocity field in the solution of the JN formulation with weight of 103

on mass conservation term at ng = 4 . . . . . . . . . . . . . . . . . . . . 155

List of Figures

List of Tables

3.1 Global errors with equal weights . . . . . . . . . . . . . . . . . . . . . . 42

3.2 L2 errors by variable with equal weights . . . . . . . . . . . . . . . . . . 43

3.3 L∞ errors by variable with equal weights . . . . . . . . . . . . . . . . . 43

3.4 H1 semi-norm errors by variable with equal weights . . . . . . . . . . . 44

3.5 Global errors with weights of 1, 1, 103, 103 . . . . . . . . . . . . . . 45






3.11 L2 errors by variable with weight of 103 on mass conservation term . . 49

3.12 H1 semi-norm errors by variable with weight of 103 on mass conservation

term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49


xvi

List of Tables xvii

3.14 Global errors with weight of 103 on mass conservation term . . . . . . . 51







3.21 Axial flow with equal weights . . . . . . . . . . . . . . . . . . . . . . . . 58

3.22 Global errors in velocity variables with equal weights . . . . . . . . . . 58


3.24 Axial flow with weights of 1, 1, 103, 103 . . . . . . . . . . . . . . . . 59

3.25 Global errors in velocity variables with weights of 1, 1, 103, 103 . . . 60








List of Tables

List of Tables xviii











3.43 H1 semi-norm errors by variable with equal equation weights . . . . . . 71



3.46 L2 errors by variable with weight of 103 on mass conservation term . . 73

3.47 Axial flow with weight of 103 on mass conservation term . . . . . . . . 73

3.48 Global errors in velocity variables with weight of 103 on mass conserva-

tion term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74



List of Tables

List of Tables xix



tion term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.53 Appropriate values of qin for given ny so that inflow and outflow match 81









tion term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89




tion term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91




List of Tables

List of Tables xx





3.72 Axial flow with weight of 103 on the mass conservation term . . . . . . 97


3.74 Axial flow with weight of 103 on the mass conservation term . . . . . . 98



3.77 Axial flow in the solution with equal weights . . . . . . . . . . . . . . . 105

3.78 Axial flow in the solution with weights of 1, 1, 103, 103 . . . . . . . 105

3.79 Errors in axial velocity with equal weights . . . . . . . . . . . . . . . . 107

3.80 Errors in axial velocity with weights of 1, 1, 103, 103 . . . . . . . . . 107



3.83 Errors in axial velocity with equal weights . . . . . . . . . . . . . . . . 112

3.84 Errors in axial velocity with weights of 1, 1, 103, 103 . . . . . . . . . 112

3.85 Axial flow with a weight of 103 on the mass conservation term . . . . . 114

3.86 Errors with a weight of 103 on the mass conservation term . . . . . . . 114

List of Tables

List of Tables xxi

3.87 Pivot ratio on a given grid versus weights . . . . . . . . . . . . . . . . . 116

3.88 H1 semi-norm errors by variable with weights of 1, 1, 101, 101 . . . 117







3.95 Variation in spectrum of null matrix with size of grid . . . . . . . . . . . 121

3.96 Global errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3.97 Global errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126









List of Tables

List of Tables xxii





B.1 Comparison of conditioning rules for S functional with equal weights . 195

B.2 Comparison of conditioning rules with weights of 1, 1, 103, 103 . . . 196

B.3 Comparison of solution times for S functional with equal weights . . . . 196

B.4 Comparison of solution times for S functional with weights of 1, 1, 103, 103

196

B.5 Comparison of conditioning rules for J functional with equal weights . 197

B.6 Comparison of conditioning rules for J functional with weight of 103 on

mass conservation term . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

B.7 Comparison of solution times for J functional with equal weights . . . . 197

B.8 Comparison of solution times for J functional with weight of 103 on mass

conservation term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

B.9 Comparison of conditioning rules for G3 functional with equal weights . 198

B.10 Comparison of conditioning rules for G3 functional with weight of 103

on mass conservation term . . . . . . . . . . . . . . . . . . . . . . . . . 198

B.11 Comparison of solution times for G3 functional with equal weights . . . 199

B.12 Comparison of solution times for G3 functional with weight of 103 on

mass conservation term . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

List of Tables

List of Tables xxiii

B.13 Comparison of conditioning rules for S functional with equal weights . 201

B.14 Comparison of conditioning rules with weights of 1, 1, 103, 103 . . . 201

B.15 Comparison of solution times with equal weights . . . . . . . . . . . . . 201

B.16 Comparison of solution times for S functional with weights of 1, 1, 103, 103

202

List of Tables

Chapter 1

The Least-Squares Finite

Element Method

The finite element technique is a frequently used process for obtaining approximate

solutions to ordinary or partial differential equations. As with many other numerical

algorithms, it has been adopted widely because the steps involved can be automated

and the computer code which is subsequently produced is efficient. Originally developed

as a tool in elasticity [61] and utilised heavily at first in structural mechanics, the range

of fields where the approach is applied has broadened to include electromagnetism, fluid

mechanics and many others.

Whilst the fundamental finite element formulation is very powerful, there are dan-

gers in too naıve and unwary a use. Although superficially it may seem that the method

can be easily altered and extended, care must be taken that the underlying mathemat-

ical foundation of any newly introduced variation is sound. In the course of this work,

we shall examine one particular finite element approach, the least-squares finite ele-

ment method. We shall see that even before formulating the method, attention must

be paid to the properties of the required solution, in particular its degree of differentia-

bility. Though least-squares is by no means unique in this respect, it does restrict the

1

Chapter 1. The Least-Squares Finite Element Method 2

attractiveness and power of the method in particular ways. In later chapters we shall

use the least-squares finite element method to solve test problems in fluid mechanics

which have been extensively studied using established finite element techniques. Use

of the least-squares finite element method does have potential advantages over other

finite element formulations, particularly in the realm of fluid mechanics.

1.1 Basic Definitions

A partial differential equation with a solution u and which holds over some region Ω

can be expressed in the general form

Lu = f (1.1)

where L is some operator, possibly non-linear. Boundary conditions must also be

specified for the problem to be properly posed, which we state in general form as

Bu = g. (1.2)

We use the symbol Lp for function spaces with elements u for which up is Lebesgue

integrable. We let f lie in the function space of square Lebesgue integrable functions

L2. Considering the case where L is a first-order linear differential operator we have

that the solution must then fall in a suitable function space. For example, if L is the

gradient operator then u must fall in the space W 12 , one of the class of spaces called

Sobolev spaces [59]. Supposing we have a function u(x1, x2, . . . , xn) then let us define

a partial differential operator D of order α such that

Dα =∂|α|

∂xα11 ∂xα2

2 . . . ∂xαnn

where α = (α1, α2, . . . , αn) is a multi-index such that

| α |= α1 + α2 + . . . + αn;

see for example [81].

1.1. Basic Definitions


Any function u defined over a region Ω ∈ <n which is in the Sobolev space Wmp has

the property that the function and all its weak partial derivatives [59] up to order m

are in Lp

| Dαu |∈ Lp ∀α ≤ m.

We can define a norm over a Sobolev space Wmp (Ω)

‖ u ‖W mp

=‖ u ‖m, p=

∫

Ω

∑

|α|≤m

| Dαu |p dΩ

1p

; (1.3)

see [59].

Semi-norms over Sobolev spaces can also be defined

| u |W mp

=| u |m, p=

∫

Ω

∑

|α|=m

| Dαu |p dΩ

1p

. (1.4)

These are positive semi-definite as they are zero when u is zero and non-negative for

u 6= 0. Furthermore, with distance measured using the norm (1.3), every Cauchy

sequence in a Sobolev space converges to a point in the space. An infinite sequence

ak, k = 1, 2, . . . on a metric space with metric | • | is a Cauchy sequence if for any

real ε there exists an integer N such that for all i, j > N

| ai − aj |< ε.

Hence Wmp is complete and is a Banach space; see [6] and [59].

In the space Wm2 there exists a bilinear, symmetric form, or inner product. One

particularly important inner product is that on the space W 02 . With two functions

u, v ∈ W 02 then the inner product (u, v) is

(u, v) =∫

Ωuv dΩ.

Since the space Wm2 (Ω) has both an inner product (u, v) and a norm ‖ u ‖W m

pwith

(u, u) =‖ u ‖2 then it is a Hilbert space. We shall henceforth denote the space Wm2 by

the symbol Hm(Ω). In other words, we can identify Sobolev spaces with appropriate



Hilbert spaces [94]. We shall usually denote the norm ‖ u ‖m, 2 by ‖ u ‖m and similarly

denote the semi-norm | u |m, 2 by | u |m. From [81] if a given set of piecewise polynomials

is in the space Hm(Ω) then it is also in the space of functions with continuous derivatives

in Ω up to order m− 1, Cm−1(Ω

), where the closed region Ω is comprised of the union

of Ω with its boundary. The converse is also true.

Successive spaces Hm, m = 1, 2, . . . are proper subsets of one another

H0 ⊃ H1 ⊃ H2 ⊃ H3 ⊃ . . . ⊃ Hm−1 ⊃ Hm . . . .

Higher order Hilbert space norms dominate lower order ones. Given an element v ∈ Hk

then

‖ v ‖k≥‖ v ‖k−1≥‖ v ‖k−2 . . . ≥‖ v ‖1≥‖ v ‖0 .

We can also define negative-order norms over dual spaces. Firstly we define the terms

covering and support; see [9]. A covering of a set S is a union of countably many sets

si, i = 1, 2, . . ., where

Ω =⋃

i=1, 2, ...

si.

If the number of sets si is finite, the covering is referred to as a finite covering and if

each si is open it is called an open covering. A subcovering is a set which is a subset of

the set si and is also a covering. Finally, a set is compact if any given open covering

contains a finite subcovering. Every compact set is closed and bounded [9] and for sets

in Euclidean spaces these designations are equivalent.

The support of a function is the closure of the domain over which it is non-zero. We

use the notation Hk0 (Ω) to represent that subspace of Hk(Ω) for which functions have

compact support in Ω. We let the set H ′k0 consist of those elements v ∈ Hk

0 for which

‖ v ‖k= 1. If we have a continuous linear functional f such that

f ∈ H00 (Ω)

then for k ≥ 0 the negative norm ‖ f ‖−k is

‖ f ‖−k= supv∈H′k

| (f, v) |; (1.5)



see [88]. It can be deduced from this definition that

‖ f ‖0≥‖ f ‖−1≥‖ f ‖−2≥ . . . ≥‖ f ‖−k≥‖ f ‖−k−1 . . .

so that as with positive k, the norm ‖ • ‖k+1 dominates the norm ‖ • ‖k for all k. The

dual space H−k(Ω) is defined as the completion of the space L2(Ω) with respect to the

norm ‖ f ‖−k; see [142]. Fractional order Sobolev norms can be obtained from integer

norms by interpolation; see [29] and [99].

A problem (1.1) with boundary conditions (1.2) may have a solution which is dif-

ficult or impossible to obtain by analytical methods, so we may have to look for an

approximate solution. The finite element method seeks an approximation uh which is

a linear combination of a finite number of linearly independent basis functions from a

finite dimensional subspace Uh of some solution space U . The solution space U need

not include the analytical solution itself. The solution space most commonly used is

the Hilbert space H1 introduced above. Functions in this space have square integrable

first derivatives. The basis functions in Uh are low order and have support over only a

small number of subdomains of the whole domain over which the equation or equation

system holds.

The first applications of the finite element technique employed the Rayleigh-Ritz

approach in which a solution is sought which minimises a particular functional; see

[81].

A minimisation approach cannot be used where there is no obvious functional which

is made stationary by the solution of the given problem. Alternatively, the finite

element technique can be considered as a variational problem. In a variational problem

the solution u is sought in a trial space U such that the weighted residual∫

Ω(Lu− f)v dΩ = 0∀v ∈ V (1.6)

The space V is called a test space. The equation (1.6) is also referred to as the weak

form of the differential equation Lu = f . When employing the finite element method,



we work with members of finite dimensional subspaces of U and V , which we shall

denote as uh ∈ Uh and vh ∈ Vh respectively.

Poisson’s equation with Dirichlet boundary conditions is

−∇2u = f in Ω, (1.7)

u = h(x, y) on Γ (1.8)

where u is the potential. In the classical Galerkin variational approach for Poisson’s

equation with homogeneous Dirichlet boundary conditions the trial space and the test

space match. In this case (1.6) takes the form∫

Ω−∇2uhvh dΩ =

∫

Ωfvh dΩ ∀vh ∈ Vh. (1.9)

Applying Green’s theorem enables us to write the equation as∫

Ω∇uh.∇vh dΩ =

∫

Ωfvh dΩ ∀vh ∈ Vh (1.10)

which turns out to be equivalent to the classical Rayleigh-Ritz approach; see [62].

If u is any function in a space with second-order Lebesgue square integrable differ-

entials, like H2, then the term ∇2u in (1.9) will fall in L2. We cannot make a similar

statement if u is merely a member of a space of first order regularity, for instance H1.

On the other hand if u ∈ H1 then each component of the left hand-side term ∇u in

(1.10) must be an element of L2. This is highly significant, because, as we shall see

shortly, the finite element spaces which are usually easiest to work with are located in

H1.

1.2 The Least-Squares Finite Element Method

The Galerkin method is not the only variational or weighted residual formulation.

Others exist, including collocation [62] and the one which shall be of most interest to

us here, the least-squares approach.

1.2. The Least-Squares Finite Element Method


1.2.1 Other Variational Formulations

For the collocation method, the test functions v ∈ V of equation (1.6) are delta func-

tions. A delta function for vertex i with coordinates xi is denoted δ(xi) and has the

property that ∫

Ωfδ(xi) dΩ = f(xi)

so that (1.6) becomes∫

Ω(Lu− f) δ(xi) dΩ = Lu(xi)− f(xi) ∀δ(xi) ∈ V.

The collocation method reduces to solving a set of simultaneous equations for the values

of the approximated functions at chosen points.

1.2.2 The Least-Squares Variational Method

As for other methods we look for a solution u from some trial space U . In the least-

squares variational approach employed here we weight the operator with test functions

Lv. The functions v come from a space V with the same continuity requirements as U

though they generally satisfy different conditions on the boundary Γ of Ω.

Theorem 1.1

Let u ∈ U and v ∈ V , where functions in U satisfy the boundary conditions

Bu = g on Γ

for the partial differential equation

Lu = f in Ω

and functions in V are zero on Γ. The function u ∈ U such that the functional∫

Ω(Lu− f)2 dΩ (1.11)



is minimised satisfies the equation∫

Ω(Lu− f) Lv dΩ = 0 ∀ v ∈ V. (1.12)

Proof

The functional in (1.11) is a minimum for u only if

0 = limt→0

d

dt

∫

Ω(L(u + tv)− f)2 dΩ

,

= limt→0

d

dt

∫

Ω

(LuLu− 2fLu + f2 + 2tLuLv − 2ftLv + t2LvLv

)dΩ

,

= limt→0

∫

Ω(2LuLv − 2fLv + tLvLv) dΩ,

=∫

Ω2Lv (Lu− f) dΩ;

see [81].

Applying (1.12) to obtain a solution for Poisson’s equation (1.7), we see that the trial

functions must lie in a space with stronger differentiability prerequisites than H1. For

Lu and Lv to be square integrable in general U must be a subset of H2 and not just H1.

Hence we cannot solve Poisson’s equation in the form (1.7) by the least-squares finite

element method without using elements which are higher order and computationally

more expensive to use.

We write down an equation system which is equivalent to Poisson’s equation (1.7),

but for which solutions may belong to the whole of the space H1. Specifically, we

rephrase the equation as a system of equations in which no derivatives are higher than

first-order; see [53], [74], [104], [83], [102], [107] and [112]. This has been the usual ap-

proach in using the least-squares finite element method to solve diverse equations and

equation systems. Examples in fluid dynamics are the Stokes and Navier-Stokes equa-

tions for incompressible flow (see for instance [17] and [73]), the convection-diffusion

equations [68], non-Newtonian flows (see [27] and [47]) and the Stokes equations for

compressible flow (see [44], [90] and [143]). Other equations and equation systems

solved as first-order systems by the least-squares finite element method include the



Helmholtz equation (see [13], [49], [69], [86] and [97]), the neutron transport equa-

tion (see [7], [105] and [106]), the Reissner-Mindlin model (see [35] and [45]) and the

equations of linear elasticity (see [82], [87], [89], [138] and [139]). Systems which are

first-order in standard form have also been considered, like the Euler equations [128]

and Maxwell’s equations of electromagnetism (see [11], [81], [98], [103] and [110]).

To write Poisson’s equation (1.7) in the plane as a first-order system we introduce

new variables ux and uy which are components of the gradient of u in the x and y

directions respectively, so that

∂ux

∂x+

∂uy

∂y= f(x, y). (1.13)

The Schwarz relation gives us a second equation

∂ux

∂y− ∂uy

∂x= 0. (1.14)

Boundary conditions can be of Neumann form

∇u.n = g1(x, y)

on the whole boundary Γ of Ω. They may also be of Dirichlet form

∇u.s = g2(x, y),

again at every point on Γ, or it may be that Neumann conditions are specified on some

subset ΓN of the boundary and Dirichlet conditions on a subset of the boundary ΓD

such that Γ = ΓD ∪ ΓN .

We introduce a trial space U with elements u = (ux, uy) which are defined and in

[H1]2 over Ω and satisfy the boundary conditions for the first-order system on Γ. We

can form a least-squares functional for a system by summing the functionals for each

individual equation. In this case we have that the solution (ux, uy) ∈ U minimises

I(ux, uy) =∫

Ω

((∂ux

∂x+

∂uy

∂y− f

)2

+(

∂ux

∂y− ∂uy

∂x

)2)

dΩ



over U . Furthermore we introduce a test space V ⊂ [H1(Ω)]2 with elements v = (vx, vy)

which are zero on the boundary. Then the minimum satisfies the relation

limt→0

d

dtI (ux + tvx, uy + tvy) = 0

for every v ∈ V . Hence∫

Ω

((∂ux

∂x+

∂uy

∂y

)(∂vx

∂x+

∂vy

∂y

)+

(∂ux

∂y− ∂uy

∂x

)(∂vx

∂y− ∂vy

∂x

))dΩ

=∫

Ωf

(∂vx

∂x+

∂vy

∂y

)dΩ ∀ v ∈ V.

The extension of this system to three dimensions is called the div-curl formulation; see

[53] and [81]. We observe that the generalisation of (1.14) to three dimensions is the

identity

∇× (∇u) = 0.

So with Cartesian coordinates (x, y, z) the div-curl system in three dimensions system

can be written as

∂ux

∂x+

∂uy

∂y+

∂uz

∂z= f(x, y, z),

∂uz

∂y− ∂uy

∂z= 0,

∂ux

∂z− ∂uz

∂x= 0,

∂uy

∂x− ∂ux

∂y= 0.

In two dimensions, the potential u can be obtained from the solution of the Cauchy-

Riemann system (1.13) and (1.14) by solving a second least-squares problem, often

called a recovery problem. The relevant system is

ux − ∂u

∂x= 0 in Ω,

uy − ∂u

∂y= 0 in Ω.

The appropriate boundary condition is

u = h(x, y) on Γ.



In three dimensions u can be found by solving the system

ux − ∂u

∂x= 0 in Ω,

uy − ∂u

∂y= 0 in Ω,

uz − ∂u

∂z= 0 in Ω

with the boundary condition

u = h(x, y, z) on Γ.

This two-stage process is sometimes referred to as a div-curl-grad formulation; see [25].

Other reformulations of Poisson’s equation are discussed elsewhere, in particular the

div-grad formulation in which the potential u is coupled to the gradients; see [81] and

[112].

Variations on the Standard Least-Squares Finite Element Method

An extension of the least-squares method for systems of equations allows for different

weighting of equation terms. We denote the component of operator L in (1.11) which

acts on equation i by Li. Similarly the component of f which is the right hand-side of

equation i is represented by fi. The functional (1.11) generalises to

∫

Ω

Neq∑

i=1

wi ‖ Liu− fi ‖2 dΩ.

This has a minimum when∫

Ω

Neq∑

i=1

wiLiuLiv dΩ =∫

Ω

Neq∑

i=1

wifiLiv dΩ

where v is an element of a test space V , with elements of the same order of continuity

as the trial solutions and which are homogeneous on the boundary Γ of Ω.

There has been some research on what are called H−1 or inverse-norm least-squares

finite element methods. These were first introduced in [28] and have been developed



for instance in [10], [18], [19], [20], [21], [30], [35] and [38]. With this technique negative

inner products appear in (1.12) (see below) which are replaced by approximate terms

which can be obtained computationally.

We consider a negative inner product expressed as

(f, g)−1 (1.15)

for elements f and g in the dual space H−1(Ω) of H10 (Ω). The form (1.15) induces a

negative norm ‖ f ‖−1 as defined in (1.5). It is observed in [25] and [28] that

‖ f ‖2−1= (Sf, f). (1.16)

Here S, which is a mapping from H−1(Ω) to H10 (Ω), is the solution operator of Poisson’s

equation with Dirichlet boundary conditions, so that given

−∇2u = f in Ω,

u = 0 on Γ,

where f ∈ H−1(Ω), then

Sf = u. (1.17)

It follows that (1.15) can be rewritten as

(f, g)−1 = (Sf, g)0 ,

= (f, Sg)0 .

We let Sh be a discrete approximation to S. The norm ‖ f ‖−1 may be approximated

by a discrete norm

‖ f ‖−1, h=(Shf, f

)0. (1.18)

In practice, a conditioned operator Bh is introduced which is spectrally equivalent to

Sh in the sense that

C(Shf, f)0 ≤ (Bhf, f)0 ≤ K(Shf, f)0 (1.19)



where C and K are constants. The favoured discretisation of ‖ f ‖−1 is then

‖ f ‖−1, h=(Shf, f

)0, (1.20)

where

Sh = h2I + Bh; (1.21)

see [25] and [31].

So it can be seen that determining the appropriate substitute for the negative norm

requires solving the Galerkin weak form of Poisson’s equation, for which there are well

established fast solution techniques, in particular multigrid; see [34] and [119]. A third

development FOSLL∗ [42] seeks to fuse the L2 and H−1 least-squares methods and

their respective advantages. It preserves the efficiency and simplicity of the former

approach yet can be applied in solving a wider class of problems.

1.3 Implementing the Finite Element Method

As mentioned above in the finite element method the basis functions have compact

support. The region Ω is divided into a number Ne of simple geometric shapes ∆i, i =

1, . . . , Ne. These are referred to as the elements. They may for instance be triangles

in <2, or cuboids in <3. The partitioning is carried out in such a manner that no two

elements have intersecting interiors and their union matches the region

∪Nei=1∆i = Ω.

Certain points of each element are designated as nodes. We introduce local basis

functions which are one at a particular node in an element and zero at all other nodes.

Over the rest of the element they vary in a manner consistent with these given values.

Linear basis functions are particularly widely used because of their simplicity. In this

case the nodes may be chosen to be the vertices of the elements. In the planar case,

the commonly used shape for the element is the triangle.

1.3. Implementing the Finite Element Method


The global basis function for a given node is equal to the local basis function for that

node in any element containing the node, and is zero in any element not containing the

node.

1.3.1 Linear Triangles

We divide the region Ω ⊂ <2 over which we wish to find a solution into a number

of non-overlapping triangular elements. If we have a triangle with vertices (x1, y1),

(x2, y2) and (x3, y3) then the area ∆ can be found by evaluating the determinant∣∣∣∣∣∣∣∣∣

1 x1 y1

1 x2 y2

1 x3 y3

∣∣∣∣∣∣∣∣∣

which is equal to 2∆.

The areal or barycentric functions Li, i = 1, 2, 3 can be defined by the equations

Li =ai + bix + ciy

2∆,

where

ai = xjyk − xkyj ,

bi = yj − yk,

ci = xk − xj ;

see [144]. In the notation we use here

j = (i + 1) mod 3,

k = (i + 2) mod 3.



1 2

3

P

Figure 1.1: Reference triangle

Given any point P in a triangle with vertices labelled 1, 2 and 3 then three interior

triangles may be formed which have vertices [2, 3, P ], [3, 1, P ] and [1, 2, P ] respec-

tively, as illustrated in Figure (1.1). We denote these interior triangles ∆1, ∆2 and ∆3.

The name areal derives from the property of the functions that

Li =∆i

∆.

Consequently Li is one at vertex i and zero at other vertices. These barycentric func-

tions are used as basis functions in determining finite element solutions over a triangular

grid.

1.3.2 An Example Finite Element Implementation of the Galerkin

Formulation of Poisson’s Equation

We look for an approximation uh to the solution u of Poisson’s equation over an open

region Ω ⊂ <2 with a connected boundary Γ. The equation is

−∇2u = f(x, y)



and we shall enforce homogeneous Dirichlet conditions

u = 0 (1.22)

at every point on Γ. Our approximate solution uh is drawn from a space of trial

functions U which we define here as

U =u ∈ H1(Ω) | u = 0 on Γ

.

We look for uh in a finite dimensional subset Uh of U . We divide the region Ω into a

number Ne of non-overlapping triangles with N vertices in total, counting each vertex

only once regardless of the number of elements it appears in. We shall call this tessel-

lation T . We form N interpolating functions MI , I = 1, . . . , N . The interpolating

function MI satisfies the properties

MI = 1 at vertex J = I,

MI = 0 at vertex J 6= I.

We also define MI to be zero on triangles which do not contain the point I and linear

on triangles for which the point I is a vertex. In fact if point I coincides with vertex i

on a particular element ∆ then

MI = Li on ∆

where Li is the barycentric function associated with vertex i. The general form for

uh ∈ Uh is a linear combination of these functions

uh =N∑

i=1

xiMi. (1.23)

The boundary condition (1.22) will give the coefficients for the interpolating functions

which are characteristic with respect to the NF points on the boundary. With homo-

geneous boundary conditions as given in (1.22) these coefficients are of course simply

zero. So Uh has as unfixed basis functions the NA = N − NF characteristic interpo-

lating functions for the vertices of T in the interior of Ω. We need to determine these



remaining free coefficients x = (x1, x2, . . . , xNA)T . As explained above uh satisfies the

relation ∫

Ω

(∂uh

∂x

∂vh

∂x+

∂uh

∂y

∂vh

∂y

)dΩ =

∫

Ωfvh dΩ ∀vh ∈ Vh. (1.24)

The space Vh is chosen to be a subspace of a test space V of functions in H1(Ω). We

let Vh be the function space spanned by the NA basis functions with maxima at the

free nodes.

We can in principle obtain the unknown coefficients by substituting in the expansion

for uh over the whole of Ω given in (1.23) into (1.24). We then substitute in place of

vh each of the N basis functions spanning V in turn. This gives a linear system of N

equations in N variables, which can be reduced to one of NA variables in NA unknowns

given the specified value of the approximation at the NF points on the boundary. For

an approach like the spectral method, where the basis functions are defined so that

they may have support over the whole region, this may become impractical as the

number of vertex points and basis functions increases. A salient advantage of the

finite element method, where the basis functions have compact support, is that we can

instead evaluate (1.24) over individual elements, one at a time.

We work with a triangular master element in a local coordinate system and with

barycentric coordinates L1, L2 and L3. The triangle has area ∆. The restriction of

(1.24) to this element is∫

∆

(∂uh

∂x

∂vh

∂x+

∂uh

∂y

∂vh

∂y

)d∆ =

∫

∆fvh d∆ ∀vh ∈ Vh. (1.25)

We obtain a 3× 3 matrix k for this element. This is calculated from the left hand-side

of (1.25). So the elements of k are

kij =∫

∆

(bibj + cicj)4∆2

d∆,

=(bibj + cicj)

4∆.

Because of the origins of the finite element method in the solution of structural engi-

neering problems, a matrix of this sort is usually called a stiffness matrix. Specifically,



this is a local stiffness matrix, defined over just one element.

In a similar manner we derive a 3× 1 vector r by substituting for vh with Lj , j =

1, 2, 3 in the right hand-side of (1.25). The elements of this vector, often called a

forcing vector, are given by the integral

rj =∫

∆f(x, y)Lj d∆.

A local node i, respectively j, of element m corresponds to a vertex point I, respectively

J , in the triangulation T of Ω. We introduce the mapping sm such that

I = sm(i), J = sm(j).

The local stiffness matrices of all of the elements are assembled together into a global

matrix. For instance row i and column j of a given local stiffness matrix are associated

respectively with row I = sm(i) and column J = sm(j) of the assembled global matrix.

Similarly row j of the local forcing vector is associated with row J of the global forcing

vector. Assembly of the contributions from the Ne elements is carried out according to

the scheme

A = 0

R = 0

for1 m = 1, . . . , Ne

for2 i = 1, . . . , 3

for3 j = 1, . . . , 3

Asm(j)sm(i) = Asm(j)sm(i) + kji

end for3

Rsm(j) = Rsm(j) + rj

end for2

end for1.

This gives a linear system of the form

AxN = R,



where xN is a column vector with element xI equal to the value of the interpolating

function at node I. We now apply the boundary conditions to the linear system and

solve.

1.3.3 An Example Least-Squares Finite Element Method

We wish to demonstrate a way in which an algorithm for solving a given problem by the

least-squares finite element method might be framed. In solving a first-order system

least-squares problem by the finite element method we seek approximate values for

nf variables across Neq equations at each of the NV vertices of some finite element

subdivision of a region. The total number of variables is thus

N = (NV )(nf ).

We wish to approximate the solution u = (u1, . . . , unf) of a collection of Neq partial

differential equations. We express this system as

Lu = f in Ω

with f = (f1, . . . , fNeq)T . We also have conditions on the boundary Γ of Ω. Using B

to denote a boundary operator these are

Bu = g(x, y) on Γ

in which g may also have multiple components. From (1.12) a least-squares variational

approximation U to u is found by solving∫

ΩLULV dΩ =

∫

Ωf(x, y)LV dΩ. (1.26)

Here U is a trial solution satisfying the boundary conditions of the original classical

problem. It is located in a function space such as [H1(Ω)]nf , with V an arbitrary

element of this same space.



We distinguish multiple components of the equation operator L and write these as

LI , I = 1, . . . , Neq. For the exact solution u we have that

LIu = fI .

Substituting the components of L into (1.26) we see that the functional has the form∫

Ω

Neq∑

I=1

LIULIV dΩ =∫

Ω

Neq∑

I=1

fI(x, y)LIV dΩ.

Here the component LI represents the operation of the equation I of the system on the

trial solution U = (U1, . . . , Unf)T and on a test function V = (V1, . . . , Vnf

)T .

Given elements with VE vertices, there are a total of Nl = VEnf degrees of freedom

in each element. As a specific illustration we consider the solution of the Cauchy-

Riemann equations (1.13) to (1.14) in the particular case where the interpolation is

carried out using linear functions MI , I = 1, . . . , N . The functions MI are such that

in any triangle containing vertex I, with a local label i, MI is equal to the barycentric

function Li and MI is zero on triangles not containing vertex I. The elements of the

local stiffness matrix k are

k2i−1, 2j−1 =bibj + cicj

4∆, i, j = 1, 2, 3,

k2i, 2j−1 =bicj − bjci

4∆, i, j = 1, 2, 3,

k2i−1, 2j =bjci − bicj

4∆, i, j = 1, 2, 3,

k2i, 2j =bibj + cicj

4∆, i, j = 1, 2, 3.

This matrix is of course symmetric. The right hand-side forcing vector has elements

r2i−1 =1

2∆

∫

∆f(x, y)bi d∆, i = 1, 2, 3,

r2i =1

2∆

∫

∆f(x, y)ci d∆, i = 1, 2, 3.

We assemble together individual element stiffness matrices and right hand-side vectors

into a corresponding global stiffness matrix A and global right hand-side vector R.

The solution coefficient matrix x is then given by solving the linear algebra problem

Ax = R. (1.27)



Just as in solving Poisson’s equation by the discrete Galerkin approach we use the

given boundary condition to eliminate NF of the N = (NV ) (nf ) unknowns so that

both x and r are column vectors of length NA = N − NF and A is an NA × NA

matrix. Gaussian elimination can be used to solve the system (1.27). However this is

slow and requires the whole of the NA × NA matrix A to be stored. Other methods

of solution exist which can be applied provided that the matrix A possesses special

properties. One of the chief advantages of the least-squares formulation over other

finite element approaches is that the matrix A in (1.27) is always both symmetric and

positive definite. This means that we have a choice from a range of fast and reliable

specialised linear equation system solvers. Two particular ones which we have made

use of are Choleski decomposition of a banded matrix [88] and the conjugate gradient

method; see [46], [72] and [83].


Chapter 2

First-Order Reformulations of

the Stokes System of Equations

2.1 The Stokes Equations for Incompressible Flow in the

Plane

The Stokes equations describe the behaviour of highly viscous fluids in laminar motion.

We restrict our attention here to flow which is steady, so that variables do not change

over time. For fluid of velocity ~u = (u1, u2) and pressure p(x, y) the Stokes equations

for incompressible flow can be expressed as

−ν∇2~u +∇p = ~f, (2.1)

∇.~u = 0. (2.2)

The first term on the left hand-side of (2.1) represents diffusion, and the second is the

pressure gradient. The parameter ν is the viscosity. The right hand-side ~f = (f1, f2)

represents the effects of the external body forces, for instance gravity, on the fluid. In

this thesis, we shall only consider situations in which no such forces act, so that ~f is

uniformly zero. In this case momentum is conserved.

22

Chapter 2. First-Order Reformulations of the Stokes System of Equations 23

Equation (2.2) is the mass balance term.

One commonly used class of boundary condition for a closed finite region Ω with

a connected boundary Γ is the enclosed flow condition, where both the normal and

tangential components of the velocity are specified

~u = g(x, y) on Γ. (2.3)

This condition leaves the pressure undetermined to an arbitrary constant [94] so this

condition is supplemented by the relation∫

Ωp dΩ = 0.

Equivalently we can fix p at a single point within Ω.

Much work has been done both on the physics modelled by the equations, for ex-

ample in [12], [58] and [108], and on their theoretical properties, for instance in [94]

and [129]. It can be appreciated that the solution of (2.1) and (2.2) by analytical

methods can be very difficult. Only in special cases, dependent on the given boundary

conditions and the shape of the region, is it possible to write down a precise solution

analytically. Much effort has been put into obtaining valid solutions by numerical

methods. Overviews and assessments of the progress made in finding solutions to these

equations by Galerkin variational approaches with finite element discretisation can be

found in [3], [59] and [88].

2.2 The Mixed Finite Element Method

Much finite element work has focussed on obtaining solutions to the so-called primitive

formulation of the Stokes system of equations (2.1) and (2.2), in which the variables are

the velocities and the pressure. A singularly significant obstacle to obtaining solutions

of the Stokes equations by Galerkin techniques arises because the variational problem

is of a saddle point nature. In order for numerical solutions to be stable the respective

2.2. The Mixed Finite Element Method


solution spaces Vh for the velocities and Qh for the pressure must satisfy what is referred

to as the LBB (Ladyzhenskaya-Babuska-Brezzi) or inf-sup condition. From [88] this

condition is

sup~vh∈Vh

(ph, ∇.~vh)‖ ~vh ‖1

≥ c ‖ ph ‖0 ∀ ph ∈ Qh

where ~vh = (uh, vh). This is a compatibility condition between these two function

spaces. It has been shown that this condition holds for certain combinations of finite

element spaces. Other spaces are incompatible. For instance if the velocity space Vh(Ω)

consists of continuous functions which are locally linear over triangular subdomains and

the elements of the pressure space Qh(Ω) are piecewise constant with respect to the

same triangulation then the condition is violated. Of particular note it has been proved

that the function spaces used to approximate pressure and the velocities cannot be the

same; see [71]. So called mixed methods are adopted in solving the equations (2.1)

and (2.2). Velocity approximations are confined to spaces lying within (H10 (Ω))2, the

subset of (H1)2 for which elements must take the value zero on the boundary of Ω.

Where enclosed flow boundary conditions of the form (2.3) are enforced on the original

problem (2.1) and (2.2) then Qh is taken to lie in L20(Ω); this means that the functions

ph ∈ Qh must satisfy the requirement

ph ∈ L2(Ω),∫

Ωph dΩ = 0.

We refer to [71] for further explanation.

Combinations of elements and function spaces commonly used include Taylor-Hood

and a range of elements over which are defined bubble functions. One such is the mini-

element, which is triangular; see [109] and [4], where it is first presented. Velocities

are approximated on an element ∆ of a tessellation Th of Ω using linear functions

augmented with a cubic bubble function B3(∆), which is zero on the boundary of ∆.

We let Pi denote the space consisting of the set of polynomial functions of at most

degree i over an element ∆. The approximate velocity space is

Vh =

(uh, vh) ∈ (C

(Ω

) ∩H10 (Ω)

)2 : uh, vh ∈ P1 (∆)⊕B3 (∆) ∀ ∆ ∈ Th



and the approximate pressure space is

Qh =qh ∈ C

(Ω

) ∩ L20 (Ω) : qh ∈ P1 (∆) ∀ ∆ ∈ Th

.

For the Taylor-Hood element, discussed for example in [77] and [109], the velocity space

consists of the piecewise quadratic functions

Vh =

(uh, vh) ∈ (C

(Ω

) ∩H10 (Ω)

)2 : uh, vh ∈ P2(∆) ∀ ∆ ∈ Th

.

The trial functions for the pressure come from the space

Qh =ph ∈ C

(Ω

) ∩ L20 (Ω) : ph ∈ P1(∆) ∀ ∆ ∈ Th

.

Another viable form of approximation is one in which the velocity is approximated

linearly on triangles and the pressure is approximated as a discontinuous linear function

on quadrilateral macro-elements; see [109], where examples of similar elements in both

two and three dimensions are also presented. The macro-elements K ∈ Mh are unions

of eight of the triangles; see Figure 2.1. The velocities are approximated at the vertices

of each triangle ∆ ∈ Th and the pressure and its derivatives are approximated at the

midpoint of each element K ∈ Mh. The function space from which the velocities are

drawn is

Vh =

(uh, vh) ∈ (C

(Ω

) ∩H10 (Ω)

)2 : uh, vh ∈ P1(∆) ∀ ∆ ∈ Th

(2.4)

whilst the pressure approximation space is

Qh =ph ∈ L2

0 (Ω) : ph ∈ P1(K) ∀ K ∈ Mh

. (2.5)

Each of these macro-elements is an approximation to a quadrilateral element on which

the velocity is a quadratic function and the pressure is a linear one; see [70] and [109].

One property of these quadrilateral elements is that mass is conserved locally over them.

Mass is also conserved locally over each macro-element K ∈ Mh used to approximate

these quadrilateral elements, so that∫

K∇. (uh, vh)T dK =

∫

Sn. (uh, vh)T dS = 0 ∀ K ∈ Mh

where n is the unit normal to the boundary S of K.



Figure 2.1: A macro-element K ∈ Mh

It is considered a great advantage of the least-squares finite element method that the

LBB condition is not a requirement. The least-squares functional is always minimised,

regardless of the characteristics of the particular equations, and the same spaces can be

used to approximate all of the variables. Mixed methods on the other hand typically

generate a linear equation system where the matrix A in (1.27) is of the form M B

BT 0

.

Unlike the corresponding matrix for the least-squares method this is not positive-

definite, which restricts the choice and power of possible solvers.

2.3 The Stress and Stream Function Reformulation

In [130], a system of equations is derived which is equivalent to (2.1) and (2.2). This

system is comprised only of terms which contain first-order derivatives. Results are

presented for the solution of this system by the least-squares finite element method.

We introduce a stream function ψ and express the velocities in terms of ψ such that

2.3. The Stress and Stream Function Reformulation


the mass conservation equation is satisfied implicitly. Specifically

u1 =∂ψ

∂y, (2.6)

u2 = −∂ψ

∂x. (2.7)

The stresses are represented by a 2× 2 tensor

σ =

σ11 σ12

σ21 σ22

.

The tensor is symmetric so that σ12 = σ21. For fluids in steady flow and in the absence

of body forces the divergence of the stress tensor is zero

∇.σ = 0. (2.8)

The divergence operator in (2.8) acts on successive columns. In component form (2.8)

can be written as

∂σ11

∂x+

∂σ12

∂y= 0,

∂σ21

∂x+

∂σ22

∂y= 0.

Postulating the existence of a so-called stress function φ and setting the planar stress

tensor equal to φyy −φxy

−φxy φxx

(2.9)

ensures that (2.8) is true.

Now for incompressible fluids

σ = −pI + 2νd (2.10)

where I is the identity matrix and p and ν have the meanings given with the definitions



of (2.1) and (2.2). The deformation tensor d is

d =12

2∂u1

∂x

∂u1

∂y+

∂u2

∂x

∂u1

∂y+

∂u2

∂x2∂u2

∂y

.

In terms of the pressure and the stress and stream functions, equation (2.10) can be

written component by component as

φyy = −p + 2νψxy,

−φxy = −νψxx + νψyy,

φxx = −p− 2νψxy.

These can be rewritten in the form

−φxx + φyy = 4νψxy, (2.11)

−φxy = −νψxx + νψyy (2.12)

so that p does not appear explicitly. We remark that this second-order system was

used in [48] to study non-Newtonian flows by spectral methods.

By defining

U1 =∂φ

∂x, U2 =

∂φ

∂y, U3 =

∂ψ

∂x, U4 =

∂ψ

∂y(2.13)

in (2.11) and (2.12) and using the Schwarz relations for φ and ψ we are able to write

the Stokes equations as a first-order system

−∂U1

∂x+

∂U2

∂y− 2ν

∂U3

∂y− 2ν

∂U4

∂x= f1, (2.14)

∂U1

∂y+

∂U2

∂x− 2ν

∂U3

∂x+ 2ν

∂U4

∂y= f2, (2.15)

∂U1

∂y− ∂U2

∂x= f3, (2.16)

2ν∂U3

∂y− 2ν

∂U4

∂x= f4. (2.17)

The terms f1 to f4 vanish but in order to facilitate the use of certain analytic techniques,

they are allowed here to be non-zero; see [130] and [131]. The system of equations (2.14)



to (2.17) is square, as the number of unknowns matches the number of equations. It

can be shown [131] that this formulation is also equivalent to the planar biharmonic

equation

∇4u = q

where u is a displacement and q represents a body force. In the classical statement of

the biharmonic problem q is zero, but as with f1 to f4 we allow it to be non-zero. To

obtain a least-squares solution we minimise the functional

S =∫

Ω

((−∂U1

∂x+

∂U2

∂y− 2ν

∂U3

∂y− 2ν

∂U4

∂x− f1

)2

+(

∂U1

∂y+

∂U2

∂x− 2ν

∂U3

∂x+ 2ν

∂U4

∂y− f2

)2

+(

∂U1

∂y− ∂U2

∂x− f3

)2

+(

2ν∂U3

∂y− 2ν

∂U4

∂x− f4

)2)

dΩ. (2.18)

We shall henceforth refer to the system of equations (2.14) to (2.17) as the S formulation

of the Stokes system and shall call (2.18) the S functional.

Four different forms of boundary condition for the system (2.14) to (2.17) are con-

sidered in [130]. These are

U1 = g1 (x, y) , U2 = g2 (x, y) on Γ, (2.19)

U3 = g1 (x, y) , U4 = g2 (x, y) on Γ, (2.20)

(U1, U2) .n = g1 (x, y) , (U3, U4) .s = g2 (x, y) on Γ, (2.21)

(U1, U2) .s = g1 (x, y) , (U3, U4) .n = g2 (x, y) on Γ. (2.22)

It is shown in [130] that each of these satisfies the Lopatinski conditions; see [50] and

[133]. The system with these boundary conditions is shown in [130] to be elliptic in

the sense of Wendland [133] in regions with smooth boundaries.

We let U ∈ [H1 (Ω)

]4. We use L to denote the equation operator for (2.14) to (2.17)

and B to symbolize the boundary operator for one of the conditions (2.19) to (2.22).



Then the inequality

‖ U ‖l+1, Ω≤‖ LU ‖l, Ω + ‖ BU ‖l+ 12, Γ +

Nc∑

i=1

| ΛiU | (2.23)

holds for l = −1, 0 and where Nc is the number of linear constraints required for a

unique solution. This estimate applies for regions with boundaries of continuity C1, 1.

A function F has continuity CM, N if all the derivatives

∂m+nF

∂xm1 ∂xn

2

, m ≤ M, n ≤ N

are continuous.

The estimate (2.23) implies that approximations on these regions using linear el-

ements will converge with error of order h, where h is a discretisation parameter,

measured in an H1 metric.

2.4 Other First-Order Reformulations of the Stokes Equa-

tions

There are other Stokes equivalent systems for which no differential term is greater than

first-order. One first-order recasting of the Stokes equations is the velocity-vorticity-

pressure formulation; see [65]. This is probably the one most frequently used in work

appearing in published studies of least-squares methods; see for example [5], [19], [23],

[38], [54], [55], [63], [81], [85], [100], [135], [140] and [141].

In deriving this system we utilise the identity

−∇2~u = ∇×∇× ~u = ∇× ω

where ω is the vorticity and ~u = (u1, u2). This identity holds if ∇.~u = 0. The system

(2.1) and (2.2) can then be written as

ν∇× ω +∇p = ~f,

2.4. Other First-Order Reformulations of the Stokes Equations


ω −∇× ~u = 0,

∇.~u = 0.

Explicitly this is

ν∂ω

∂y+

∂p

∂x= fx, (2.24)

−ν∂ω

∂x+

∂p

∂y= fy, (2.25)

ω +∂u1

∂y− ∂u2

∂x= 0, (2.26)

∂u1

∂x+

∂u2

∂y= 0. (2.27)

We shall use the symbol J to denote this system and the corresponding least-squares

functional

J =∫

Ω

((ν

∂ω

∂y+

∂p

∂x− f1

)2

+(−ν

∂ω

∂x+

∂p

∂y− f2

)2

+(

ω +∂u1

∂y− ∂u2

∂x− f3

)2

+(

∂u1

∂x+

∂u2

∂y− f4

)2)

dΩ. (2.28)

A slight modification of (2.28) is

Jν =∫

Ω

((ν

∂ω

∂y+

∂p

∂x− f1

)2

+(−ν

∂ω

∂x+

∂p

∂y− f2

)2

+

ν2

(ω +

∂u1

∂y− ∂u2

∂x− f3

)2

+ ν2

(∂u1

∂x+

∂u2

∂y− f4

)2)

dΩ; (2.29)

see [38], [56] and [63]. A variety of boundary conditions are known to be compatible

with this system. A list can be found in [81]. For easy comparison with the results for

the stress and stream formulation, and because these are the boundary conditions for

the primitive system (2.1) and (2.2), in our experiments we shall enforce the enclosed

flow boundary conditions

~u = g(x, y) (2.30)

on the whole of the boundary, with the pressure fixed at a single point somewhere

in the region. From [81] the approximate solutions to the system with homogeneous

boundary conditions of this form satisfy the inequality

‖ ~u ‖1 + ‖ p ‖0 + ‖ ω ‖0≤ C ‖ f ‖0 . (2.31)



and the extension of this inequality to cases in which the boundary conditions are inho-

mogeneous is straightforward. So approximations to the velocity using linear elements

should converge at order h in H1, whilst approximations in the other variables should

converge at order h in L2.

In the following work we shall usually choose to refer to the four variables of the J

formulation as

U1 = u1, (2.32)

U2 = u2, (2.33)

U3 = p, (2.34)

U4 = ω. (2.35)

An alternative planar formulation is the velocity-velocity gradient-pressure formulation,

as proposed in [40]; see also [102]. A very similar reformulation has been used in the

application of least-squares methods to the solution of the linear elasticity equations,

which are related to the Stokes equations; see [40], in which both are considered as

special cases of the system

−ν∇2~u +∇p = ~f, (2.36)

∇.~u + δp = g. (2.37)

The parameter δ is zero for the Stokes equations and equal to the inverse of a second

parameter λ for the linear elasticity equations. A new variable U is introduced

U = ∇~uT =

∂u1

∂x

∂u1

∂y

∂u2

∂x

∂u2

∂y

.

For the Stokes system U is the velocity gradient. The equations (2.36) and (2.37) can

be written in the form

−ν (∇.U)T +∇p = ~f, (2.38)

∇.~u + δp = g, (2.39)

U−∇~uT = 0. (2.40)



Following [40], we shall designate this form of the system and its associated least-squares

functional G1. This is

G1(U, ~u, p) =‖ −ν (∇.U)T +∇p− ~f ‖2−1 +ν2 ‖ ∇.~u + δp− g ‖2

0

+ ν2 ‖ U−∇~uT ‖20 . (2.41)

This functional is defined even if the components of ~f are only in H−1(Ω). Appropriate

boundary conditions for this functional are

~u = gb(x, y). (2.42)

We note that trace U = ∇.~u. The supplementary equations

∇×U = 0,

∇ ( trace U + δp− g) = 0

lead to the least-squares functional G2

G2(U, ~u, p) =‖ −ν (∇.U)T +∇p− ~f ‖20 +ν2 ‖ ∇.~u + δp− g ‖2

0

+ ν2 ‖ U−∇~uT ‖20 +ν2 ‖ ∇ ×U ‖2

0 +ν2 ‖ ∇ ( trace U + δp− g) ‖20 . (2.43)

In solving this functional (2.42) must be supplemented with the boundary conditions

U× n = Gb(x, y). (2.44)

We let D be the distance between an element and the nearest vertex of Ω. Then the

further equation

D−1 ( trace U− g) = 0 (2.45)

taken together with the equations of G2 for the Stokes case gives the system from which

is generated the least-squares functional which is designated G3 in [40]. The solution

of the G3 formulation is found by looking for the values of U, ~u and p which minimise

the functional

G3(U, ~u, p) = ‖ −ν (∇.U)T +∇p− ~f ‖20 +ν2 ‖ ∇.~u− g ‖2

0

+ν2 ‖ U−∇~uT ‖20 +ν2 ‖ ∇ ×U ‖2

0

+ν2 ‖ ∇ ( trace U− g) ‖20 +ν2 ‖ D−1 ( trace U− g) ‖2

0 . (2.46)



The appropriate boundary conditions are again (2.42) and (2.44). We shall refer to

these conditions as the enclosed flow conditions for a G2 or G3 formulation.

The boundary condition (2.42) is often referred to as a displacement condition when

describing elastic materials. In [37] there is consideration of a Neumann boundary

condition, specifically

n.νU− pn = 0.

In [40] coercivity and continuity bounds are obtained for the case in which ~f = 0 and

g = 0. Following [40] we introduce the spaces

U0 =(V ∈ H1 (Ω)n2

: n×V = ~0 on Γ)

,

U1 = (V ∈ V0 : δ trace V ∈ L2 (Ω)) ,

V1 = L2 (Ω)n2 ×H10 (Ω)n × L2

0 (Ω) ,

V2 = U0 ×H10 (Ω)n × L2

0 (Ω) ,

V3 = U1 ×H10 (Ω)n × (

H1 (Ω) \ <)

so that V3 ⊂ V2 ⊂ V1. For the planar case the appropriate spaces are given by setting

n = 2, whilst n = 3 in the three dimensions. It is proved in [40] that the G1 functional

(2.41) has the bounds

1C

(ν2 ‖ U ‖2

0 +ν2 ‖ ~u ‖21 + ‖ p ‖2

1

) ≤ G1(U, ~u, p) ∀ (U, ~u, p) ∈ V1. (2.47)

and

G1(U, ~u, p) ≤ C(ν2 ‖ U ‖2

0 +ν2 ‖ ~u ‖21 + ‖ p ‖2

1

) ∀ (U, ~u, p) ∈ V1. (2.48)

The functional G1 is not fully H1 coercive; see [25]. The G2 functional (2.43) satisfies

the bounds

1C

(ν2 ‖ U ‖2

1 +ν2 ‖ ∇ trace U ‖20 +ν2 ‖ ~u ‖2

1 + ‖ p ‖21

) ≤ G2(U, ~u, p)

∀ (U, ~u, p) ∈ V2 (2.49)

and

G2(U, ~u, p) ≤ C(ν2 ‖ U ‖2

1 +ν2 ‖ ∇ trace U ‖20 +ν2 ‖ ~u ‖2

1 + ‖ p ‖21

)

∀ (U, ~u, p) ∈ V2. (2.50)



The coercivity bounds (2.49) and (2.50) hold if the boundary of the region has conti-

nuity C1, 1. The appropriate bounds for the G3 functional (2.46) are

1C

(ν2 ‖ U ‖2

1 +ν2 ‖ D−1 trace U ‖20 +ν2 ‖ ~u ‖2

1 + ‖ p ‖21

)

≤ G3(U, ~u, p) ∀ (U, ~u, p) ∈ V3 (2.51)

and

G3(U, ~u, p) ≤ C(ν2 ‖ U ‖2

1 +ν2 ‖ D−1 trace U ‖20

+ν2 ‖ ~u ‖21 + ‖ p ‖2

1

) ∀ (U, ~u, p) ∈ V3. (2.52)

The relations (2.51) and (2.52) are valid in convex polygons; see [40]. In what follows

we shall often refer to a member of the collection of systems G1, G2 and G3 as simply

a G formulation of the Stokes equations.

We shall henceforth generally symbolise the variables appearing in a G formulation

with the notation

U1 = u1, (2.53)

U2 = u2, (2.54)

U3 =∂u1

∂x, (2.55)

U4 =∂u2

∂x, (2.56)

U5 =∂u1

∂y, (2.57)

U6 =∂u2

∂y, (2.58)

U7 = p. (2.59)

Using these designations the twelve equations of the full G3 functional can be written

as

−∂U1

∂x+ U3 = f1,

−∂U2

∂x+ U4 = f2,

−∂U1

∂y+ U5 = f3,



−∂U2

∂y+ U6 = f4,

−ν∂U3

∂x− ν

∂U5

∂y+

∂U7

∂x= f5,

−ν∂U4

∂x− ν

∂U6

∂y+

∂U7

∂y= f6,

∂U1

∂x+

∂U2

∂y= f7,

−∂U3

∂y+

∂U5

∂x= f8,

−∂U4

∂y+

∂U6

∂x= f9,

∂U3

∂x+

∂U6

∂x= f10,

∂U3

∂y+

∂U6

∂y= f11,

D−1U3 + D−1U6 = f12.

In the following chapter we shall compare results obtained by minimising the G2 and

G3 functionals as well as the J functional with those arrived at by minimisation of the

S functional presented previously.

Other first-order reformulations are discussed in the literature. One such is the

velocity-pressure-stress formulation; see [24] and [25]. Another is the acceleration-

pressure formulation [50]. Here the variables are

φ1 =∂u1

∂x= −∂u2

∂y,

φ2 =∂u1

∂y,

φ3 =∂u2

∂x

together with the pressure p. The variables φ1, φ2 and φ3 are velocity gradients, not

actually accelerations, and hence the system is also termed the constrained velocity

gradient-pressure formulation in [25].

The Stokes equations can be written using these variables as

−ν

(∂φ1

∂x+

∂φ2

∂y

)+

∂p

∂x= f1,



−ν

(∂φ3

∂x− ∂φ1

∂y

)+

∂p

∂y= f2,

∂φ2

∂x− ∂φ1

∂y= 0,

∂φ1

∂x+

∂φ3

∂y= 0.

Appropriate boundary conditions on this system are

φ1n1 + φ3n2 = 0,

φ2n1 − φ1n2 = 0,∫

Ωp dΩ = 0

where n = (n1, n2). The velocities ~u = (u1, u2) for a fluid in incompressible flow can

be recovered by solving the system

∂u2

∂x− ∂u1

∂y= φ3 − φ2 in Ω,

∂u1

∂x+

∂u2

∂y= 0 in Ω

~u.n = g(x, y) on Γ.

For studies of this formulation we refer to [57], [136] and [140]. The recasting of the

equations of linear elasticity in these variables is analysed in [138] using the theory

presented in [133].


Chapter 3

Experimental Comparison of

First-Order Stokes Systems

The paper [130] presented results for the solution of Poiseuille flow [58] in a square

region [0, 1]2 obtained by minimising the S functional (2.18). Two forms of boundary

conditions were considered: the enclosed flow boundary conditions, for which U3 and

U4 are fixed all around the boundary, and the downstream stress or symmetry bound-

ary conditions, for which U2 and U3 are fixed on portions of the boundary with U3 and

U4 fixed on the remaining sections. It was noted in [130] that mass is not generally

conserved in finite element solutions of the S formulation. There have also been re-

ports of poor mass conservation for other first-order formulations when enclosed flow

boundary conditions are applied; we refer to [55] and [63] for results obtained using

the J formulation and to [26] and [123] for studies of solutions of the G formulations.

With the aim of more strongly enforcing mass conservation for arbitrary permissable

boundary conditions, we weight the term corresponding to the equation (2.27) in the

J least-squares functional (2.28) and the term in the appropriate G functional (2.43)

or (2.46) corresponding to equation (2.39); see [26] and [63]. In dealing with the S

functional, we follow [130] and [131] in weighting not only the residual of the mass-

conservation equation (2.17) but also the residual of (2.16), so that the weighted S

38

Chapter 3. Experimental Comparison of First-Order Stokes Systems 39

functional takes the form

Sw =∫

Ω

((−∂U1

∂x+

∂U2

∂y− 2ν

∂U3

∂y− 2ν

∂U4

∂x− f1

)2

+(

∂U1

∂y+

∂U2

∂x− 2ν

∂U3

∂x+ 2ν

∂U4

∂y− f2

)2

+ w3

(∂U1

∂y− ∂U2

∂x− f3

)2

+ w4

(2ν

∂U3

∂y− 2ν

∂U4

∂x− f4

)2)

dΩ. (3.1)

We apply the same numerical weight to both terms, setting w3 = w4. Weighting both

the latter terms seems natural as they are of the same form.

3.1 Poiseuille Flow in a Square Region

We have used the three formulations G, J and S to obtain solutions for Poiseuille flow

in the square region [0, 1]2. The velocity ~u = (ux, uy) for Poiseuille flow in this region

has a parabolic profile so that ux = y(1−y) and uy = 0. In this case the exact solution

in the variables of the S formulation is

U1 = ν(x2 + y2 − 2x− y

), (3.2)

U2 = ν (2xy − 2y + 1− x) , (3.3)

U3 = 0, (3.4)

U4 = y(1− y); (3.5)

see [26] and [123]. The exact form in the four variables of the J formulation can also

be written down in a simple form as can the solution in the seven variables of the G

formulation; see [123].

3.1.1 The Finite Element Grid Used in the Solution of Poiseuille Flow

The elements we use are triangular and the interpolation is linear. The finite element

grid configuration used is that which is frequently referred to as the Union Jack. We

3.1. Poiseuille Flow in a Square Region


denote the degree of refinement of the grid by the parameters nx and ny. There are

nx + 1 nodes along lines of constant y and ny + 1 nodes along lines of constant x. We

may also combine these parameters in the format nx × ny. An 8 × 4 Union Jack grid

is shown in Figure 3.1.

Figure 3.1: Union Jack grid of size 8× 4

3.1.2 Error Measurements

We show how the error changes as the grid is refined, using a number of different norms.

The L2 error is the difference between the exact solution U and the computed one Uh,

measured in the norm

‖ U − Uh ‖0,2=

√∫

Ω(U − Uh)2 dΩ.

In practice we calculate the L2 norm using the mid-side quadrature rule, which is exact

for polynomials of degree two; see [88]. We let U i, j denote the solution U at vertex i of

element j of our Ne elements and let the approximate solution at this point be denoted

U i, jh . Then the error at this vertex is

U i, je =| U i, j

h − U i, j |

Furthermore we use the symbol ∆j to denote the area of element j. Then the L2 error

we use is given by the equation

‖ U − Uh ‖0,2=

√√√√√√Ne∑

j=1

∆j

((U1, j

e + U2, je

)2+

(U2, j

e + U3, je

)2+

(U3, j

e + U1, je

)2)

12.



The discrete L∞ error is the maxU i, je , i = 1, 3, j = 1, Ne. We also compute the H1

semi-norm error, defined by

| U − Uh |1,2=

√√√√∫

Ω

((∂ (U − Uh)

∂x

)2

+(

∂ (U − Uh)∂y

)2)

dΩ.

This integral is determined by the midpoint rule, which is exact for polynomials of

degree one; see [88]. Specifically

| U − Uh |1,2=

√√√√√Ne∑

j=1

(U1, j

e b1 + U2, je b2 + U3, j

e b3

)2+

(U1, j

e c1 + U2, je c2 + U3, j

e c3

)2

12∆j.

(3.6)

3.1.3 Results for the S Formulation

We work with two classes of boundary conditions in obtaining an approximation to the

solution given by (3.2) to (3.5).

Enclosed Flow Boundary Conditions

Our first form of boundary condition is enclosed flow for which we fix the velocities all

around the boundary. In particular in this case on the inlet line x = 0 and the outlet

line x = 1 we have that

U3 = 0, U4 = y(1− y). (3.7)

The lines y = 0 and y = 1 are walls and the velocities are fixed to zero on them by the

no-slip boundary condition

U3 = 0, U4 = 0. (3.8)

We require three linear constraints; see [130]. We fix U1 = 0 and U2 = ν = 1 at the

corner (0, 0) and U2 = 0 at the opposite corner (1, 1).



Downstream Stress Boundary Conditions

The downstream stress boundary conditions consist of (3.7) on the line x = 0 and (3.8)

on the walls together with the outlet condition

U2 = 0, U3 = 0.

A single linear constraint suffices in this case; see [130]. We set U1 = 0 at the origin.

Results obtained with Enclosed Flow Boundary Conditions

Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

4× 4 0.18281 0.29247 0.45718

8× 8 0.07737 0.12135 0.22549

16× 16 0.02954 0.04579 0.10568

32× 32 0.01030 0.01588 0.04954

64× 64 0.00335 0.00515 0.02382

128× 128 0.00103 0.00159 0.01170

Table 3.1: Global errors with equal weights



L2 Errors by Variable

nx × ny U1 U2 U3 U4

4× 4 0.09896 0.15214 0.00190 0.02180

8× 8 0.04522 0.06233 0.00123 0.00742

16× 16 0.01802 0.02329 0.00057 0.00225

32× 32 0.00641 0.00803 0.00019 0.00063

64× 64 0.00211 0.00259 0.00006 0.00017

128× 128 0.00066 0.00080 0.00002 0.00004

Table 3.2: L2 errors by variable with equal weights

L∞ Errors by Variable


4× 4 0.13532 0.29247 0.00465 0.02517

8× 8 0.05932 0.12135 0.00276 0.01047

16× 16 0.02264 0.04579 0.00124 0.00325

32× 32 0.00772 0.01588 0.00044 0.00090

64× 64 0.00245 0.00515 0.00013 0.00024

128× 128 0.00075 0.00159 0.00004 0.00006

Table 3.3: L∞ errors by variable with equal weights



H1 Semi-Norm Errors by Variable


4× 4 0.29598 0.31374 0.01862 0.15043

8× 8 0.13801 0.16252 0.01287 0.07224

16× 16 0.06075 0.07898 0.00620 0.03467

32× 32 0.02704 0.03786 0.00231 0.01689

64× 64 0.01259 0.01841 0.00075 0.00834

128× 128 0.00609 0.00909 0.00023 0.00415

Table 3.4: H1 semi-norm errors by variable with equal weights

For a C1, 1 boundary we expect H1 convergence at the rate h; see equation (2.23).

It appears from the results presented in Table 3.1 that this is the asymptotic rate of

convergence. The convergence rates in L2 and L∞ are approximately the same as each

other. Both converge faster than the H1 semi-norm does, though not at a rate of h2,

which would be the optimal rate in these two norms.

It can be seen from Tables 3.2, 3.3 and 3.4 that the errors are far larger in U1 and

U2 than in the other two variables.



Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

4× 4 0.13843 0.20009 0.42206

8× 8 0.04745 0.07152 0.21662

16× 16 0.01520 0.02356 0.10852

32× 32 0.00465 0.00733 0.05417

64× 64 0.00137 0.00220 0.02705

128× 128 0.00040 0.00064 0.01352

Table 3.5: Global errors with weights of 1, 1, 103, 103

Table 3.5 gives the errors in the solution of the enclosed flow problem by the S

formulation with equation weights of 1, 1, 103, 103. The L2 errors and L∞ errors

are all smaller in magnitude than the corresponding ones given in Table 3.1 but the

same cannot be claimed for the H1 semi-norm errors. Indeed the H1 errors on the finer

grids are in fact larger than the equivalent ones computed without equation weighting.

The rate of convergence in H1 is in this case almost exactly order h between all of the

grids. The actual convergence rates in L2 and L∞ are comparable with those obtained

by minimising the unweighted functional.



Results Obtained with Downstream Stress Boundary Conditions

Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

4× 4 0.43676 0.57361 0.89468

8× 8 0.18559 0.24918 0.38944

16× 16 0.06109 0.08142 0.14341

32× 32 0.01798 0.02362 0.05578

64× 64 0.00507 0.00655 0.02465

128× 128 0.00140 0.00178 0.01180




4× 4 0.65197 0.57567 0.02848 0.20782

8× 8 0.27691 0.25669 0.01582 0.09404

16× 16 0.09521 0.09961 0.00651 0.03921

32× 32 0.03330 0.04110 0.00220 0.01756

64× 64 0.01348 0.01883 0.00067 0.00842

128× 128 0.00620 0.00913 0.00019 0.00416


Table 3.6 shows the global errors in the solution obtained by the unweighted S func-

tional with downstream boundary conditions on the outlet. The rate of convergence

in H1 is better than h. The convergence rates in L2 and L∞ appear to get closer to

h2 as the grid is refined. This behaviour is similar to that observed for enclosed flow

boundary conditions from Table 3.1. The absolute errors are larger here, though the



difference is much less pronounced on the finer grids. From Table 3.7 it can be seen

that the error in U1 influences the global error much more than the errors in the other

variables.

Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

4× 4 0.19232 0.25418 0.48942

8× 8 0.05720 0.07680 0.22676

16× 16 0.01621 0.02148 0.10977

32× 32 0.00450 0.00586 0.05431

64× 64 0.00124 0.00158 0.02707

128× 128 0.00034 0.00042 0.01352


Table 3.8 show the errors in the solution satisfying the downstream boundary con-

ditions but with weights on the third and fourth equations. The rate of convergence

in H1 in this case is almost exactly order h. The convergence rates in H1 are slower

with weighting than without; compare Table 3.6 with Table 3.8. Convergence rates in

L2 and L∞ are greater than order h, though not quite order h2. The errors in these

two norms are considerably reduced with weighting.

3.1.4 Results for the J Formulation

The results for the J formulation presented here have been obtained by enforcing

enclosed flow boundary conditions. For enclosed flow as in equation (2.30) the velocities

U1 and U2 are specified on every part of the boundary of the region. We fix the pressure

U3 at the point (0, 0).





4× 4 0.01487 0.00002 0.09267 0.04510

8× 8 0.00413 0.00003 0.02801 0.01275

16× 16 0.00109 0.00003 0.00914 0.00343

32× 32 0.00030 0.00002 0.00368 0.00099

64× 64 0.00008 0.00000 0.00160 0.00031

128× 128 0.00002 0.00000 0.00066 0.00010




4× 4 0.13855 0.00022 0.15805 0.15810

8× 8 0.06748 0.00029 0.04562 0.04567

16× 16 0.03330 0.00029 0.01349 0.01352

32× 32 0.01656 0.00021 0.00511 0.00512

64× 64 0.00826 0.00011 0.00223 0.00223

128× 128 0.00412 0.00004 0.00092 0.00092


From (2.31) the variables U3 and U4 should converge at order h in L2. In fact from

Table 3.9 they converge faster than this. The vorticity U4 converges somewhat faster

than the pressure U3. The velocities also converge rapidly in L2. Convergence in U1

seems to be order h2 between the 64×64 and 128×128 grids. Also we expect from the

inequality (2.31) that U1 and U2 converge at order h in H1. It seems from Table 3.10

that the axial velocity converges at this rate. The variables U3 and U4 both seem to

converge in the H1 semi-norm at the same rate as each other. This rate is greater than



order h.



4× 4 0.01233 0.00000 0.07444 0.03610

8× 8 0.00308 0.00000 0.02003 0.00903

16× 16 0.00077 0.00000 0.00611 0.00228

32× 32 0.00019 0.00000 0.00217 0.00059

64× 64 0.00005 0.00000 0.00082 0.00016

128× 128 0.00001 0.00000 0.00030 0.00004

Table 3.11: L2 errors by variable with weight of 103 on mass conservation term



4× 4 0.14431 0.00003 0.12705 0.12713

8× 8 0.07215 0.00002 0.03258 0.03262

16× 16 0.03607 0.00001 0.00899 0.00901

32× 32 0.01804 0.00000 0.00297 0.00295

64× 64 0.00902 0.00000 0.00113 0.00113

128× 128 0.00451 0.00000 0.00042 0.00042

Table 3.12: H1 semi-norm errors by variable with weight of 103 on mass conservation

term

Increasing the weight on the mass conservation term reduces the magnitude of all

of the errors; see Tables 3.11 and 3.12. The approximations in the velocity variables

U1 and U2 are particularly accurate. As in the unweighted solution they converge

at order h2 in L2 and order h in H1. From Table 3.11 we see that U3 and U4 are

converging faster than order h in L2. The convergence rate in U3 is greater than in the



unweighted solution whilst the convergence rate in U4 is order h2 between the 64× 64

and 128 × 128 grids. Both these variables converge at the same rate as each other in

H1, just as they do in the unweighted solution. The convergence rate is somewhat

greater in the weighted solution.

3.1.5 Results for the G Formulations

The boundary conditions we have applied are those for enclosed flow, expressly

U1 = y(1− y), U2 = 0, U5 = 1− 2y, U6 = 0 on the line x = 0,


U1 = 0, U2 = 0, U3 = 0, U4 = 0 on the line y = 0,

U1 = 0, U2 = 0, U3 = 0, U4 = 0 on the line y = 1.

We present detailed results for the G3 formulation. Some results which have been

obtained using the G2 formulation are shown, mainly for comparison with the results

for the G3 formulation.

Results for the G2 Formulation



Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

4× 4 0.01378 0.01277 0.13822

8× 8 0.00494 0.00667 0.06820

16× 16 0.00178 0.00281 0.03375

32× 32 0.00058 0.00099 0.01676

64× 64 0.00018 0.00031 0.00835


Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

4× 4 0.01542 0.01504 0.14606

8× 8 0.00530 0.00726 0.07282

16× 16 0.00183 0.00290 0.03626

32× 32 0.00059 0.00100 0.01807

64× 64 0.00018 0.00031 0.00902

Table 3.14: Global errors with weight of 103 on mass conservation term

The L2 and L∞ errors on the finest grids are almost the same with or without

weighting. Convergence rates in the solution of both the unweighted and the weighted

formulations as measured in L2 and L∞ appear to be approaching order h2 as the

grid is refined but this rate is not reached for the grids studied. Weighting the mass

conservation term increases the H1 error substantially at all grid levels. The rate of

convergence in H1 is almost exactly order h with or without weighting.



Results for the G3 Formulation

Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

4× 4 0.01352 0.01208 0.13808

8× 8 0.00448 0.00571 0.06805

16× 16 0.00141 0.00207 0.03367

32× 32 0.00040 0.00062 0.01673

64× 64 0.00011 0.00017 0.00834



nx × ny U1 U2 U3 U4 U5 U6 U7

4× 4 0.01126 0.00009 0.00062 0.00007 0.00251 0.00019 0.00701

8× 8 0.00291 0.00001 0.00033 0.00010 0.00094 0.00014 0.00326

16× 16 0.00074 0.00000 0.00011 0.00005 0.00028 0.00006 0.00116

32× 32 0.00019 0.00000 0.00003 0.00002 0.00008 0.00002 0.00034

64× 64 0.00005 0.00000 0.00001 0.00000 0.00002 0.00000 0.00009





nx × ny U1 U2 U3 U4 U5 U6 U7

4× 4 0.13682 0.00089 0.00335 0.00113 0.01382 0.00160 0.01181

8× 8 0.06758 0.00022 0.00172 0.00126 0.00539 0.00118 0.00535

16× 16 0.03355 0.00008 0.00063 0.00065 0.00171 0.00049 0.00187

32× 32 0.01672 0.00003 0.00018 0.00021 0.00047 0.00015 0.00054

64× 64 0.00834 0.00001 0.00005 0.00006 0.00012 0.00004 0.00014


With equal weights on all the equation terms of the G3 functional the L2 and L∞ errors

reduce at a rate of almost order h2 between the most refined grids; see Table 3.15.

Table 3.16 gives the L2 errors component by component. Most of these converge at

approximately the global rate. By far the greatest errors are in the pressure variable U7.

Also from Table 3.15 we see that the magnitude of the H1 semi-norm error decreases

at order h as the grid is refined. Table 3.17 shows that the errors in U1 are the

largest amongst all of the variables and the convergence rate for this component is

approximately order h; this is the theoretical convergence rate implied by relations

(2.51) and (2.52). The other variables converge at a rate of almost order h2.

By comparing Table 3.15 with Table 3.13 we can assess the benefit of incorporating

the extra equation (2.45) in G3 as compared with G2. We see that the errors can be

reduced by incorporating the extra term. The solutions of the G3 formulation are more

accurate than those of the G2 formulation, particularly on finer meshes.



Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

4× 4 0.01510 0.01420 0.14588

8× 8 0.00483 0.00626 0.07266

16× 16 0.00148 0.00220 0.03619

32× 32 0.00042 0.00065 0.01805

64× 64 0.00010 0.00018 0.00902


The errors are slightly increased by applying these weights; compare Table 3.18 with

Table 3.15. The convergence rates are however roughly the same in all three metrics,

although we remark that the convergence rate in H1 is even closer to order h with

weighting. Also the difference in the errors in the L2 and L∞ norms is much less

pronounced for the finer grids. In fact it appears that as in the solutions obtained with

the G2 functional, the errors in L2 and L∞ are approximately the same in the solutions

of both the unweighted and weighted formulations.

The errors in Table 3.18 are smaller than the corresponding ones in Table 3.14. The

errors in L2 and L∞ are noticeably smaller. Convergence in these two norms is faster,

indicating that the weighted G3 formulation is more effective than the weighted G2

one.

3.1.6 Summary of Results in the Square Region

We have modelled Poiseuille flow in a square channel. We have examined the effect

of applying different weights to particular equations. In H1 we have obtained conver-

gence of order h in the solution obtained by minimising the unweighted S functional and



almost of order h2. The convergence rate in H1 for solutions of the unweighted J for-

mulation is approximately order h in the velocities, which is the theoretically expected

rate. The rate of convergence in H1 is somewhat greater in the other two variables. In

solutions obtained by minimising the G3 formulation the rate of convergence in H1 is

order h in the axial velocity and almost order h2 in the other variables. Convergence

in L2 is close to order h2 in the solutions of the G3 formulation. It is somewhat slower

than order h2 in the solutions of the S functional. In solutions of the J formulation,

the velocities may be converging at order h2 in L2 between the finest grids. The other

two variables are converging more slowly than this between these grids but the rate of

convergence is still greater than the theoretical one. Comparing the magnitudes of the

errors in the velocity variables for the three formulations, we find that the solution of

the unweighted S formulation is much less accurate than the solution of the unweighted

J formulation, which is in turn less accurate than the solution of the unweighted G

formulation.

Weighting appropriate terms reduces the magnitudes of the errors in the S formu-

lation quite markedly and the magnitudes of the errors in the J formulation are also

significantly reduced. The solutions of the G3 formulation are however only slightly

more accurate with weighting than without. For the most part the convergence rates

do no change significantly after the application of weights. The errors in the velocity

variables are all very small in magnitude in the solutions of the weighted formulations.

However we can observe that the errors in the velocity variables in the solution of

the weighted J formulation are smaller than those in the solution of the weighted S

formulation.

3.2 Poiseuille Flow in a Long Channel

We now extend the channel to a length L > 1, so that we determine flow in the region

[0, L] × [0, 1]. Specifically we choose L = 20. We also increase the number of points

in the horizontal direction on our grid, so that nx = Lny. Using the trapezium rule,

3.2. Poiseuille Flow in a Long Channel


which captures the value of the integral of the piecewise linear approximation exactly,

we determine the mass flow across lines of constant x, at intervals of five units along the

x-axis. This gives an indication of how strongly mass is conserved using least-squares

methods.

3.2.1 Exact solution in the S Formulation

When U4 = y(1− y) as we have in (3.5) then we use the modified functions

U1 = ν(x2 + y2 − 2Lx− y

),

U2 = ν (2xy − 2Ly + L− x)

so that U2 = 0 on the outlet as it is in the square region.

The exact solutions for the J and G formulations are the same as those over the

square channel, with the domain of definition extended. In particular we note that at

ν = 1, which we choose for the examples here, the pressure varies linearly from zero on

the line x = 0 to −2L on the line x = L.

3.2.2 Results for the S Formulation

We give results for four combinations of boundary conditions: enclosed flow as in (2.20),

downstream stress, normal velocities with tangential stresses as in (2.21) and tangential

velocities with normal stresses as in (2.22).


For enclosed flow

U3 = 0, U4 = y(1− y) on the line x = 0,

U3 = 0, U4 = y(1− y) on the line x = L,



U3 = 0, U4 = 0 on the line y = 0,

U3 = 0, U4 = 0 on the line y = 1.

We also fix U1 and U2 at (0, 0) and U2 at the point (L, 1).

Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 324.18940 98.73032 71.66404

160× 8 307.51242 94.35631 67.87390

320× 16 255.66478 79.25964 56.36035

640× 32 152.73801 47.65190 33.66220

1280× 64 58.48968 18.28780 12.89087




80× 4 50.75109 50.54039 0.06840 2.39266

160× 8 48.06872 47.86804 0.05831 2.14606

320× 16 39.91835 39.75046 0.04206 1.80906

640× 32 23.83899 23.74212 0.02360 1.07558

1280× 64 9.12901 9.09207 0.00892 0.41230




Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15625 0.00121 0.00002 0.00121 0.15625

160× 8 0.16406 0.01034 0.00130 0.01034 0.16406

320× 16 0.16602 0.04153 0.01870 0.04153 0.16602

640× 32 0.16650 0.09433 0.07403 0.09433 0.16650

1280× 64 0.16663 0.13931 0.13061 0.13931 0.16663

Table 3.21: Axial flow with equal weights

Global Errors in the Velocity Variables

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 0.75294 0.24997 2.39364

160× 8 0.69802 0.24802 2.21483

320× 16 0.57080 0.22185 1.80954

640× 32 0.33930 0.13886 1.07584

1280× 64 0.12981 0.05404 0.41230

Table 3.22: Global errors in velocity variables with equal weights

We see from Tables 3.19 and 3.20 that the errors in the three norms reduce by a

factor of 2.6 between the 640 × 32 and 1280 × 64 grids. From theory we only expect

convergence in H1 to be order h but the convergence rate in L2 and L∞ are still short

of the optimal rate of h2. From Table 3.22 we see that the local convergence rates in

the velocities are approximately the same as the global convergence rates. Table 3.21

shows just how poorly mass conservation is enforced in the solution of the unweighted

S formulation. On the 80 × 4 grid, the net flow through the middle of the channel is

only about 0.0128% of that through the ends. Even with four grid refinements 21.6%

of the flow is lost between the inlet or outlet and the line x = 10.



Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 76.48424 23.38740 16.92595

160× 8 21.98771 6.73303 4.92615

320× 16 5.70879 1.74806 1.34270

640× 32 1.44267 0.44157 0.39799

1280× 64 0.36664 0.11195 0.14465


Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15625 0.14505 0.14140 0.14505 0.15625

160× 8 0.16406 0.16074 0.15963 0.16074 0.16406

320× 16 0.16602 0.16515 0.16486 0.16515 0.16602

640× 32 0.16650 0.16628 0.16621 0.16628 0.16650

1280× 64 0.16663 0.16657 0.16655 0.16657 0.16663

Table 3.24: Axial flow with weights of 1, 1, 103, 103

Table 3.24 shows that mass is conserved along the length of the channel much better

than it is when all of the equation weights are equal; see Table 3.21. The errors shown

in Table 3.23 are considerably smaller than those in Table 3.19. Furthermore it appears

that the errors in L2 and L∞ reduce at a rate of around h2 as the grid is refined, whilst

the H1 errors reduce at a rate greater than order h. Convergence in the long channel

is in this instance actually faster than convergence in the square region; see Table 3.5.




nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 0.10060 0.02505 0.66861

160× 8 0.02739 0.00678 0.32644

320× 16 0.00700 0.00174 0.16176

640× 32 0.00176 0.00044 0.08068

1280× 64 0.00044 0.00011 0.04031

Table 3.25: Global errors in velocity variables with weights of 1, 1, 103, 103

The convergence rates for the velocity errors shown in Table 3.25 are much greater

than those for the errors in the unweighted case shown in Table 3.22. In L2 and L∞

the velocities converge at order h2. In H1 they converge at order h.


The second boundary condition is the downstream stress condition, for which the ve-

locities are as per enclosed flow on the inlet and the walls, namely

U3 = 0, U4 = y(1− y) on the line x = 0,

U3 = 0, U4 = 0 on the line y = 0,

U3 = 0, U4 = 0 on the line y = 1

but with the outlet condition

U2 = 0, U3 = 0 on the line x = L.

We must also set U1 = 0 at the origin.



Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 1300.25189 398.73017 145.33467

160× 8 1283.70710 394.31133 143.19109

320× 16 1218.23592 376.76959 135.58888

640× 32 1011.97018 316.03517 112.43785

1280× 64 603.17481 189.47308 66.98214


Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15625 0.00121 0.00001 0.00000 0.00000

160× 8 0.16406 0.01030 0.00065 0.00004 0.00001

320× 16 0.16602 0.03944 0.00941 0.00236 0.00106

640× 32 0.16650 0.08141 0.04109 0.02330 0.01828

1280× 64 0.16663 0.12229 0.09429 0.07886 0.07394


The convergence rates between the coarser grids for the data displayed in Table 3.26

are very poor. They do increase with grid refinement. Convergence between the 640×32

and 1280× 64 grids is stronger. In particular the H1 error reduces by a factor of 1.7,

close to order h. We see from Table 3.27 that most of the mass is lost between the

inlet and the outlet, except on the 1280× 64 grid. In the solution on the 160× 8 grid

the flow through the line x = 20 is 6.10× 10−3% of that through the line x = 0. Even

in the solution on the 640 × 32 grid the flow through the outlet is only 11.0% of that

through the inlet. The solution on the 1280× 64 grid is of better quality, but 55.6% of

the mass is lost between the inlet and the outlet.




nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 0.78550 0.25000 2.48960

160× 8 0.75957 0.24999 2.40540

320× 16 0.70217 0.24840 2.22234

640× 32 0.57277 0.22255 1.81239

1280× 64 0.33955 0.13905 1.07448


Table 3.28 shows the errors in the velocity variables. Though the errors in the

solution on the 80 × 4 grid are only slightly greater than the corresponding ones for

enclosed flow, which were shown in Table 3.22, they converge more slowly. The errors

do seem to reduce at a rate of almost order h between the 640×32 and 1280×64 grids

in all three metrics.

Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 489.80384 152.25323 54.47820

160× 8 161.90217 50.46723 18.01332

320× 16 43.85023 13.67868 4.89465

640× 32 11.19607 3.49308 1.26670

1280× 64 2.82072 0.88007 0.33574




Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15625 0.13456 0.11988 0.11137 0.10859

160× 8 0.16406 0.15612 0.15157 0.14850 0.14748

320× 16 0.16602 0.16402 0.16261 0.16176 0.16148

640× 32 0.16650 0.16600 0.16563 0.16540 0.16534

1280× 64 0.16663 0.16650 0.16641 0.16635 0.16633


The errors displayed in Table 3.29 are much smaller than the equivalent ones for

the unweighted solution shown in Table 3.26. They are greater than the ones in the

weighted solution satisfying enclosed flow boundary conditions which have been shown

in Table 3.23. In this case the convergence rates in all three norms are almost h2.

It can be seen from Table 3.27 that although setting w3 = w4 = 103 in equation

(3.1) leads to much less mass being lost the amount of mass lost is still substantial on

the coarser grids. In the solution on the 80 × 4 grid over 33.2% of the mass entering

the inlet does not reach the outlet. Even at the next grid level, the quantity of mass

leaving is only around 89.9% of that entering. With two further refinements however

the mass on the line x = 20 is approximately 99.3% of that on the line x = 0.




nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 0.21660 0.07738 0.85639

160× 8 0.07001 0.02529 0.37412

320× 16 0.01884 0.00683 0.16925

640× 32 0.00416 0.00174 0.08168

1280× 64 0.00121 0.00044 0.04044


The convergence rates in the velocity are almost order h2 in L2 and L∞ and are

order h in H1; see Table 3.31.

3.2.3 Normal Velocities and Tangential Stresses

Here, we enforce the boundary conditions (U1, U2).n and (U3, U4).s. For Poiseuille

flow in a long channel these conditions are

U1 = νy(y − 1), U4 = y(1− y) on the line x = 0,

U1 = ν(y (y − 1)− L2

), U4 = y(1− y) on the line x = L,

U2 = L− x, U3 = 0 on the line y = 0,

U2 = x− L, U3 = 0 on the line y = 1.

No linear constraints are required; see [130].



Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 0.11660 0.02081 1.65847

160× 8 0.02912 0.00520 0.82920

320× 16 0.00728 0.00130 0.41459

640× 32 0.00182 0.00033 0.20729

1280× 64 0.00046 0.00008 0.10365




80× 4 0.08361 0.06588 0.00000 0.04759

160× 8 0.02087 0.01647 0.00000 0.01189

320× 16 0.00521 0.00412 0.00000 0.00297

640× 32 0.00130 0.00103 0.00000 0.00074

1280× 64 0.00033 0.00026 0.00000 0.00019




80× 4 0.02081 0.00000 0.00000 0.00747

160× 8 0.00520 0.00000 0.00000 0.00187

320× 16 0.00130 0.00000 0.00000 0.00047

640× 32 0.00033 0.00000 0.00000 0.00012

1280× 64 0.00008 0.00000 0.00000 0.00003






80× 4 0.85630 1.29099 0.00000 0.59211

160× 8 0.42816 0.64550 0.00000 0.29593

320× 16 0.21408 0.32275 0.00000 0.14793

640× 32 0.10704 0.16137 0.00000 0.07396

1280× 64 0.05352 0.08069 0.00000 0.03698


Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15625 0.15625 0.15625 0.15625 0.15625

160× 8 0.16406 0.16406 0.16406 0.16406 0.16406

320× 16 0.16602 0.16602 0.16602 0.16602 0.16602

640× 32 0.16650 0.16650 0.16650 0.16650 0.16650

1280× 64 0.16663 0.16663 0.16663 0.16663 0.16663


We see from Table 3.36 that no mass is lost with these boundary conditions, even

though no term in the functional is weighted differently from any other. Tables 3.32,

3.33, 3.34 and 3.35 show the errors. These errors are small. In L2 and L∞ the solution

converges at a rate of h2; see Tables 3.32, 3.33 and 3.34. In H1 the solution converges

at a rate of h; see Tables 3.32 and 3.35. The trivial function U3 is captured exactly.

As we see from Table 3.34 the value of the bilinear function U2 is captured exactly at

the nodes, so the L2 and H1 errors in this variable shown in Table 3.33 and Table 3.35

respectively are just the interpolation errors.

We do not show results obtained for these boundary conditions with weights of



1, 1, 103, 103 here, but we have performed investigations of this case. With these

weights, axial flow is conserved as in the unweighted case. Though the solution in U4

is more accurate, the solution in U1 is less so. The rates of convergence do not change

with weighting.

3.2.4 Tangential Velocities and Normal Stresses

The boundary conditions (U1, U2).s and (U3, U4).n for the long channel are

U2 = νL(1− 2y), U3 = 0 on the line x = 0,

U2 = 0, U3 = 0 on the line x = L,

U3 = 0, U4 = 0 on the line y = 0,

U3 = 0, U4 = 0 on the line y = 1.

The solution of the S formulation with these boundary conditions is unique and there

is no need to apply further linear constraints; see [130].

Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 0.12096 0.02447 1.67784

160× 8 0.02966 0.00601 0.83383

320× 16 0.00734 0.00150 0.41575

640× 32 0.00183 0.00038 0.20758

1280× 64 0.00046 0.00009 0.10372






80× 4 0.08874 0.06588 0.00000 0.04917

160× 8 0.02151 0.01647 0.00000 0.01209

320× 16 0.00529 0.00412 0.00000 0.00300

640× 32 0.00130 0.00103 0.00000 0.00075

1280× 64 0.00033 0.00026 0.00000 0.00019




80× 4 0.02447 0.00000 0.00000 0.00756

160× 8 0.00601 0.00000 0.00000 0.00187

320× 16 0.00150 0.00000 0.00000 0.00047

640× 32 0.00038 0.00000 0.00000 0.00012

1280× 64 0.00009 0.00000 0.00000 0.00003






80× 4 0.88683 1.29099 0.00000 0.60167

160× 8 0.43544 0.64550 0.00000 0.29832

320× 16 0.21591 0.32275 0.00000 0.14853

640× 32 0.10750 0.16137 0.00000 0.07411

1280× 64 0.05364 0.08069 0.00000 0.03702


Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15387 0.15387 0.15387 0.15387 0.15387

160× 8 0.16376 0.16376 0.16376 0.16376 0.16376

320× 16 0.16598 0.16598 0.16598 0.16598 0.16598

640× 32 0.16650 0.16650 0.16650 0.16650 0.16650

1280× 64 0.16663 0.16663 0.16663 0.16663 0.16663


We observe from Table 3.41 that mass is conserved exactly in the solution which

satisfies these boundary conditions. The solutions are highly accurate, with the values

of U2 and U3 captured exactly at the nodes. The L2 and L∞ errors converge at almost

the optimal rate of h2; see Tables 3.37, 3.38 and 3.39. The H1 errors converge at order

h; see Tables 3.37 and 3.40.



3.2.5 Summary of Results Obtained by the S Formulation in the Long

Channel

We consider first the solutions satisfying either enclosed flow or downstream stress

boundary conditions. With all four equations weighted equally, a very large amount of

flow is lost, although progressively less so as the grid is refined. This can be observed

for both the situation with enclosed flow boundary conditions, as in Table 3.21, and

when downstream boundary conditions are specified on the outflow; see Table 3.27.

Table 3.19 shows that the global convergence is poor with the enclosed flow boundary

conditions and it can be seen from Table 3.26 that convergence is even slower with

downstream boundary conditions. The enclosed flow solutions are the canonical ones

for the primitive Stokes formulation (2.1) and (2.2). That the unweighted S functional

does not give reasonable solutions with these boundary conditions in particular is a

serious disadvantage.

When the third and fourth equations are weighted there is little loss of flow, even

on the coarsest grids. This is illustrated in Tables 3.24 and 3.30. Convergence rates in

L2 and L∞ with either form of boundary condition appear to be around order h2 as

shown in Tables 3.23 and 3.29. In H1 and with enclosed flow boundary conditions the

global solution converges faster than order h whilst in the velocity variables the rate

of convergence in this metric is order h. With downstream stress boundary conditions

the convergence rate is close to order h2 in the stresses and order h in the velocity.

With either tangential or normal velocity conditions applying, but not both, the

solutions without weighting are highly accurate. Convergence in L2 and L∞ is order

h2 and convergence in H1 is order h. Of special note is that there is no loss of mass.

However these boundary conditions are not as widely applicable as the enclosed flow

ones. For many fluid problems, the velocities along the boundaries are often determined

by the nature of the problem together with the no-slip conditions. However U1 and U2

are functions of the velocity gradients and pressure, which cannot usually be deduced

from the specification of a given problem.



3.2.6 Results for the J Formulation

We enforce enclosed flow boundary conditions. These are

U1 = y(1− y), U2 = 0 on the line x = 0,

U1 = 0, U2 = 0 on the line y = 1,

U1 = y(1− y), U2 = 0 on the line x = L,

U1 = 0, U2 = 0 on the line y = 0.

We also set the pressure U3 = 0 at (0, 0).



160× 8 0.56249 0.00465 70.08062 1.77727

320× 16 0.40713 0.00382 50.16926 1.28790

640× 32 0.19842 0.00192 24.32808 0.62790

1280× 64 0.06523 0.00064 7.99197 0.20643




160× 8 1.78739 0.03096 6.16433 6.16482

320× 16 1.29260 0.02522 4.46798 4.46809

640× 32 0.63119 0.01269 2.17849 2.17850

1280× 64 0.20942 0.00424 0.71627 0.71627

Table 3.43: H1 semi-norm errors by variable with equal equation weights



Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

160× 8 0.16406 0.04355 0.02112 0.04355 0.16406

320× 16 0.16602 0.07930 0.05645 0.07930 0.16602

640× 32 0.16650 0.12472 0.11172 0.12472 0.16650

1280× 64 0.16663 0.15295 0.14847 0.15295 0.16663



nx × ny ‖ u− uh ‖0,2 ‖ u− uh ‖∞ | u− uh |1,2

160× 8 0.56251 0.21790 1.78766

320× 16 0.40469 0.16505 1.28447

640× 32 0.19843 0.08229 0.63131

1280× 64 0.06523 0.02725 0.20947


Table 3.42 shows that the errors in L2 reduce by a factor of about three in all four

variables between the 640× 32 and 1280× 64 grids. This is greater than the expected

rate of convergence for U3 and U4; see the inequality (2.31). From Table 3.43 we see

that the errors in H1 also shrink by approximately a factor of three between these two

grids. The theoretical expectation is that convergence is of order h in H1 for U1 and

U2; see (2.31). Table 3.44 shows that quite a substantial proportion of the flow is lost

when the equations terms are all weighted equally, particularly on the coarser grids.

The results still compare favourably with those obtained for the corresponding problem

using the unweighted S functional which were shown in Table 3.21.

The errors in the velocities as measured in L2, L∞ and H1 reduce by a factor of



around 3.1 between the 640× 32 and 1280× 64 grids; see Table 3.45. Convergence in

the velocity is more rapid in this case than it is in the solutions of the unweighted S

formulation. Comparing Table 3.45 with Table 3.22 we see that the errors are less in

the solution of the unweighted J formulation than in the solution of the unweighted S

formulation, especially on the finer grids.



160× 8 0.01810 0.00006 2.31663 0.05673

320× 16 0.00453 0.00001 0.58530 0.01423

640× 32 0.00113 0.00000 0.14895 0.00356

1280× 64 0.00028 0.00000 0.03837 0.00089

Table 3.46: L2 errors by variable with weight of 103 on mass conservation term

Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

160× 8 0.16406 0.16290 0.16252 0.16290 0.16406

320× 16 0.16602 0.16572 0.16563 0.16572 0.16602

640× 32 0.16650 0.16643 0.16641 0.16643 0.16650

1280× 64 0.16663 0.16661 0.16660 0.16661 0.16663

Table 3.47: Axial flow with weight of 103 on mass conservation term

When a weight of 103 is applied to the mass conservation term we see from comparing

Table 3.46 with Table 3.42 that the errors are reduced in magnitude very substantially,

more so as the mesh is refined. The errors decrease at a rate closer to order h2 than in

the solutions found with the unweighted functional. Table 3.47 shows the axial flow.

Comparing this with Table 3.44 we see that in this case only a small quantity of mass

is lost. Table 3.24 gave the axial flow in the solution arrived at by minimising the



weighted S functional (3.1). On any given grid the flow lost in the solution of the

weighted S formulation is more than the flow lost in the solution of the weighted J

formulation.


nx × ny ‖ u− uh ‖0,2 ‖ u− uh ‖∞ | u− uh |1,2

160× 8 0.01810 0.00235 0.32314

320× 16 0.00453 0.00058 0.16138

640× 32 0.00113 0.00015 0.08067

1280× 64 0.00028 0.00004 0.04033

Table 3.48: Global errors in velocity variables with weight of 103 on mass conservation

term

The velocity converges to the analytical solution at approximately order h2 in L2

and L∞ and order h in H1; see Table 3.48. The convergence rate is high even between

the coarsest grids. The errors here are lower in magnitude than those in the minimum

of the weighted S functional; see Table 3.25.

3.2.7 Results for the G3 Formulation

The boundary conditions imposed are those of enclosed flow. For this region they are


U1 = y(1− y), U2 = 0, U5 = 1− 2y, U6 = 0 on the line x = L,

U1 = 0, U2 = 0, U3 = 0, U4 = 0 on the line y = 0,

U1 = 0, U2 = 0, U3 = 0, U4 = 0 on the line y = 1.

In addition the pressure U7 is forced equal to zero at the point (0, 0).



Unweighted G3 Functional

Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15625 0.02770 0.00735 0.02770 0.15625

160× 8 0.16406 0.04915 0.02060 0.04915 0.16406

320× 16 0.16602 0.08917 0.06145 0.08917 0.16602

640× 32 0.16650 0.13278 0.11845 0.13278 0.16650


It can be seen from Table 3.49 that considerable flow is lost. Even on the 320 ×16 grid the flow through the central line is less than half that through the end two

lines. Nevertheless the results still compare very favourably with those obtained by

minimising the unweighted S functional; see Figure 3.21. In the solutions on the

finer grids, the flows are greater than those obtained by minimising the unweighted J

functional which were shown in Table 3.44.

Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1,2

80× 4 8.63509 3.25241 1.24607

160× 8 2.23076 0.84217 0.42381

320× 16 0.56274 0.21275 0.17554

640× 32 0.14113 0.05339 0.08250




Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15625 0.15192 0.15047 0.15192 0.15625

160× 8 0.16406 0.16291 0.16253 0.16291 0.16406

320× 16 0.16602 0.16573 0.16563 0.16573 0.16602

640× 32 0.16650 0.16643 0.16641 0.16643 0.16650



nx × ny ‖ u− uh ‖0,2 ‖ u− uh ‖∞ | u− uh |1,2

80× 4 0.07128 0.00972 0.64894

160× 8 0.01807 0.00234 0.32313

320× 16 0.00453 0.00058 0.16138

640× 32 0.00113 0.00015 0.08067


term

The flow profile of the solution obtained when the mass conservation term is weighted

can be seen in Table 3.51. Only a small portion of the mass on the inflow and outflow

is lost, with progressively less being lost on the finest grids. Less flow is lost on a given

grid than in the solution of the weighted S formulation; see Table 3.24. The rate of

convergence is almost order h2 in L2 and L∞ and order h in H1; see Table 3.50 and

Table 3.52. The magnitudes of the errors in the velocities are approximately the same

as or very slightly less than those achieved with the weighted J formulation; see Table

3.48. They are considerably less than those in the solution obtained by minimising the

weighted S functional; see Table 3.25.



3.2.8 Summary of Results in the Long Channel for the Three Formu-

lations

We see that, regardless of the system employed, a large amount of mass is lost when

enclosed flow boundary conditions are enforced. The picture is similar with downstream

stress boundary conditions specified on the outlet in the S formulation. Progressively

greater quantities are lost further away from lines on which flow is fixed. The loss

of mass is particularly acute in solutions on less refined grids. Weighting appropriate

terms reduces the amount of mass lost considerably, but we do not see the complete

mass conservation that can be achieved by standard mixed methods.

With weighting the solutions of all three formulations converge at a rate of h2 in L2

and L∞. Across all the variables the solutions converge faster than order h in H1 and

the velocity variables converge at about order h.

Flow is preserved perfectly in the solution of the S formulation when normal veloc-

ities and tangential stresses (2.21) or tangential velocities and normal stresses (2.22)

are specified on the boundary. We have found that axial flow is also conserved in

the solution of the J formulation when either the normal or the tangential velocity is

specified at every point on the boundary, together with the pressure; see also [23], [25]

and [63]. In [23] it is shown that when normal velocities and pressure are specified the

system of equations as a whole satisfies the complementing condition with all of the

equation indices equal; see [2] and Appendix A. The system with enclosed flow bound-

ary conditions does not satisfy this condition unless the equation and unknown indices

are unequal; see [23]. Hence the system with enclosed flow boundary conditions fails

to satisfy the Lopatinski conditions; see [23] and [133]. In [63] it is pointed out that

the principal part [2] of the J formulation with appropriately chosen equal equation

indices and equal unknown indices, so that the principal part consists of the terms of

highest order, decomposes into two systems, firstly one in terms only of ω and p

ν∇× ω +∇p = 0



and secondly the Cauchy-Riemann system

∇× ~u = 0,

∇.~u = 0.

With enclosed flow boundary conditions the first system is under-determined and the

second is over-determined.

By their definitions the variables U1 and U2 of the S formulation are the gradients

of a stress function φ, whilst U3 and U4 are the gradients of a stream function ψ; see

(2.13). From [130], these satisfy

∇2ψ = F1, (3.9)

∇2φ = F2. (3.10)

In terms of the variables U1 to U4 defined in (2.13) the equation (3.9) decomposes into

the Cauchy-Riemann system

∂U1

∂x+

∂U2

∂y= F1, (3.11)

∂U1

∂y− ∂U2

∂x= 0 (3.12)

and (3.10) decomposes into the system

∂U3

∂x+

∂U4

∂y= F2, (3.13)

∂U3

∂y− ∂U4

∂x= 0. (3.14)

Given enclosed flow boundary conditions both U3 and U4 are specified at all points on

the boundary. Hence the system (3.11) and (3.12) is over-determined. On the other

hand the variables U1 and U2 are specified only at a finite set of points and so the

system (3.13) and (3.14) is under-determined. With the boundary conditions (2.21) or

(2.22) both systems are well posed.



3.3 Backward Facing Step

A much examined Stokes flow geometry in the finite element literature in general and

the least-squares literature in particular is that over the planar backward facing step,

for which the region is not convex but has a re-entrant corner; see for example [111].

Our region is A ∪B where

A = [−2, 0]× [−1, 0], B = [0, 6]× [−1, 1].

We shall also call section A the inlet channel and section B the outlet region.

A sample Union Jack grid for our region is illustrated in Figure 3.2. We indicate the

level of refinement of the grids by the value of a parameter ny, where ny+1 is the number

of nodes on the line AB. Progressive grid refinements are labelled ny = 4, 8, 16, . . ..

Unless explicitly stated otherwise mesh refinement is uniform throughout the region.

A

B C

DE

O

Figure 3.2: Planar backward facing step grid at ny = 2

In the particular case we consider here fluid enters across the line AB. The fluid

there is in Poiseuille flow, namely

ux = −y (1 + y) , uy = 0. (3.15)

It leaves through the line CD. The fluid here should have settled back into Poiseuille

flow if section B is long enough with velocities

ux = 0.125(1− y2

), uy = 0. (3.16)

3.3. Backward Facing Step


The lines BC, DE, EO and OA are walls on which the no-slip boundary condition holds

so that both components of velocity vanish. The solution throughout the region cannot

be determined analytically from the Stokes equations.

For examples of other studies of this particular problem using least-squares methods

we refer the reader to [81], where approximate solutions are given in both two and

three dimensions. Deang and Gunzburger [63] solve the velocity-vorticity-pressure

formulation to approximate a known exact solution with a singularity in a symmetric

L-shaped region. They use both a regular directional triangles grid and a grid with

refinement in elements close to the re-entrant corner.

We calculate total flow across a number of lines parallel to the y axis in order to

demonstrate the degree to which mass is conserved by each of the methods. We use

the trapezium rule to obtain the value of this integral.

With enclosed flow boundary conditions the velocities are set on both the inlet line

AB and the outlet line CD. The total flow in balances with the total flow out, that is∫ 0

−1−y(1 + y) dy =

16,

18

∫ 1

−1

(1− y2

)dy =

16.

In the numerical solution these flows do not balance exactly because our approximations

are piecewise linear. On the inlet line and the outlet line the nodal values match those

for the quadratic functions ux in (3.15) and (3.16) respectively but the integrals do

not. We force the integral of the piecewise linear approximation to the inflow to be

equal to the integral of the piecewise linear approximation to the outflow. We do this

by multiplying the values of the inflow specified at the nodes on the inlet line by qin.

The value of qin is a constant for a given grid. We let yini , i = 0, . . . , nin be the nodes

on the inlet line and youti , i = 0, . . . , nout be those on the outlet line. We have that

U4(yini ) = −qinyin

i (1 + yini )

and

U4(youti ) = (1 + yout

i )(1− youti ).



The interval between the points yini−1 and yin

i we write as ∆ii−1 and that between yout

i−1

and youti we denote by δi

i−1. The constant qin is such that

nin∑

i=1

12

(U4(yin

i−1) + U4(yini )

)∆i

i−1 =nout∑

i=1

12

(U4(yout

i−1) + U4(youti )

)δii−1.

Table 3.53 shows the appropriate value of qin on a given grid.

ny qin Flow

4 2120 0.16406

8 8584 0.16602

16 341340 0.16650

32 13651364 0.16663

Table 3.53: Appropriate values of qin for given ny so that inflow and outflow match

3.3.1 Boundary Conditions and Results for the S Formulation

Specifically the boundary conditions on the inflow are that

U3 = 0, U4 = −qiny(1 + y).

On the walls they are

U3 = 0, U4 = 0.

We consider two forms of boundary condition on the outlet. The first is an enclosed

flow, so that

U3 = 0, U4 = 0.125(1− y2

). (3.17)

In this circumstance three linear constraints are required for the problem and we impose

U1 = 0 and U2 = 0 at A and U2 = 0 at D. Our second form of boundary condition is a

downstream stress condition. In this case we specify on the outlet that

U2 = 0, U3 = 0



and we also take one constraint. Here we take U1 = 0 at the point A. We set qin = 1

in obtaining results for these boundary conditions.


Axial Flow

ny qin x = −2 x = 0 x = 3 x = 6

4 2120 0.16406 0.02685 0.06918 0.16406

8 8584 0.16602 0.06460 0.10409 0.16602

16 341340 0.16650 0.10860 0.13326 0.16650

32 13651364 0.16663 0.13857 0.15104 0.16663


Table 3.54 shows the axial flow in the unweighted solution. As for channel flow we see

that a large proportion of the flow is lost, especially on the coarser grids. When ny = 4

the flow through the line x = 0 is 16.4% of that through the inlet and outlet whilst on

x = 3 it is 42.2%. The equivalents at ny = 16 are 65.2% and 80.0% respectively and

at ny = 32 they are 83.2% and 90.6%.

Figure 3.3: Velocity field with equal weights at ny = 8




Figures 3.3 and 3.4 show the velocity fields in the unweighted solution at ny = 8

and ny = 16 respectively. They give a graphical indication of how much of the flow is

lost, particularly in the portion of the region which is close to the re-entrant corner.

Axial Flow

ny qin x = −2 x = 0 x = 3 x = 6

4 2120 0.16406 0.16254 0.16331 0.16406

8 8584 0.16602 0.16547 0.16572 0.16602

16 341340 0.16650 0.16629 0.16639 0.16650

32 13651364 0.16663 0.16654 0.16658 0.16663


Figure 3.5: Velocity field with weights of 1, 1, 103, 103 at ny = 8




From Table 3.55 we observe that far less mass flow is lost with weighting. At ny = 4

the flow through the line x = 0 is 99.1% of the inflow whilst the flow through the line

x = 3 is 99.5% of the inflow. At ny = 16 the flow through both the line x = 0 and

the line x = 3 is approximately 99.9% of the inflow. Figures 3.5 and 3.6 show the

velocity fields in the weighted solution at ny = 8 and ny = 16. We see that there is

much more flow than in the corresponding unweighted solutions shown in Figures 3.3

and 3.4, especially in the part of the region which is close to the re-entrant corner.


ny qin ‖ u 132− uh ‖0,2 ‖ u 1

32− uh ‖∞ | u 1

32− uh |1,2

4 2120 0.02958 0.05202 0.33016

8 8584 0.00935 0.03142 0.15901

16 341340 0.00280 0.01951 0.07470


Table 3.56 shows the difference between the linear solution in the velocity variables

on a given grid and the solution in the velocity variables on the grid for which ny = 32.

Between ny = 8 and ny = 16 this error reduces by a factor of 3.3 in L2, 1.6 in L∞ and

2.1 as measured in the H1 semi-norm. So asymptotic convergence may be order h2 in

L2 and around order h in H1.




Axial Flow

ny x = −2 x = 0 x = 3 x = 6

4 0.15625 0.02255 0.01069 0.00783

8 0.16406 0.05723 0.04326 0.03954

16 0.16602 0.10145 0.09295 0.09080

32 0.16650 0.13417 0.13063 0.12986


The degradation of flow in the unweighted solution is even worse with these boundary

conditions than it is for enclosed flow, as was seen for the results in the straight channel.

From Table 3.57 we see that at ny = 4 only 14.4% of the inflow remains at the line

x = 0. Only 5.0% of the flow entering the region leaves it. The results at ny = 8 and

ny = 16 are better; the net flow on exit is 24.1% of that on entry in the former case and

54.7% in the latter. At ny = 32 the equivalent figure is 78.0%, which is a substantial

improvement.





Illustrations of the velocity fields in the unweighted solution at ng = 8 and ng = 16

can be seen in Figures 3.7 and 3.8. These show graphically the extent to which flow is

lost.


ny ‖ u 132− uh ‖0,2 ‖ u 1

32− uh ‖∞ | u 1

32− uh |1,2

4 0.03965 0.05354 0.33745

8 0.01108 0.03159 0.16509

16 0.00302 0.01956 0.08100


Axial Flow

ny x = −2 x = 0 x = 3 x = 6

4 0.15625 0.15464 0.15476 0.15479

8 0.16406 0.16342 0.16336 0.16335

16 0.16602 0.16576 0.16574 0.16573

32 0.16650 0.16640 0.16639 0.16639




When using the weighted functional (3.1) the flow is conserved quite well; see Ta-

ble 3.59. Even in this case though the mass loss is greater than for the enclosed flow

boundary conditions as can be seen from comparing Table 3.59 with Table 3.55. We see

for instance that at ny = 16 approximately 0.2% of the flow is lost in the inlet channel,

whilst at ny = 32 less than 0.1% of the mass disappears. We can see from Table 3.58

that, as we saw with enclosed flow in Table 3.56, convergence in H1 is a little greater

than order h. (The algorithm used to estimate the error for this problem only gives an

approximate figure. It may be that this algorithm gives an order of convergence greater

than the true one, particularly in H1.) The errors in L2 here are greater in magnitude

than those in the solutions satisfying enclosed flow boundary conditions, particularly

on the coarser grids. Convergence in L2 is closer to order h2. The rate of convergence

in L∞ is almost the same as in the solution satisfying enclosed flow conditions.



Figures 3.9 and 3.10 show plots of the velocity field in the weighted solution at

ng = 8 and ng = 16 respectively. These are much more plausible representations of the



expected flow fields than the plots of the unweighted solutions shown in Figures 3.7

and 3.8. Hardly any of the flow appears to be lost.

3.3.2 Boundary Conditions and Results for the J Formulation

For enclosed flow boundary conditions the fixed variables on the inlet are

U1 = −qiny(1 + y), U2 = 0.

On the walls we set

U1 = 0, U2 = 0.

The conditions on the outlet are that

U1 = 0.125(1− y2

), U2 = 0.

Additionally we set the pressure U3 at the point B equal to zero.

Axial Flow

ny qin x = −2 x = 0 x = 3 x = 6

4 2120 0.16406 0.02934 0.08492 0.16406

8 8584 0.16602 0.04116 0.09576 0.16602

16 341340 0.16650 0.05373 0.10487 0.16650

32 13651364 0.16663 0.06989 0.11431 0.16663





ny qin ‖ u 132− uh ‖0,2 ‖ u 1

32− uh ‖∞ | u 1

32− uh |1,2

4 2120 0.03005 0.05205 0.32859

8 8584 0.00969 0.03124 0.15767

16 341340 0.00293 0.01943 0.07539


term

Axial Flow

ny qin x = −2 x = 0 x = 3 x = 6

8 8584 0.16602 0.16477 0.16537 0.16602

16 341340 0.16650 0.16578 0.16612 0.16650

32 13651364 0.16663 0.16619 0.16639 0.16663


The results in Table 3.60 can be compared with those in Table 3.54, which gives the

equivalent results for the S formulation. Though the solution of the J formulation on

the lines x = 0 and x = 3 appears more accurate on the coarsest grid, the S formulation

rapidly overtakes it for accuracy as the space between the nodes is reduced. For instance

at ny = 32 only 41.9% of the inflow passes through the line x = 0, as compared with

83.2% in the solution of the S formulation. Somewhat more mass is conserved in the

solution of the J formulation with weighting than in the solution of the S formulation

with weighting; compare Table 3.62 with Table 3.55. The quiver plots of the velocity

field look roughly similar to those for the S formulation. The errors in the velocity

fields are shown in Table 3.61. These errors and the convergence rates are very similar

to those in the solution of the weighted S formulation; see Table 3.56.



3.3.3 Boundary Conditions and Results for the G Formulations

We specify enclosed flow boundary conditions. On the inlet

U1 = −qiny(1 + y), U2 = 0, U5 = −(1 + 2y), U6 = 0.

By the no-slip boundary condition we have that on the walls

U1 = 0, U2 = 0.

On the walls BC, EO and ED the boundary requirement U× n = Gb(x, y) implies

U3 = 0, U4 = 0.

On the wall EO, the condition on the tangential gradient of the velocity means

U5 = 0, U6 = 0.

The boundary conditions on the outlet are

U1 = 0.125(1− y2

), U2 = 0, U5 = −0.25y, U6 = 0.

We set the pressure U7 at B equal to zero. The results presented below have been

obtained by solving the G3 system of equations. The only points we consider as vertices

when calculating D in (2.45) are E and O as it is only at infinity that there is truly

undisturbed Poiseuille flow on the inlet and outlet.

Axial Flow

ny qin x = −2 x = 0 x = 3 x = 6

8 8584 0.16602 0.02353 0.07463 0.16602

16 341340 0.16650 0.02337 0.07477 0.16650

32 13651364 0.16663 0.02388 0.07438 0.16663




We see from Table 3.63 that the solution obtained with no weighting of the velocity

divergence term does not appear to converge as the grid is refined. For instance the

flow through the line x = 0 is 14.2% of the flow through the inlet x = −2 at ny = 8

and 14.0% at ny = 16. The right hand-side of (2.38) is in L2. However the gradients

of the velocity, which are the variables U1 through to U6 of this formulation, are not

in H1 in the analytical solution of this problem, but this is the space in which we are

approximating. Furthermore in a region with a re-entrant corner the estimates (2.49)

and (2.50) for solutions of the G2 formulation and (2.51) and (2.52) for solutions of

the G3 formulation may not apply; see [40]. So for this problem we cannot expect

convergence of even order h in H1 in the velocities. We note that the results we have

obtained by minimising the unweighted G2 functional are similar to those obtained by

minimising the unweighted G3 functional.


ny qin ‖ u 132− uh ‖0,2 ‖ u 1

32− uh ‖∞ | u 1

32− uh |1,2

4 2120 0.02888 0.04997 0.32407

8 8584 0.00901 0.02970 0.15374

16 341340 0.00267 0.01843 0.07193


term

Axial Flow

ny qin x = −2 x = 0 x = 3 x = 6

4 2120 0.16406 0.16079 0.16240 0.16406

8 8584 0.16602 0.16338 0.16460 0.16602

16 341340 0.16650 0.16418 0.16523 0.16650

32 13651364 0.16663 0.16446 0.16542 0.16663




With weighting the flow is preserved much better; see Table 3.65. For instance at

ny = 4 the flow through the line x = 0 is 98.0% of the imposed flow through the inlet

whilst at ny = 16 it is 98.6%. We also observe convergence; see Table 3.64. The errors

in the velocity variables are slightly smaller than those in the solutions of the S and J

formulations; see Tables 3.56 and 3.61. Convergence rates are about the same as those

in the solutions of the two other formulations where these boundary conditions are

satisfied. The solutions obtained using the weighted G2 functional are almost identical

to these here for the weighted G3 functional.

3.3.4 Effect of Further Refinement near the Re-entrant Corner

There is a singularity in the solution at the corner O of the L-shaped region shown in

Figure 3.2 and therefore the grid near this point perhaps requires more refinement than

for other portions of the region. We developed a grid where the elements and vertices

are denser closer to the corner O. This grid is illustrated in Figure 3.11.

Figure 3.11: Planar backward facing step grid with further refinement close to the

re-entrant corner at ny = 2



Enclosed Flow Boundary Conditions in the S Formulation with Variable qin

on a Grid with Refinement Close to the Re-entrant Corner

Axial Flow

ny qin x = −2 x = 0 x = 3 x = 6

4 2120 0.16406 0.16260 0.16332 0.16406

8 8584 0.16602 0.16549 0.16573 0.16602

16 341340 0.16650 0.16630 0.16639 0.16650

32 13651364 0.16663 0.16654 0.16658 0.16663


Table 3.66 shows the axial flow in the solution of the weighted S formulation on meshes

of the form illustrated in Figure 3.11. Comparing Table 3.66 with Table 3.55, the

equivalent one for the regular grid, we see that the results are almost identical although

we would expect to see an improvement in the quality of the solution close to the re-

entrant corner.

3.3.5 Summary of Results on Grid with Refinement Near to the Re-

entrant Corner

We have examined solutions on the form of grid illustrated in Figure 3.11 in all three

formulations. From our investigations, flow is conserved significantly better in the

solutions of the unweighted formulations considering the relatively small amount of

extra refinement near the corner which we have introduced here. It would appear

that the extra effort required in developing and solving on a grid with even more

refinement close to the corner could be rewarded, in so far as a solution of an unweighted

formulation is desirable.

However, the weighted solutions do not improve significantly, taking into account

the extra work required in the refinement, as we have shown for the solution of the

S functional with enclosed flow boundary conditions. We have already demonstrated



that it is the weighted solutions which are generally to be preferred for their accuracy.

3.3.6 Summary of Results for Stokes Problems in Backward Facing

Step Region Obtained with Linear Triangles

None of the three formulations give solutions where mass is preserved adequately unless

the mass conservation term is weighted. This is especially so for results on coarser grids.

In particular the solution obtained by minimising unweighted G functionals does not

appear to be converging. The solution of the unweighted S formulation is more accurate

than the solution of the unweighted J formulation.

We have found that for this problem the velocities converge at a broadly similar

rate in the weighted solutions of all three formulations. The convergence rates on the

grids considered are somewhat greater than order h in H1 and not too far from order

h2 in L2. Convergence in L∞ is slower than convergence in H1.

3.4 Flow over a Backward Facing Step with a Long Out-

flow Region Modelled Using Quadratic Triangles

The results presented so far have been obtained using continuous piecewise linear basis

functions on triangles. We shall now give results using quadratic approximation on

triangles; see [88]. The solution for parabolic Poiseuille flow in a straight channel

is captured exactly using this method, as are the inflow and outflow in the solution

over the backward facing step region. Hence there is no need to modify the specified

nodal values of the axial velocity on the inlet to force the integral of the inflow to be

equal to the integral of the outflow. Both are equal to the integral of the analytical

solution. We present and discuss results obtained using this form of interpolation to

approximate flow over a backward facing step with a long outflow region; the whole

region is [−2, 0]× [−1, 0] ∪ [0, 20]× [−1, 1].

3.4. Flow over a Backward Facing Step with a Long Outflow Region Modelled Using

Quadratic Triangles


3.4.1 Results in the S Formulation


Axial Flow

ny x = −2 x = 0 x = 5 x = 10 x = 15 x = 20

4 0.16667 0.08817 0.10492 0.12552 0.14613 0.16667

8 0.16667 0.11869 0.12871 0.14137 0.15402 0.16667

16 0.16667 0.14042 0.14592 0.15283 0.15975 0.16667

32 0.16667 0.15327 0.15608 0.15961 0.16314 0.16667


Axial Flow

ny x = −2 x = 0 x = 5 x = 10 x = 15 x = 20

4 0.16667 0.16617 0.16628 0.16641 0.16654 0.16667

8 0.16667 0.16646 0.16651 0.16656 0.16661 0.16667

16 0.16667 0.16658 0.16660 0.16662 0.16664 0.16667

32 0.16667 0.16662 0.16663 0.16664 0.16666 0.16667


The flow in the solution obtained using the unweighted S formulation is shown in

Table 3.67. For a given value of ny, the flow through the line x = 0 is considerably

greater than the flow through the same line shown in Table 3.54. However there is still

a significant amount of mass lost between the inlet and this line. With weighting, we

see from Table 3.68 that there is very little loss of flow.


Quadratic Triangles



Axial Flow

ny x = −2 x = 0 x = 5 x = 10 x = 15 x = 20

4 0.16667 0.08633 0.08263 0.08263 0.08263 0.08263

8 0.16667 0.11700 0.11442 0.11442 0.11442 0.11442

16 0.16667 0.13929 0.13789 0.13789 0.13789 0.13789

32 0.16667 0.15263 0.15192 0.15192 0.15192 0.15192


Axial Flow

ny x = −2 x = 0 x = 5 x = 10 x = 15 x = 20

4 0.16667 0.16615 0.16613 0.16613 0.16613 0.16613

8 0.16667 0.16645 0.16644 0.16644 0.16644 0.16644

16 0.16667 0.16657 0.16657 0.16657 0.16657 0.16657

32 0.16667 0.16662 0.16662 0.16662 0.16662 0.16662


With equal weights on each equation term, the flow decreases substantially between

the inflow line x = −2 and the line x = 0; see Table 3.69. A small quantity of mass is

also lost between the line x = 0 and x = 5 but then the flow is constant along the rest

of the length of the region. This is because the axial velocity of the fluid has settled

back into a form which varies quadratically in y, and this is captured exactly by the

elements used. The numerical value of the outflow is however considerably less than

the inflow, more so on the coarser grids. It ranges from 49.6% of the inflow at ny = 4

to 91.2% at ny = 32.

Table 3.70 shows the corresponding solution obtained by the weighted S formulation.

A tiny fraction of the flow is lost between x = −2 and x = 0. On the two coarser grids


Quadratic Triangles


an even smaller portion is lost between x = 0 and x = 5. The flow is then constant

until it reaches the outlet.

3.4.2 Results in the J Formulation

We enforce enclosed flow boundary conditions.

Axial Flow

ny x = −2 x = 0 x = 5 x = 10 x = 15 x = 20

4 0.16667 0.02852 0.05916 0.09507 0.13097 0.16667

8 0.16667 0.03370 0.06354 0.09794 0.13234 0.16667

16 0.16667 0.04613 0.07352 0.10458 0.13563 0.16667

32 0.16667 0.06150 0.08532 0.11244 0.13956 0.16667


Axial Flow

ny x = −2 x = 0 x = 5 x = 10 x = 15 x = 20

4 0.16667 0.16364 0.16435 0.16512 0.16590 0.16667

8 0.16667 0.16270 0.16366 0.16466 0.16566 0.16667

16 0.16667 0.16394 0.16460 0.16529 0.16598 0.16667

32 0.16667 0.16503 0.16542 0.16584 0.16625 0.16667

Table 3.72: Axial flow with weight of 103 on the mass conservation term

With equal weights much mass is lost between the lines x = −2 and x = 0. There

is more mass lost here than was the case with the unweighted S formulation; compare

Tables 3.67 and 3.71. It appears that convergence to a reasonable solution is slow.

Though most of the flow is preserved in the solution by the weighted J formulation, as

can be seen from Table 3.72, more mass is lost than in the solution by the weighted S

functional; we refer back to Table 3.68.


Quadratic Triangles


3.4.3 Results in the G3 Formulation

Enclosed flow boundary conditions hold.

Axial Flow

ny x = −2 x = 0 x = 5 x = 10 x = 15 x = 20

4 0.16667 0.02206 0.03807 0.08161 0.12584 0.16667

8 0.16667 0.02145 0.03642 0.08028 0.12483 0.16667

16 0.16667 0.02239 0.03558 0.07959 0.12428 0.16667


Axial Flow

ny x = −2 x = 0 x = 5 x = 10 x = 15 x = 20

4 0.16667 0.16053 0.16191 0.16350 0.16508 0.16667

8 0.16667 0.15804 0.16004 0.16225 0.16446 0.16667

16 0.16667 0.16011 0.16158 0.16328 0.16497 0.16667

Table 3.74: Axial flow with weight of 103 on the mass conservation term

The solution obtained by minimising the unweighted G3 functional does not converge

as the grid is refined; see Table 3.73. This is the behaviour that we would expect from

our results on linear triangles. Though the solution obtained with weighting is far

more accurate we note that nevertheless more of the flow is lost than in the solutions

of the weighted S and J formulations with the same boundary conditions. For instance

between the line x = −2 and the line x = 0 we find that 3.9% of the flow is lost

at ng = 16; see Table 3.74. This compares with only 0.05% in the solution of the S

formulation and 1.0% in the solution of the J formulation; see Tables 3.68 and 3.72

respectively.


Quadratic Triangles


We have also obtained solutions of each of the three formulations in the same region

using biquadratic quadrilateral elements; see [62]. We see the same features as in

the solutions consisting of quadratic interpolations on triangles. The solutions arising

from the unweighted functionals are poor; the solutions obtained with functionals in

which appropriate terms are weighted are much more accurate. The solution of the

unweighted G3 functional does not converge, just as is the case with linear or quadratic

triangles.

3.5 Flow around a Cylindrical Obstruction

In this section we examine least-squares solutions for the Stokes problem of flow around

a solid circular cylinder. We consider the region [0, 10]× [−2.5, 2.5] from which points

satisfying (x − 3)2 + y2 < 1 are excluded. In other words the cylinder is centred

at the point (3, 0) and has unit radius. The geometry of the cross-section is shown in

Figure 3.12. We solve the problem in which the cylinder is travelling at some velocity V

through an expanse of fluid which is at rest at infinity. This is a very frequently studied

problem in the computational fluid dynamics literature and finite element techniques

have been used to obtain a solution. In particular, discussion of solutions obtained

using the first-order least-squares J formulation can be found in [55] and [63]. By

a straightforward change of reference frames they alter the specification so that they

model the flow around a stationary cylinder in an oblong domain on the exterior edges

of which the fluid is moving at velocity V .

3.5. Flow around a Cylindrical Obstruction


A B

CD

P

Q

P"

Q" Y

Y"

Figure 3.12: Geometry of region for flow around solid circular cylinder

We experimented generating Delaunay triangulations for this region using MATLAB

PDE Toolbox. An example of one such grid is shown in Figure 3.13. However the

bandwidths of the stiffness matrices for problems on the grids produced by this program

are usually unreasonably high1.

Figure 3.13: Grid generated by MATLAB PDE Toolbox

Instead we work with grids of the form displayed in Figures 3.14, 3.15 and 3.16. The1As we subsequently obtained solutions on these grids using code written in FORTRAN

we were unable to take advantage of the features of the MATLAB solvers which reduce thebandwidth.



number and spacing of the points are functions of a single parameter which we call ng.

Figure 3.14: Mesh for region around a cylindrical obstruction at ng = 1

Figure 3.15: First refinement at ng = 2

Figure 3.16: Second refinement at ng = 4



Boundary Conditions for Flow around a Cylinder Moving through Fluid at

Rest at Infinity in the J Formulation

The boundary conditions in this case are that

U1 = 1, U2 = 0

on the lines AB, BC, CD and DA and on the surface of the cylinder that

U1 = 0, U2 = 0.

The pressure is fixed at the midpoint of the line AD.

Results Obtained Modelling Flow around a Cylinder Moving through Fluid

at Rest at Infinity in the J Formulation

Axial Flow

ng AD P”Q” PQ YY” BC

1 5.00000 0.74627 0.74627 2.85242 5.00000

2 5.00000 0.86385 0.86385 2.97726 5.00000

4 5.00000 1.21536 1.21536 3.40644 5.00000

8 5.00000 1.79458 1.79458 4.13324 5.00000




Axial Flow


1 5.00000 2.25850 2.25850 4.68971 5.00000

2 5.00000 2.44986 2.44986 4.93442 5.00000

4 5.00000 2.48731 2.48731 4.98348 5.00000

8 5.00000 2.49716 2.49716 4.99634 5.00000


The flow is not well conserved in the solution of the unweighted J formulation though

less of the mass is lost on finer grids; see Table 3.75. In particular a large quantity

of mass is lost near the cylinder. We see for example that 70.1% of the flow is lost

between the line AD and the lines PQ and P”Q” at ng = 1 compared with 28.2% at

ng = 8. There is much less mass lost in the solution of the weighted J formulation,

even on coarse grids; see Table 3.76. For instance at ng = 1 only 9.7% of the fluid

is lost between the line AD and the lines PQ and P”Q”. We have also obtained the

solution of the problem in which the lines AB and CD are walls whilst flow enters

in parabolic Poiseuille flow through the line AD and leaves through the line BC with

the same profile. We see in that case that the flows above and below the cylinder in

the unweighted solution only very slowly converge to the correct values as the grid is

refined whilst not much flow is lost in the weighted solution at all grid levels.

3.5.1 Solution of S Formulation for Cylinder Moving through Fluid

at Rest at Infinity in Symmetric Half Region

In obtaining solutions of the S formulation for the cylinder moving through fluid at

rest at infinity we can enforce symmetry conditions on the horizontal centreline and

just solve in the region shown in Figure 3.17. On the line AF we have that x = 0 whilst

x = 10 on the line DE. On EF we have y = 2.5 whilst y = 0 on the lines AB and CD.



On the line joining Y and Y” we have that x = 6. The cylinder is of unit radius with

centre at (3, 0).

A B C D

EF

P

Q

A B C D

EF

P

Q

Y

Y"

Figure 3.17: Geometry of symmetric half cylinder problem

We solve on the upper half of the grid illustrated in Figure 3.14 and its refinements.

We have obtained results at ng = 1, 2, 4, 8 and 16.

On the lines EF, AF and DE the boundary conditions are

U3 = 0, U4 = 1.

As boundary conditions on the intervals AB and CD we have the symmetry conditions

U2 = 0, U3 = 0.

Fluid in contact with the cylinder is at rest, which gives us the conditions

U3 = 0, U4 = 0.



The pointwise constraints are that U1 is fixed to zero at both A and E.

Results for Symmetric Half Cylinder Solution of S Formulation for Cylinder

Moving through Fluid at Rest at Infinity

Axial Flow

ng AF PQ YY” DE

1 2.50000 0.71943 1.39352 2.50000

2 2.50000 0.71283 1.33154 2.50000

4 2.50000 0.70965 1.30670 2.50000

8 2.50000 0.70857 1.29901 2.50000

16 2.50000 0.70826 1.29691 2.50000

Table 3.77: Axial flow in the solution with equal weights

Axial Flow

ng AF PQ YY” DE

1 2.50000 2.23178 2.32635 2.50000

2 2.50000 2.40286 2.43531 2.50000

4 2.50000 2.43147 2.45406 2.50000

8 2.50000 2.43825 2.45853 2.50000

16 2.50000 2.43965 2.45946 2.50000

Table 3.78: Axial flow in the solution with weights of 1, 1, 103, 103



Figure 3.18: Unweighted solution at ng = 4

Figure 3.19: Weighted solution at ng = 4

We observe from Table 3.77 and Figure 3.18 that much flow is lost in the solution

obtained with equal weights. Not only this, but also the flow is not converging to the

correct value as the grid is refined. Table 3.78 and Figure 3.19 show that considerably

less mass is lost with weighting. There is much more flow around the cylinder.

The errors shown in Tables 3.79 and 3.80 are obtained by comparing the axial

velocity uh in the solution on a given grid obtained using the S formulation with the

axial velocity uh in the solution of a mixed problem. We use the method presented in

[109] and referred to earlier in this thesis. We recall from (2.4) that for this method

the approximation to the velocity is linear and defined on triangular elements. The

solution consists of the value of the velocity at the vertex of each triangle. We solve

for the velocities uh on the same grids as we solve the least-squares problems on to

obtain the solution uh, namely the upper portions of the grids illustrated in Figures

3.14, 3.15 and 3.16 or their refinements. The pressure space is defined by (2.5). The

approximation to the pressure is linear on quadrilateral macro-elements. We construct



these macro-elements from sets of eight neighbouring triangles on the grid used in

solving for the velocity. An example of how to construct one of these macro-elements

is shown in Figure 2.1. The solution consists of the value of the pressure and its

derivatives at the midpoints of each of the macro-elements. We recall that mass is

conserved over each quadrilateral pressure element with this method. For the mixed

problem both velocities are fixed on the inlet, the outlet, the line EF and the cylinder.

The transverse component of the velocity is set equal to zero on the lines AB and CD.

The pressure is specified at the interior point closest to the bottom left hand corner.

The approximations to the velocity in the Galerkin solution should converge at order

h2 in L2 and we regard this solution as being close to the analytical one; see [26].

Errors in Axial Velocity

ng ‖ uh − uh ‖0,2 ‖ uh − uh ‖∞ | uh − uh |1,2

1 2.83635 1.90956 5.11488

2 2.85985 1.77298 4.96352

4 2.87260 1.75544 4.93015

8 2.87732 1.75050 4.92354

16 2.87875 1.75030 4.92225

Table 3.79: Errors in axial velocity with equal weights


ng ‖ uh − uh ‖0,2 ‖ uh − uh ‖∞ | uh − uh |1,2

1 0.86128 0.97935 3.09705

2 0.51684 0.62075 2.14012

4 0.18322 0.26138 1.01890

8 0.09798 0.07618 0.39344

16 0.09570 0.05699 0.19715

Table 3.80: Errors in axial velocity with weights of 1, 1, 103, 103



We see little evidence of convergence in Table 3.79. The reduction in error in L∞

and H1 is negligible between all the grids except ng = 1 and ng = 2 and the error in L2

slightly increases as the grid is refined. With weighting, the convergence rate between

the grids at ng = 4 and ng = 8 is greater than order h in L∞ and H1; see Table 3.80.

It is approximately order h in L2. Between the grids for ny = 8 and ny = 16 the

convergence rate is much less.

We recall the warning concerning algorithms which only give approximations of the

error in the discussion of Table 3.58. It may be that the algorithm we use here and

later to estimate errors in solutions of flow around a cylinder or over a half-cylinder

does not give a true estimate of the error. In particular, the rate of convergence in H1

may appear greater than it is.

Figures 3.20 and 3.21 show how the axial velocity varies along the line PQ on the

grids ng = 1 and ng = 16 respectively. On both grids the variation in the unweighted

solution is almost linear.

1 1.5 2 2.50

0.5

1

1.5

2

2.5

y

u x

unweightedweighted

Figure 3.20: Plot of ux on line PQ at ng = 1



1 1.5 2 2.50

0.5

1

1.5

2

2.5

y

u x

unweightedweighted

Figure 3.21: Plot of ux on line PQ at ng = 16

3.6 Poiseuille Flow over a Semicylindrical Restriction

We also model Poiseuille flow in the region illustrated in Figure 3.17. The upper and

lower extremes of the region, including the surface of the semicylindrical restriction,

are walls, and we set the flow on the inlet line AF to be parabolic. The line DE is an

outlet line. The flow there should settle down so that it is also parabolic.

Boundary Conditions and Results in the S Formulation

In this case the boundary conditions on the inlet line AF are

U3 = 0, U4 = 0.16y (2.5− y) .

The fluid velocity is zero on the walls AB, CD and EF. It is also zero on the surface of

the cylinder, so that

U3 = 0, U4 = 0.

The results shown here are for enclosed flow boundary conditions. In this case the

3.6. Poiseuille Flow over a Semicylindrical Restriction


conditions on the outlet are

U3 = 0, U4 = 0.16y (2.5− y)

with linear constraints

U1 = 0 at A, U1 = 0 at E, U2 = 0 at E.

We compare the least-squares solution uh on a given grid with the mixed solution uh

obtained with continuous and piecewise linear velocity approximations from the space

(2.4) and discontinuous pressure approximations from the space (2.5). The mixed

solution satisfies enclosed flow boundary conditions, with the pressure fixed at a single

point in the region.

Axial Flow

ng AF PQ YY” DE

1 0.40509 0.05054 0.10687 0.40509

2 0.41377 0.14101 0.20311 0.41377

4 0.41594 0.27899 0.31424 0.41594

8 0.41649 0.37107 0.38313 0.41649

16 0.41662 0.40430 0.40759 0.41662




Axial Flow

ng AF PQ YY” DE

1 0.40509 0.36006 0.37576 0.40509

2 0.41377 0.40433 0.40755 0.41377

4 0.41594 0.41369 0.41446 0.41594

8 0.41649 0.41602 0.41618 0.41649

16 0.41662 0.41654 0.41657 0.41662


Setting ng = 1 much fluid is lost in the unweighted solution, around 87.5% between

the inlet AF and the line PQ; see Table 3.81. Even with weighting 11.1% of the mass is

lost between those two lines on this grid; see Table 3.82. Much less mass is lost on the

more highly refined grids. At ng = 16, the flow in the unweighted solution through the

line PQ is 97.0% of that through the inlet AF. In the weighted solution the loss of flow

between the two lines on this grid is negligible. Summing up, we see that weighting the

appropriate terms in the S functional reduces the loss of mass between AF and PQ,

though the difference is less significant on more refined grids. A similar picture is seen

comparing the flow through the line YY” to that through the outlet line DE, though

the loss of mass is somewhat less acute between these two lines.




ng ‖ uh − uh ‖0,2 ‖ uh − uh ‖∞ | uh − uh |1,2

1 0.63378 0.37822 1.11428

2 0.46517 0.26966 0.81304

4 0.22938 0.13405 0.40150

8 0.07570 0.04441 0.13287

16 0.02052 0.01205 0.03637

Table 3.83: Errors in axial velocity with equal weights


ng ‖ uh − uh ‖0,2 ‖ uh − uh ‖∞ | uh − uh |1,2

1 0.16524 0.20019 0.64266

2 0.10858 0.12726 0.44746

4 0.03504 0.05290 0.21839

8 0.00767 0.01441 0.07875

16 0.00128 0.00268 0.02421

Table 3.84: Errors in axial velocity with weights of 1, 1, 103, 103

In all three metrics, the convergence rates shown in Table 3.83 are approaching order

h2. Whilst the errors in L2 and L∞ reduces by more than a factor of four between

ng = 8 and ng = 16 in the weighted solution, the convergence rate in H1 between these

two grids is actually less than it is in the unweighted solution; see Table 3.84. The

magnitudes of the errors are of course much less with weighting.

Figure 3.22 shows how the value of ux in the solutions obtained on the grid ng = 1

varies along the line PQ. The solution is much closer to zero in the unweighted case.

The weighted solution is less symmetric than the unweighted one. Figure 3.23 shows



the corresponding plots at ng = 16. Here the unweighted solution is much closer to the

weighted one. Both are quite symmetric.

1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

y

u x

unweightedweighted

Figure 3.22: Plot of ux in solutions on line PQ at ng = 1

1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

y

u x

unweightedweighted

Figure 3.23: Plot of ux in solutions on line PQ at ng = 16



Boundary Conditions and Results in the J Formulation

We have enclosed flow boundary conditions, so that on the lines AF and DE

U1 = 0.16y (2.5− y) , U2 = 0.

On the walls, including the surface of the cylinder

U1 = 0, U2 = 0.

We fix the pressure U3 as zero at the interior vertex closest to the origin.

Axial Flow

ng AF PQ YY” DE

1 0.40509 0.35970 0.37582 0.40509

2 0.41377 0.40361 0.40715 0.41377

4 0.41594 0.41324 0.41418 0.41594

8 0.41649 0.41586 0.41608 0.41649

16 0.41662 0.41649 0.41654 0.41662

Table 3.85: Axial flow with a weight of 103 on the mass conservation term


ng ‖ u− uh ‖0,2 ‖ u− uh ‖∞ | u− uh |1,2

1 0.16659 0.20129 0.64612

2 0.11245 0.12939 0.45820

4 0.03706 0.05393 0.22414

8 0.00821 0.01465 0.07972

16 0.00139 0.00271 0.02442

Table 3.86: Errors with a weight of 103 on the mass conservation term



At ng = 1 the flow through the line PQ is 88.8% of the total imposed flow and at

ng = 8 it is 99.8%; see Table 3.85. These are slightly less than the comparable amounts

in the solution of the weighted S formulation shown in Table 3.84, which we recall are

88.9% at ng = 1 and 99.9% at ng = 8.

The errors between this solution and the mixed solution on a given grid are shown in

Table 3.86. The errors in Table 3.86 are slightly greater than the ones in the solution

of the S formulation of the same problem which were given in Table 3.84, but the

convergence rates are roughly the same.

We have found that the unweighted J functional does not perform as well as the

unweighted S functional in this region. We note that the unweighted S functional also

performed better than the unweighted J functional in modelling flow over a backward

facing step which was another problem in a concave domain.

3.7 Other Means of Overcoming the Lack of Mass Con-

servation

We have presented results of experiments using three very different first-order refor-

mulations of the planar Stokes equations. With all three mass is not conserved as the

flow advances.

We have shown that mass can be conserved reasonably well provided that appropri-

ate terms in the least-squares functionals, in particular the one corresponding to the

mass continuity equation, are multiplied by suitably sized factors.

Similar problems have already been highlighted in the literature. As stated previ-

ously Chang and Nelson [55] modelled the Stokes flow through a rectangular region in

which there was a hard, non-porous cylindrical obstruction using the velocity-vorticity-

pressure formulation. They made special reference to the mass lost between the walls

and the poles of the obstruction. They proposed introducing terms having Lagrange

3.7. Other Means of Overcoming the Lack of Mass Conservation


undetermined multipliers to the functional to ameliorate this problem and named this

new method the restricted LSFEM. Though altering the functional in this way gave

a much better solution, the linear system was no longer positive definite, one of the

major attractions of the standard least-squares method.

In [23], [29], [55], [63] and elsewhere, researchers have used or suggested mesh de-

pendent weighting of particular terms. The theory behind this is most fully developed

in [8]. We refer to Appendix A.

3.8 The Null Matrix Least-Squares Finite Element Method

At the end of this section we shall present a new method for obtaining a least-squares

finite element solution and apply this to the S formulation. Firstly we investigate the

effect of varying the numerical value of the weights w3 and w4 in the functional (3.1)

when determining flow in the long channel. We solve on the region [0, 20]× [0, 1] with

a regular Union Jack grid configuration.

Grid

Weights 80× 4 160× 8 320× 16 640× 32 1280× 64

1, 1, 1, 1 2.3× 105 5.9× 105 1.3× 106 1.9× 106 2.1× 106

1, 1, 101, 101 8.4× 105 1.4× 106 1.8× 106 2.0× 106 2.1× 106

1, 1, 102, 102 4.2× 106 4.8× 106 5.2× 106 5.5× 106 5.7× 106

1, 1, 103, 103 3.3× 107 3.7× 107 3.9× 107 4.2× 107 4.4× 107

1, 1, 106, 106 3.2× 1010 3.5× 1010 3.8× 1010 4.0× 1010 3.8× 1010

1, 1, 109, 109 3.2× 1013 3.4× 1013 3.4× 1013 2.5× 1013 4.0× 1012

Table 3.87: Pivot ratio on a given grid versus weights

3.8. The Null Matrix Least-Squares Finite Element Method




80× 4 46.10704 45.92960 0.06647 2.09562

160× 8 34.59434 34.45773 0.03795 1.52077

320× 16 17.27580 17.20742 0.01706 0.76221

640× 32 5.74949 5.72704 0.00554 0.26506

1280× 64 1.56746 1.56172 0.00150 0.07895

Table 3.88: H1 semi-norm errors by variable with weights of 1, 1, 101, 101

Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15625 0.02036 0.00516 0.02036 0.15625

160× 8 0.16406 0.06104 0.03709 0.06104 0.16406

320× 16 0.16602 0.11516 0.09981 0.11516 0.16602

640× 32 0.16650 0.14966 0.14420 0.14966 0.16650

1280× 64 0.16663 0.16204 0.16053 0.16204 0.16663






80× 4 27.83047 27.74178 0.03557 1.19924

160× 8 11.20718 11.17209 0.01089 0.52555

320× 16 3.29980 3.28931 0.00280 0.20170

640× 32 0.87272 0.87042 0.00071 0.08617

1280× 64 0.23101 0.23085 0.00018 0.04079


Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15625 0.09032 0.07166 0.09032 0.15625

160× 8 0.16406 0.13657 0.12782 0.13657 0.16406

320× 16 0.16602 0.15789 0.15522 0.15789 0.16602

640× 32 0.16650 0.16438 0.16367 0.16438 0.16650

1280× 64 0.16663 0.16609 0.16591 0.16609 0.16663






80× 4 11.98586 11.91272 0.00652 0.17542

160× 8 3.50364 3.42088 0.00134 0.05061

320× 16 0.96915 0.88705 0.00030 0.01312

640× 32 0.29801 0.22385 0.00007 0.00331

1280× 64 0.11347 0.05623 0.00002 0.00084




80× 4 8.69593 8.69614 0.00001 0.64550

160× 8 2.43900 2.43915 0.00000 0.32275

320× 16 0.75729 0.75677 0.00000 0.16137

640× 32 1.81122 1.80438 0.00000 0.08069


Axial Flow

nx × ny x = 0 x = 5 x = 10 x = 15 x = 20

80× 4 0.15625 0.15624 0.15623 0.15624 0.15625

160× 8 0.16406 0.16406 0.16406 0.16406 0.16406

320× 16 0.16602 0.16601 0.16601 0.16601 0.16602

640× 32 0.16650 0.16650 0.16650 0.16650 0.16650




We discuss the preceding tables in conjunction with Table 3.21 and Table 3.24,

which show the axial flow with equal weights and with weights of 1, 1, 103, 103respectively, and Table 3.20, which shows the H1 error by variable in the unweighted

solution. With weights of 1, 1, 101, 101 mass is not well conserved except on the

640 × 32 and 1280 × 64 grids; see Table 3.89. Even with weights of 1, 1, 102, 102there is a lot of flow lost in the solutions on the less refined grids; see Table 3.91. The

errors in the solution obtained with weights of 1, 1, 101, 101 on the 80× 4 grid are

not much smaller than those obtained with equal weights; compare Table 3.88 with

Table 3.20. However the convergence rates with these weights are faster than they

are for the unweighted solution. Between the 640 × 32 and 1280 × 64 grids the rate

of convergence in H1 is almost h2 and furthermore is approximately the same in all

four variables. With weights of 1, 1, 102, 102 or 1, 1, 103, 103 the convergence

rates in H1 of the solution in U2 and U3 and U4 are around order h2; see Tables 3.90

and 3.92. The convergence rate in U4 is approximately order h, more so with weights

of 1, 1, 103, 103 than with weights of 1, 1, 102, 102. Also the convergence

rate in U1 in Table 3.90 is greater than that for the same variable in Table 3.92 even

though the errors are smaller in the latter. Raising the weights on the third and fourth

equations above 103 enforces mass conservation more strongly; see Table 3.94. In some

cases the accuracy of the solution improves, particularly in the velocity variables; see

Table 3.93. But in general as the weights are increased the approximation in U1 and

U2 tends to become less accurate as we also see from Table 3.93. The solution may

appear to be converging toward the analytical one between the coarser grids but as the

grid is refined further the error actually increases. We have observed that eventually

even the approximations in U3 and U4 diminish in quality and the solution process

frequently fails outright. As the order of magnitude of the weights on the third and

fourth equation is raised the pivot ratio increases; see Table 3.87. The pivot ratio is

also generally somewhat larger for matrices arising from finer grids. The pivot ratio is

an indication of the condition number of the linear system. It is the ill-conditioning and

consequent numerical instability of the system of linear equations to be solved which

leads to reduced accuracy in the solution.



Concerns over condition number aside it seems that the larger the weights on the

third and fourth equations, the better mass is conserved and the more precisely channel

flow is modelled. We infer that an accurate solution with no loss of mass can in principle

be obtained by letting these weights tend to infinity.

Equivalent to letting the weights on the third and fourth equations tend to infinity

is multiplying the contributions from the first and second equations in the least-squares

functional (3.1) by zero. In this case, the global assembled matrix is singular even after

rows and columns corresponding to fixed nodes have been eliminated.

We assemble the global stiffness matrix for an enclosed flow problem on [0, 1]2 with

the analytical solution (3.2) to (3.5). In this instance however we take as weights

w1 = 0, w2 = 0, w3 = 1, w4 = 1. Table 3.95 shows the total number of eigenvalues and

the number of trivial eigenvalues upon successive refinements of the mesh. Eigenvalues

are deemed trivial here if they are smaller than 10−8. The small proportion of zero

e-values encouraged us to believe that we could obtain a solution by devising a means

to solve the linear algebra system in the limiting case.

Grid Size Number of e-values Trivial e-values

4× 4 65 19

8× 8 257 67

16× 16 1025 259

Table 3.95: Variation in spectrum of null matrix with size of grid

We present here an algorithm by which we can obtain a solution of the linear system

of which this singular matrix forms a part.

Let wi, i = 1, . . . , 4 be the weight on the term corresponding to equation i in the

least-squares functional. Consider the assembled stiffness matrix from which the rows

and columns corresponding to the fixed nodes have been deleted, and the remaining



elements modified appropriately.

Let the matrix for which wi = 1, i = 1, . . . , 4 be denoted A and the corresponding

right-hand side be written r. The solution x of the least-squares linear algebra system

is found by minimising the expression

AT Ax− 2Ar + rT r.

This is equivalent to solving the equation

AT Ax = AT r.

The matrix AT A can be decomposed into the two component matrices

AT A = AT0 A0 + AT

1 A1

where AT0 A0 is equal to AT A in the case where the weights are w1 = 0, w2 = 0, w3 =

1, w4 = 1 and AT1 A1 is equal to AT A with weights w1 = 1, w2 = 1, w3 = 0, w4 = 0.

Similarly the right hand side AT r can be expressed in the form

AT r = AT0 r0 + AT

1 r1

where AT0 r0 is zero if the weights are w1 = 1, w2 = 1, w3 = 0, w4 = 0 and AT

1 r1 is

zero if the weights are w1 = 0, w2 = 0, w3 = 1, w4 = 1.

As AT0 A0 is singular, then provided that the equation

AT0 A0γ = AT

0 r0 (3.18)

has at least one solution γ for a given r0 it must have a continuum of solutions. As AT0 A0

is singular it will have a reduced range of dimension l and so a corresponding null space

of dimension n− l = m. Let γi, i = 1, . . . , m be a set of linearly independent n× 1

column vectors which span the null space of AT0 A0. Furthermore, let c be a column

vector of length m with elements ci, i = 1, . . . , m, c ∈ <m. Given an arbitrary

solution γ0 of equation (3.18), then any column vector γ which can be written as

γ = γ0 +m∑

i=1

ciγi



will also be a solution.

Our first aim is to obtain γ0. Now the square roots of the eigenvalues e1, e2, . . . , en

of a matrix are referred to as the singular values of the matrix. At this stage, we

introduce Λ, a diagonal n × n matrix for which the first l diagonal entries are equal

to the non-trivial singular values e1, 1, . . . , l of the matrix AT0 A0. The remaining

m elements on the diagonal and all the other elements of the matrix are zero. We

resort to singular value decomposition (SVD), which can be performed on any matrix.

A matrix is designated as normal if it commutes with its adjoint. In this context by

adjoint we mean the Hermitian conjugate, the transpose of the matrix A, such that Aij

is the complex conjugate of Aij . A normal matrix A can be written in the form

A = BΛB∗

where B is a unitary matrix and B∗ is its Hermitian conjugate, so that BB∗ = I. The

diagonal matrix Λ is of the same form as Λ. We say that Λ is similar to A. In particular

for a real symmetric matrix Asym

Asym = MΛMT .

In this case M ∈ <n×n and MT are orthogonal matrices so that M−1 = MT .

If Asym is non-singular then all of its eigenvalues are non-zero and we can write

down the relation

A−1sym = MT Λ−1M (3.19)

in which Λ−1 is another diagonal matrix with entries Λ−1ii .

If Λ has some diagonal elements which are zero then we can define a diagonal matrix

Λs according to the rules

Λsii =

1Λii

if Λii 6= 0,

Λsii = 0 if Λii = 0.

Replacing Λ with Λs in (3.19) gives us an expression for a matrix As which is a pseudo-

inverse of the matrix A. This matrix As is referred to as the singular value decompo-

sition of A.



We find the singular value decomposition As of the matrix AT0 A0, using an obvious

choice for the orthogonal matrix, specifically the matrix with columns which are the

eigenvectors of A20, indexed to the corresponding e-values. The column vector γ0 can

then be found since

γ0 = AsA0r0.

We use Γ to symbolise the n×m matrix the columns of which consist of the null space

vectors γi, i = 1, . . . , m. We have that

Γc =m∑

i=1

ciγi. (3.20)

The appropriate coefficients for the expansion (3.20) are given by minimising

(A1x− r1)2 =

(A1

(γ0 +

m∑

i=1

ciγi

)− r1

)2

=

(A1

(m∑

i=1

ciγi

)+ A1γ0 − r1

)2

= (A1γc + A1γ0 − r1)2 .

So c can be found by solving the linear system

ΓT AT1 A1Γc = ΓT AT

1 (−A1γ0 + r1).

3.9 Solutions of the S Formulation by the Null Space

Method

We have obtained solutions of the S formulation by the null space method with weights

of 0, 0, 1, 1. Specifically we obtain the solution for Poiseuille flow in the long channel

[0, 20]× [0, 1]. Our mesh consists of linear triangles in a Union Jack configuration. We

recall that the analytical solution is

U1 = ν(x2 + y2 − 2x− y

),

U2 = ν (2xy − 2y + 1− x) ,

3.9. Solutions of the S Formulation by the Null Space Method


U3 = 0,

U4 = y(1− y).

3.9.1 Results obtained with Enclosed Flow Boundary Conditions

In enforcing enclosed flow boundary conditions we fix U3 and U4 all around the bound-

ary and also fix U1 and U2 at (0, 0) and U2 at (20, 1).

Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1, 2

40× 2 165.66558 50.25000 36.62877

80× 4 55.27534 16.75367 12.31023

160× 8 15.10013 4.57649 3.45591

Table 3.96: Global errors

We find that axial flow is preserved without loss and there are no errors at the

nodes in the approximations to the velocity variables. Therefore the errors in the

approximations to the velocities are just the interpolation errors. The error reduces by

a factor of about 3.7 in L2 and L∞ between the 80× 4 grid and the 160× 8 grid. The

error in the H1 semi-norm reduces by a factor of 3.6; see Table 3.96. We can conclude

that the convergence rates in these three metrics seem to approach h2. This apparent

rate of convergence may be due to the inaccuracy in the algorithm we use to calculate

the error in the H1 semi-norm, as defined in (3.6). For a given grid the errors are

considerably smaller than the errors in the solution with weights of 1, 1, 103, 103;we refer to Tables 3.24 and Table 3.92.



3.9.2 Results obtained for Long Channel with Downstream Stress

Boundary Conditions

We recall that the downstream stress boundary conditions for this problem are

U3 = 0, U4 = y(1− y) on the line x = 0,

U3 = 0, U4 = 0 on the line y = 0,

U3 = 0, U4 = 0 on the line y = 1,

U2 = 0, U3 = 0 on the line x = L.

In addition, we fix U1 as zero at (0, 0).

Global Errors

nx × ny ‖ U − Uh ‖0,2 ‖ U − Uh ‖∞ | U − Uh |1, 2

40× 2 654.04772 200.25000 73.08671

80× 4 218.05921 66.75367 24.41294

160× 8 59.49087 18.21285 6.70831

Table 3.97: Global errors

The characteristics of the solution which satisfies these boundary conditions are

almost identical to those of the solution satisfying enclosed flow boundary conditions.

The approximation in the velocity variables is exact at the nodes and there is no loss

of axial flow in the approximation. Convergence rates are nearly the same; compare

Table 3.96 with Table 3.97. However the errors in the approximation in L2 and L∞

with these boundary conditions are around four times greater than they are in the

solution obtained by the same method with enclosed flow boundary conditions. The

errors are around twice as large in H1.

The errors are much smaller than those in the solution obtained with weights of

1, 1, 103, 103; see Table 3.29.



3.9.3 Commentary on the Null Matrix Least-Squares Finite Element

Method

Solutions by the null matrix least-squares finite element method exhibit ideal mass

conservation. They are very accurate. For the problem studied here, the value taken

at the nodes by the velocity field in the analytical solution is obtained exactly. However

determining the eigenvalues and eigenvectors of even a real, symmetric, banded matrix

is very expensive in computer time. Furthermore the process requires the inversion of a

full matrix which is even more costly. The improvements in accuracy of the solution do

not justify the huge amount of extra work required. A better solution can generally be

obtained by the conventional weighted method simply by refining the mesh just once

or by using higher order interpolation. However, this method is of theoretical interest.

It affirms that a solution to the S formulation with enclosed flow or downstream stress

boundary conditions in which mass is preserved perfectly on a given grid can in principle

be obtained by setting weights on appropriate terms to infinity.

3.10 Summary of Results for Planar Stokes Flow

We have used the least-squares finite element method to solve approximately three

different first-order formulations of the planar Stokes equations. With all equations

weighted equally, the solutions we obtained with these three formulations were poor

when we enforced enclosed flow boundary conditions, the natural ones for the Stokes

equations. Poor conservation of mass was a particular problem.

We used the three formulations to approximate Poiseuille flow in a convex polygonal

domain. In this instance solutions of the unweighted S formulation are less accurate

than solutions of the unweighted J formulation. The difference is more pronounced

on a square grid, where the loss of mass which is displayed in the solution of both

formulations is less acute. Solutions of the unweighted J formulation are in turn less

accurate than those solutions obtained of the unweighted G formulation on a given

3.10. Summary of Results for Planar Stokes Flow


grid, although when considering the value of this latter approach, it must be borne

in mind that more variables and equations are involved, and therefore computational

storage requirements and program execution time are much greater.

We showed that the quality of the solution improved when certain terms in each

system were weighted. In particular mass is conserved much better when the mass

conservation term is weighted with a very large factor.

The results obtained with the S formulation in concave regions were generally su-

perior to those obtained with the other two formulations, particularly in comparison

with the solutions obtained by the G formulation.

We have demonstrated that the S formulation does give results which are more accu-

rate when particular forms of boundary conditions are specified. The convergence rates

in the solution of the unweighted functional improve and there is complete mass conser-

vation in the approximation. These boundary conditions are not however compatible

with canonical ones for the primitive Stokes system.

We have observed that the solution obtained by minimising the unweighted S func-

tional does not converge in domains which are not simply connected. We consider a

function ~f = (f1, f2) and let S be a finite region of the plane bounded by a curve C

with the unit tangent t . We note that Stokes’ theorem for planar functions∫

S∇× ~f dS =

∮

C

~f.t ds (3.21)

cannot be applied to functions defined in such regions; see [118]. We recall that the

variables of the S formulation are defined as the components of the gradients of the

stress and stream functions φ and ψ; see (2.13). Then by definition∇×∇φ = 0, which is

equivalent to equation (2.16) and∇×∇ψ = 0, from which we obtain equation (2.17). In

simply connected domains equation (3.21) holds with ~f replaced by∇φ or∇ψ. However

in multiply connected domains we cannot invoke (3.21) so that the line integrals of ∇φ

and ∇ψ may not be independent of path. Hence φ and ψ may not be single-valued.

3.10. Summary of Results for Planar Stokes Flow

Chapter 4

A First-Order Reformulation of

the Navier-Stokes Equations for

Steady Flow Using Stress and

Stream Functions

4.1 The Planar Navier-Stokes Equation

The Navier-Stokes equation system for an incompressible fluid in steady flow is

− 1Re∇2~u + ~u.∇~u +∇p = ~f in Ω, (4.1)

∇.~u = 0 in Ω. (4.2)

The boundary conditions are the same as those for the Stokes equations, namely

~u = g on Γ,∫

Ωp dΩ = 0.

129

Chapter 4. A First-Order Reformulation of the Navier-Stokes Equations for Steady


The quantity Re is the Reynolds number, which we define as being the inverse of the

viscosity parameter ν. Its value is critical in influencing the character of Navier-Stokes

flow. This flow is non-linear because of the presence of the term ~u.∇~u in (4.1), which

is the advection or convection term. For a fluid of velocity ~u = (u1, u2) in a Cartesian

coordinate frame with axes x and y

∇~u =

∂u1

∂x

∂u2

∂x

∂u1

∂y

∂u2

∂y

. (4.3)

Then (4.1) can be written explicitly as

− 1Re

(∂2u1

∂x2+

∂2u1

∂y2

)+

∂p

∂x+ u1

∂u1

∂x+ u2

∂u1

∂y= fx, (4.4)

− 1Re

(∂2u2

∂x2+

∂2u2

∂y2

)+

∂p

∂y+ u1

∂u2

∂x+ u2

∂u2

∂y= fy. (4.5)

We wish to develop a first-order formulation of the non-linear system (4.1) and (4.2).

To this end we introduce R, the Reynolds stress tensor [120], defined as

R =

u2

1 u1u2

u1u2 u22

. (4.6)

This matrix has a divergence with two components

∇.R =

2u1∂u1

∂x+ u2

∂u1

∂y+ u1

∂u2

∂y

2u2∂u2

∂y+ u1

∂u2

∂x+ u2

∂u1

∂x

.

When equation (4.2) holds this can be simplified to

∇.R =

u1∂u1

∂x+ u2

∂u1

∂y

u1∂u2

∂x+ u2

∂u2

∂y

.

The equation (2.10) can be modified so that it takes the form

σR = −pI + 2νd−R. (4.7)

Given ~f = 0 then by conservation of momentum

∇.σR = 0.

4.1. The Planar Navier-Stokes Equation



In this case by taking the divergence of (4.7) then we arrive at the Navier-Stokes set

of equations (4.1) and (4.2).

By introducing a stress function φ as defined by (2.9), we can write (4.7) as the

system

φyy = −p + 2ν∂u1

∂x− u2

1,

−φxy = ν

(∂u1

∂y+

∂u2

∂x

)− u1u2,

φxx = −p + 2ν∂v

∂y− u2

2.

Substituting for the velocities as in the definitions of the stream function (2.6) and

(2.7), we get

φyy = −p + 2νψxy − ψ2y ,

−φxy = ν(ψyy − ψxx) + ψxψy,

φxx = −p− 2νψxy − ψ2x.

We eliminate the pressure p and make the substitutions

U1 = φx, U2 = φy, U3 = ψx, U4 = ψy

to obtain

−∂U1

∂x+

∂U2

∂y= 4ν

∂U3

∂y+ U2

3 − U24 ,

−∂U1

∂y− ∂U2

∂x= 2ν

(∂U4

∂y− ∂U3

∂x

)+ 2U3U4.

So the Navier-Stokes equations (4.1) and (4.2) can be written in first-order form as

−∂U1

∂x+

∂U2

∂y− 2ν

∂U3

∂y− 2ν

∂U4

∂x− U2

3 + U24 = f1, (4.8)

∂U1

∂y+

∂U2

∂x− 2ν

∂U3

∂x+ 2ν

∂U4

∂y+ 2U3U4 = f2, (4.9)

∂U1

∂y− ∂U2

∂x= f3, (4.10)

2ν∂U3

∂y− 2ν

∂U4

∂x= f4. (4.11)




Like (4.1), equations (4.8) and (4.9) contain non-linear terms, specifically U23 , U2

4 and

U3U4. The least-squares formulation of the weighted residual finite element method is

difficult to implement with non-linear equations. In particular, a linear set of equations

for the unknown nodal values is far easier to solve than a corresponding non-linear one,

with a much greater range of computational techniques available. Before moving on to

develop a least-squares functional for the set of equations (4.8) to (4.11), we will first

linearise them.

A number of linearisation techniques have been employed in the finite element lit-

erature, as highlighted by Jiang in [81]. The one we shall use is Newton’s linearisation

method; see for example [17], [22], [81] and [85]. This is an iterative technique, with

an updated solution [U ]n to be obtained from an estimate or previous iterate [U ]n−1.

Applying Newton’s linearisation method to the system (4.8) to (4.11) we obtain the

equations[−∂U1

∂x+

∂U2

∂y− 2ν

∂U3

∂y− 2ν

∂U4

∂x

]

n

− 2[U3]n[U3]n−1 + 2[U4]n[U4]n−1

= f1 − [U3]n−1[U3]n−1 + [U4]n−1[U4]n−1

≡ f∗1 , (4.12)[∂U1

∂y+

∂U2

∂x− 2ν

∂U3

∂x+ 2ν

∂U4

∂y

]

n

+ 2[U3]n[U4]n−1 + 2[U3]n−1[U4]n

= f2 + 2[U3]n−1[U4]n−1

≡ f∗2 , (4.13)[∂U1

∂y− ∂U2

∂x

]

n

= f3 ≡ f∗3 , (4.14)[2ν

∂U3

∂y− 2ν

∂U4

∂x

]

n

= f4 ≡ f∗4 . (4.15)

We write this system of equations as L∗U = F ∗, with L∗ a linear operator, UT =

(U1, U2, U3, U4) and F ∗T = (f∗1 , f∗2 , f∗3 , f∗4 ). The corresponding least-squares func-

tional is ∫

Ω(L∗U − F ∗)2 dΩ (4.16)

with Ω the domain on which the equations (4.12) to (4.15) hold. We introduce a test-

space V ⊂ [H1 (Ω)]4 with elements V such that V T = (V1, V2, V3, V4). The elements of




V are zero on the boundary Γ of Ω. We seek the function U from an appropriate trial

space U ⊂ [H1 (Ω)]4 such that the functional in (4.16) is a minimum. The elements of

the trial-space satisfy the boundary conditions for the system (4.12) to (4.15). Given

that the function U minimises the functional in (4.16) then

limt→0

d∫Ω(L∗U + tL∗V − F ∗)2 dΩ

dt= 0 ∀ V ∈ V

and therefore ∫

ΩL∗UL∗V dΩ =

∫

ΩL∗V F ∗ dΩ ∀ V ∈ V ; (4.17)

see Theorem 1.1. In obtaining finite element solutions we work with a finite dimensional

subset Uh of the trial-space U and a finite dimensional subset Vh of the test-space V .

The finite element solution Uh satisfies the relation∫

ΩL∗UhL∗Vh dΩ =

∫

ΩL∗VhF ∗ dΩ ∀ Vh ∈ Vh.

We present solutions of the set of equations (4.12) to (4.15). We call these the equations

of the SN formulation. As a particular finite dimensional space, we chose the set of

piecewise continuous linear functions defined on a triangulation of the region Ω as

introduced in an earlier chapter. Local and global stiffness matrices and right-hand side

vectors can be generated and assembled in the usual way to give a linear system for the

unknown nodal values. The linear systems arising from the finite element solution of

this system at each iteration are symmetric and positive-definite. The Jacobian matrix

is symmetric and positive-definite.

4.2 The Planar Navier-Stokes Equations in Terms of Ve-

locity, Vorticity and Pressure

The velocity-vorticity-pressure equations can be extended to incorporate the non-linear

term ~u.∇~u which appears in (4.1). The equations are

ν∂ω

∂y+

∂p

∂x+ u1

∂u1

∂x+ u2

∂u1

∂y= fx,

4.2. The Planar Navier-Stokes Equations in Terms of Velocity, Vorticity and Pressure



−ν∂ω

∂x+

∂p

∂y+ u1

∂u2

∂x+ u2

∂u2

∂y= fy,

ω − ∂u2

∂x+

∂u1

∂y= 0,

∂u1

∂x+

∂u2

∂y= 0.

Solutions of a backward facing step problem using this formulation can be found in [79]

and the driven cavity problem is solved in [79], [81], [84] and [85]. Further results on

driven cavity flow using this formulation can be found in [47], where non-Newtonian

flows are studied as well. Other solutions of this system for driven cavity flow and flow

over an obstacle which also incorporate time-dependence can be found in [124] and

[125]. Work discussed in [134] concerns time-dependent two-fluid flow. Consideration

is given to thermal effects in [125], [126], [127] and [143]. Results obtained using an

extension of this formulation for compressible flow are presented in [143].

4.3 The Planar Navier-Stokes Equations in Terms of Ve-

locity, Vorticity and Head

The Navier-Stokes equations for incompressible flow with body forces f = (fx, fy) can

be rewritten as a first-order system in terms of the velocities ~u = (u1, u2), the vorticity

ω and the pressure head b = p + 12

(u2

1 + u22

)as

ν∂ω

∂y− u2ω +

∂b

∂x= fx,

−ν∂ω

∂x+ u1ω +

∂b

∂y= fy,

ω − ∂u2

∂x+

∂u1

∂y= 0,

∂u1

∂x+

∂u2

∂y= 0;

see [22].

We make the substitutions

U1 = u1, U2 = u2, U3 = b, U4 = ω

4.3. The Planar Navier-Stokes Equations in Terms of Velocity, Vorticity and Head



so that the system can be written as

∂U3

∂x+ ν

∂U4

∂y− U2U4 = f1,

∂U3

∂y− ν

∂U4

∂x+ U1U4 = f2,

∂U1

∂y− ∂U2

∂x+ U4 = f3,

∂U1

∂x+

∂U2

∂y= f4.

Bochev [17] uses this formulation to model flow in a unit square. Comparisons of this

method with the velocity-vorticity-pressure formulation as ways of solving the driven

cavity problem are made in [84]. We shall refer to this system as the JN formulation.

A solution can be found by minimising the functional

JNν =

∫

Ω

((ν

∂U4

∂y+

∂U3

∂x− U2U4 − f1

)2

+(−ν

∂U4

∂x+

∂U3

∂y+ U1U4 − f2

)2

+ν2

(U4 +

∂U1

∂y− ∂U2

∂x− f3

)2

+ ν2

(∂U1

∂x+

∂U2

∂y− f4

)2)

dΩ; (4.18)

see [63].

4.4 Backward Facing Step

Our region is [−2, 0]× [−1, 0] ∪ [0, 6]× [−1, 1] as was illustrated in Figure 3.2.

4.4.1 Enclosed Flow Boundary Conditions in the SN Formulation for

Backward Facing Step Geometry

The boundary conditions are the same as for Stokes flow. We modify the boundary

conditions on the inlet so that the interpolations of the inflow and outflow balance. We

recall that on the inlet AB we have

U3 = 0, U4 = −qiny (1− y)




where Table 3.53 shows the appropriate value of qin for the grid parameter ny.

From the specification for the Stokes problem we also have on the outlet CD that

U3 = 0, U4 = 0.125(1− y2

).

On the walls BC, DE and AO the velocity variables are specified as

U3 = 0, U4 = 0.

The linear constraints are that U1 = 0 and U2 = 0 at the point B and that U2 = 0 at

the point D. The viscosity parameter ν is set equal to 10−2.

Axial Flow

ny x = −2 x = 0 x = 3 x = 6

4 0.16406 0.02087 0.05167 0.16406

8 0.16602 0.04304 0.07728 0.16602

16 0.16650 0.07471 0.10479 0.16650

32 0.16663 0.11071 0.13063 0.16663







Axial Flow

ny x = −2 x = 0 x = 3 x = 6

4 0.16406 0.15279 0.15755 0.16406

8 0.16602 0.16328 0.16437 0.16602

16 0.16650 0.16577 0.16605 0.16650

32 0.16663 0.16643 0.16650 0.16663







Much mass is lost when the equations are weighted equally; see Table 4.1. This

loss of mass is sharper than in the corresponding Stokes solution, especially on the

finer grids; compare Table 4.1 with Table 3.54. Table 4.2 shows that weighting the

equations significantly reduces the quantity of flow lost. Contrasting this table with

Table 3.55, which shows the flow in the Stokes solution obtained by the S formulation,

we see that although flow through any given line is less in the Navier-Stokes solution

than the Stokes solution, with weighting this is less apparent on the finer grids.

Figure 4.5: Velocity field with weights of 1, 1, 103, 103 at ny = 16 close to corner E




Figure 4.5 displays the velocity field close to the corner labelled E in Figure 3.2 in

the weighted solution at ny = 16. Recirculation can be seen clearly.

4.4.2 Downstream Stress Boundary Conditions in the SN Formulation

for Backward Facing Step Geometry

In this case the inlet conditions are

U3 = 0, U4 = y(y − 1)

and the fluid is of course fixed on the walls. The second partial derivative with respect

to y of the stress function in the non-linear case is

∂U2

∂y=

∂2φ

∂y2= σxx = −p + ν

∂u1

∂x− u2

1. (4.19)

From the outflow condition on the linear equations of the S formulation we have

−p + ν∂u1

∂x= 0.

But as in this case the outflow is of such a form that u1 = 0.125(1− y2

)then (4.19)

implies that on the outlet line x = 6

∂U2

∂y= − 1

64(1− y2

)2.

This gives a value for the trace of U2 on this abscissa of

U2 = − 1960

(15y − 10y3 + 3y5

)+ f(x).

We choose f(x) = 0, so that our downstream stress boundary conditions are

U2 = − 1960

(15y − 10y3 + 3y5

), U3 = 0.




Axial Flow

ny x = −2 x = 0 x = 3 x = 6

4 0.15625 0.02100 0.05749 0.14626

8 0.16406 0.04316 0.08215 0.15646

16 0.16602 0.07497 0.10866 0.16056

32 0.16650 0.11093 0.13264 0.16238


Axial Flow

ny x = −2 x = 0 x = 3 x = 6

4 0.15625 0.14537 0.14473 0.14517

8 0.16406 0.16097 0.16054 0.16058

16 0.16602 0.16515 0.16499 0.16499

32 0.16650 0.16626 0.16621 0.16621


As we would have expected from our observations of the linear solutions, a greater

proportion of flow is lost between any two lines with downstream stress boundary

conditions on the outlet than is the case with enclosed flow boundary conditions on the

outlet; compare Table 4.3 with Table 4.1 and Table 4.4 with Table 4.2.

We recall Table 3.57 and Table 3.59 which show the flow obtained in the linear case

with equal weighting and weighting respectively. Without weighting the proportion of

mass lost in the non-linear solution between the lines x = −2 and x = 0 is greater,

particularly on the more refined grids. This is consistent with the results for enclosed

flow boundary conditions shown in Table 4.1. However for values of x > 0 the flow

in the Navier-Stokes solution is much greater than that in the corresponding Stokes




solution. The boundary conditions on the outlet in the former case are inhomogeneous.

With weighting the flow is better conserved on the Stokes solution, but the difference

between the flow in the Stokes solution and the flow in the Navier-Stokes solution is

less great on the finer grids. Again this is the same pattern as we saw for enclosed flow

boundary conditions; see Table 4.2.








Figures 4.6, 4.7, 4.8 and 4.9 show plots of the velocity field. We see particularly in

the weighted solutions that there is a recirculation region close to the corner labelled

E in Figure 3.2.

Figure 4.10: Velocity field with weights of 1, 1, 103, 103 at ny = 16 close to corner

E

In Figure 4.10 we show the velocity field close to the corner E in the weighted

solution at ny = 16. This highlights the recirculation in that region.




4.4.3 Enclosed Flow Boundary Conditions in the JN Formulation for

Backward Facing Step Geometry

We specify the velocities on the inlet AB as

U1 = −qiny (1− y) , U2 = 0,

where the appropriate value for qin for a particular grid is given in Table 3.53. On the

outlet CD the boundary conditions are

U1 = 0.125(1− y2

), U2 = 0.

The velocity variables are specified as zero on the walls and the pressure is set equal

to zero at B. The viscosity parameter ν is 10−2.

Axial Flow

ny x = −2 x = 0 x = 3 x = 6

4 0.16406 0.00621 0.02820 0.16406

8 0.16602 0.01030 0.04292 0.16602

16 0.16650 0.01829 0.06245 0.16650


Figure 4.11: Velocity field for the enclosed flow solution of the JN formulation with

equal weights at ny = 8




Figure 4.12: Velocity field for the enclosed flow solution of the JN formulation with

equal weights at ny = 16

Table 4.5 shows the axial flow in the solution which minimises the functional (4.18)

with equal weights. There is convergence to the correct values as the grid is refined but

it is slow. The loss of mass on a given grid is substantially greater than in the solution

of the equivalent Stokes problem; see Table 3.60. It is also more pronounced than it is

in the solution of the SN formulation; see Table 4.1.

Axial Flow

ny x = −2 x = 0 x = 3 x = 6

4 0.16406 0.15781 0.15925 0.16406

8 0.16602 0.16049 0.16219 0.16602

16 0.16650 0.16213 0.16389 0.16650


Table 4.6 shows that most of the flow is conserved in the solutions which minimise

the functional (4.18) with a weight on the mass conservation term. However the loss of

mass is slightly greater than it is in the solution we found by minimising the weighted

J functional for the Stokes problem; see Table 3.62. On the finer grids there is more

mass lost in the solution of the weighted JN formulation than in the solution of the

weighted SN formulation.




Figure 4.13: Velocity field in the enclosed flow solution of the JN formulation with

weight of 103 on mass conservation term at ny = 16

Figure 4.14: Velocity field with weight of 103 on mass conservation term at ny = 16

close to corner E

Figure 4.13 is a quiver plot of the velocity field in the solution of the weighted JN

formulation at ny = 16. Figure 4.14 is a magnification of the velocity field close to the

corner E, showing recirculation. The recirculation effect is not as pronounced as it is

in the solution of the weighted SN formulation and does not stretch as far away from

the corner; see Figure 4.5.




4.4.4 Conclusion

Without weighting of equation terms we observe loss of flow which is even more acute

than that which we saw in the solution of the Stokes equations over the same region.

With weighting of appropriate terms, we seem to get a reasonably accurate solution

with either formulation, particularly on more refined grids. Some recirculation can be

seen.

4.5 Flow over a Semicylindrical Restriction

We model flow in the region illustrated in Figure 3.17 using the upper half of the grid

shown in Figure 3.14 and its refinements. The dimensions are the same as those for

the Stokes flow simulations. Fluid crosses the inlet line AF with velocity ~u = (ux, uy)

such that

ux = 0.16y (2.5− y) , uy = 0

and leaves across the outlet line DE with the same velocity. The fluid is stationary on

the walls AB, CD and EF as well as on the restriction itself. The viscosity parameter

ν is put equal to 10−2. In this case we found that the SN formulation required three

linear constraints. We set U1 = 0 at A and U1 and U2 as zero at E.

4.5. Flow over a Semicylindrical Restriction



4.5.1 Results in the SN formulation

Axial Flow

ng AF PQ YY” DE

1 0.40509 0.04938 0.12432 0.40509

2 0.41377 0.07924 0.14934 0.41377

4 0.41594 0.12013 0.18685 0.41594

8 0.41649 0.17901 0.23873 0.41649

16 0.41662 0.25224 0.29755 0.41662


Axial Flow

ng AF PQ YY” DE

1 0.40509 0.23617 0.28888 0.40509

2 0.41377 0.33318 0.35888 0.41377

4 0.41594 0.38965 0.39807 0.41594

8 0.41649 0.41003 0.41212 0.41649

16 0.41662 0.41515 0.41563 0.41662


The flow in the solution obtained with the unweighted SN formulation is in general

substantially less than that obtained with the unweighted S formulation, given the

same boundary conditions, geometry and grid; compare Table 3.83 with Table 4.7.




Figure 4.15: Velocity field in solution with equal weights at ng = 4

Figure 4.16: Velocity field in solution with weights of 1, 1, 103, 103 at ng = 4

The flow fields at ng = 4 are illustrated in Figures 4.15 and 4.16. At this level of

refinement there is no separation or recirculation visible even in the weighted solution.

There is some recirculation in the weighted solution at ng = 8 and ng = 16 in the

region to the immediate right of the cylinder, close to the corner labelled C in Figure

3.17. An enlargement of the solution on the grid ng = 8 in this portion of the region is

shown as Figure 4.17.

Figure 4.17: Recirculation in solution with weights of 1, 1, 103, 103 at ng = 8




4.6 Navier-Stokes Flow around a Cylindrical Obstruction

We present results for Navier-Stokes flow in the region shown in Figure 3.12. Fluid

in touch with the cylinder is at rest and that on the edges of the region has velocity

(ux, uy) = (1, 0). The dimensions are as for the Stokes flow problem. Our grids are

shown in Figures 3.14, 3.15 and 3.16. We set ν = 10−2.

4.6.1 Navier-Stokes Flow around a Cylindrical Obstruction Modelled

Using the SN Formulation

On the lines AB, BC, CD and AD we enforce the boundary conditions

U3 = 0, U4 = 1.

On the surface of the cylinder

U3 = 0, U4 = 0.

The linear constraints are that U1 = 0 and U2 = 0 at A and U2 = 0 at C.

Axial Flow


1 5.00000 0.98131 0.98131 4.29620 5.00000

2 5.00000 0.97312 0.97312 4.21577 5.00000

4 5.00000 0.95543 0.95543 4.10787 5.00000

8 5.00000 0.96602 0.96602 3.95059 5.00000


4.6. Navier-Stokes Flow around a Cylindrical Obstruction



Axial Flow


1 5.00000 1.08487 1.08487 3.36882 5.00000

2 5.00000 1.26699 1.26699 3.47156 5.00000

4 5.00000 1.63083 1.63083 3.85807 5.00000

8 5.00000 1.99812 1.99812 4.27270 5.00000


1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

u x

unweightedweighted

Figure 4.18: Plot of ux on the line PQ at ng = 1




1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

y

u x

unweightedweighted


This is a multiply connected region and Table 4.9 shows that the axial flow in the

unweighted solution does not seem to be converging to the correct value as the grid is

refined. The unweighted solution in U4 along PQ is shown in Figures 4.18 and 4.19. It

is not in this case a linear interpolant, as it was in the Stokes solution; see Figures 3.20

and 3.21. The flows are smaller in magnitude on the finer grids than on the coarser

ones, perhaps suggesting that the flow profile tends towards a linear interpolant on

lines of constant x as the grid is refined further.

The flow in the weighted solution appears to be converging to the correct value; see

Table 4.10. Nevertheless convergence is slow and the loss of mass on the grids shown

here is substantial. For a given grid the flow is much less than that in the solution of

the weighted S formulation; see Table 3.80. The flow through the line YY” is actually

greater in the unweighted solution of the SN formulation than in the weighted solution

except at ng = 8.




Figure 4.20: Velocity field in the solution of the SN formulation with equal weights at

ng = 4

Figure 4.21: Velocity field in the solution of the SN formulation with weights of

1, 1, 103, 103 at ng = 4

Figures 4.20 and 4.21 show the velocity fields at ng = 4 in the unweighted and

weighted solutions respectively. There seems to be some indication of separation both

to the right and to the left of the cylinder but there no recirculation is visible.




4.6.2 Uniform Navier-Stokes Flow around a Cylindrical Obstruction

Modelled Using the JN Formulation

Our unweighted functional is (4.18). On the lines AB, BC, CD and AD

U1 = 1, U2 = 0,

whilst on the cylinder

U1 = 0, U2 = 0.

The pressure is fixed at the midpoint of the line AD.

Axial Flow


1 5.00000 0.81811 0.81811 4.15073 5.00000

2 5.00000 0.80847 0.80847 4.06262 5.00000

4 5.00000 0.79502 0.79502 3.79216 5.00000

8 5.00000 0.81694 0.81694 3.37767 5.00000


Axial Flow


1 5.00000 2.23339 2.23339 4.65555 5.00000

2 5.00000 2.40335 2.40335 4.87151 5.00000

4 5.00000 2.43199 2.43199 4.90900 5.00000

8 5.00000 2.43929 2.43929 4.91874 5.00000





1 1.5 2 2.50

0.5

1

1.5

2

2.5

y

u x

unweightedweighted


1 1.5 2 2.50

0.5

1

1.5

2

2.5

y

u x

unweightedweighted





Figure 4.24: Velocity field in the solution of the JN formulation with equal weights at

ng = 4

Figure 4.25: Velocity field in the solution of the JN formulation with weight of 103 on

mass conservation term at ng = 4

The flow in the solution of the unweighted JN formulation is not converging to the

correct value as the grid is refined; see Table 4.11 and Figure 4.24. With weighting the

flow is better preserved; see Table 4.25 and Figure 4.25. Plots of ux on the line PQ

are shown in Figures 4.22 and 4.23. The weighted solution on this line appears closer

to the solution of the Stokes problem obtained with the S formulation than it does to

the solution of the Navier-Stokes problem obtained with the SN formulation. We draw

attention to the similarities of Figures 4.22 and 4.23 to Figures 3.20 and 3.21, which

plot the axial velocity on this line in the solutions of the S formulation. We contrast

them with Figures 4.18 and 4.19, which are the plots of the axial velocity on this line




in the solutions of the SN formulation. The solutions shown in Figures 4.24 and 4.25

also look more like the solutions of the Stokes problem by the S formulation shown in

Figures 3.18 and 3.19 than the solutions of the SN formulation shown in Figures 4.20

and 4.21. In particular we see little evidence of recirculation or separation close to the

cylinder.

4.7 Conclusion

We have obtained solutions of the SN formulation for a number of problems on various

geometries using enclosed flow boundary conditions and downstream stress boundary

conditions. As with the Stokes solutions of the S, J and G formulations we see that

a large proportion of the mass is lost in the solutions found by minimising unweighted

functionals. The loss of mass is more pronounced than it is in the simulations of Stokes

flow. Much less mass is lost with the weighting of appropriate terms.

Our results suggest that the unweighted SN formulation fails in multiply connected

regions. We recall that the unweighted S formulation also failed in a multiply connected

region. The solution of the unweighted JN formulation used here also does not converge

to the correct form for our model problem in a multiply connected region. The solution

of the unweighted J formulation for the equivalent Stokes problem does converge to

the correct form.

4.7. Conclusion

Chapter 5

First-Order Reformulations of

the Stokes and Navier-Stokes

Equations in Three Dimensions

5.1 Vector Calculus in Three Dimensions

The result of the divergence operator ∇. acting on a column vector ~v = (v1, v2, v3)T

is defined as

∇.~v =∂v1

∂x+

∂v2

∂y+

∂v3

∂z

which gives a single scalar value. The divergence operator acts on a second rank tensor

columnwise so that given

V =

V11 V12 V13

V21 V22 V23

V31 V32 V33

then the divergence of V is the vector

∇.V =(

∂V11

∂x+

∂V12

∂y+

∂V13

∂z

∂V21

∂x+

∂V22

∂y+

∂V23

∂z

∂V31

∂x+

∂V32

∂y+

∂V33

∂z

)T

.

157

Chapter 5. First-Order Reformulations of the Stokes and Navier-Stokes Equations in

Three Dimensions 158

5.2 The Stokes and Navier-Stokes Equations in a Three-

Dimensional Cartesian Coordinate System

The velocity field for flow in three dimensions with a Cartesian coordinate system can

be written as

~u (x, y, z) = (u1 (x, y, z) , u2 (x, y, z) , u3 (x, y, z))T .

In terms of ~u and pressure p(x, y, z) the Stokes equations are

−ν∇2~u +∇p = ~f(x, y, z), (5.1)

∇.~u = r(x, y, z) (5.2)

where ~f = (fx, fy, fz). Further the Navier-Stokes equations are

−ν∇2~u + ~u.∇~u +∇p = ~f,

∇.~u = g.

5.3 A Reformulation of the Stokes Equations in Three

Dimensions in Terms of Stress Functions and Veloci-

ties

We set ~f(x, y, z) = 0. For incompressible flow, so that the density of the fluid ρ is a

constant with respect to space and time, conservation of momentum and conservation

of mass respectively give us

∇.σ = 0, (5.3)

∇.~u = 0. (5.4)

5.2. The Stokes and Navier-Stokes Equations in a Three-Dimensional Cartesian

Coordinate System



The three-dimensional stress tensor can be expressed in the form

σ =

σxx σxy σxz

σxy σyy σyz

σxz σyz σzz

.

Cassidy [48] shows how stress and stream functions can be used to rephrase the Stokes

equations holding in cylindrical domains with axial symmetry. The velocity can be

expressed in terms of a single stream function, but recasting the stress tensor requires

two stress functions. He shows that the representation of the stress tensor using these

new functions is not unique, giving four examples.

We can express the three dimensional Cartesian stress tensor σ in terms of stress

functions so that for all choices of these functions the divergence of σ is identically zero

and (5.3) is satisfied.

Two such ways of writing down the stress tensor are named after Maxwell [101] and

Morera [122]. These have been used to some extent in solving problems in elasticity;

for examples see [14], [15], [93] and [132]. We also refer the interested reader to [16].

Maxwell introduced three stress functions which we shall label φ, ζ and ξ. We can

then write the stress tensor as

σ =

σxx σxy σxz

σxy σyy σyz

σxz σyz σzz

=

φyy + ζzz −φxy −ζxz

−φxy φxx + ξzz −ξyz

−ζxz −ξyz ζxx + ξyy

. (5.5)

Morera also used three stress functions but expressed the stress tensor as

σ =

ϕyz −12 (ϕzx + ηyz − χzz) −1

2 (ϕxy − ηyy + χyz)

−12 (ϕzx + ηyz − χzz) ηzx −1

2 (−ϕxx + ηxy + χxz)

−12 (ϕxy − ηyy + χyz) −1

2 (−ϕxx + ηxy + χxz) χxy

.

In fact both of these forms of the stress tensor are special cases of a more general one.

5.3. A Reformulation of the Stokes Equations in Three Dimensions in Terms of

Stress Functions and Velocities



This is given in [121] as

σ =

φyy + ζzz − 2ϕyz ϕzx + ηyz − χzz − φxy χyz + ϕxy − ηyy − ζzx

ϕzx + ηyz − χzz − φxy φxx + ξzz − 2ηzx ηxy + χxz − ϕxx − ξyz

χyz + ϕxy − ηyy − ζzx ηxy + χxz − ϕxx − ξyz ξyy + ζxx − 2χxy

.

We choose to express the stress tensor using the Maxwell stress functions.

We introduce the parameter Poisson’s ratio which we shall designate as RP . This

is the ratio of longitudinal to transverse displacement. In general

∇.~u = − (1− 2RP ) p;

see [31]. For incompressible materials

RP =12.

The three Maxwell stress functions can be shown to be related. In particular it is

established in [78] that

∂2

∂y2

(Φ− (1 + RP )∇2φ

)+

∂2

∂z2

(Φ− (1 + RP )∇2ζ

)= 0,

∂2

∂z2

(Φ− (1 + RP )∇2ξ

)+

∂2

∂x2

(Φ− (1 + RP )∇2φ

)= 0,

∂2

∂x2

(Φ− (1 + RP )∇2ζ

)+

∂2

∂y2

(Φ− (1 + RP )∇2ξ

)= 0,

∂2

∂y∂z

(Φ− (1 + RP )∇2ξ

)= 0,

∂2

∂x∂z

(Φ− (1 + RP )∇2ζ

)= 0,

∂2

∂x∂y

(Φ− (1 + RP )∇2φ

)= 0,

where

Φ = ∇2φ +∇2ζ +∇2ξ − ∂2ξ

∂x2− ∂2ζ

∂y2− ∂2φ

∂z2,

= trace σ.





It can be shown, as in [113], that the above equations imply that

∇2φ =trace σ

1 + RP, (5.6)

∇2ζ =trace σ

1 + RP, (5.7)

∇2ξ =trace σ

1 + RP. (5.8)

Eliminating the common right hand side then (5.6) to (5.8) give

∇2 (φ− ζ) = 0, (5.9)

∇2 (φ− ξ) = 0. (5.10)

By introducing two stream functions ψ and η the velocity can be written as

~u = (u1, u2, u3) = (ψz, ηz, −ψx − ηy)

and its divergence is identically zero. It is not however necessary for us to make use

of stream functions as the equations (5.3) and (5.4) can also be written as a first-

order system using the derivatives of the stress functions φ, ζ and ξ together with the

velocities themselves.

The stresses and the velocities are related by the tensor equation

σ = −pI + 2νd (5.11)

where p is the pressure and ν is the viscosity parameter, both of which are scalar

quantities, I is the 3× 3 identity matrix and d is the deformation tensor, which is

d =12

2∂u1

∂x

∂u1

∂y+

∂u2

∂x

∂u1

∂z+

∂u3

∂x

∂u1

∂y+

∂u2

∂x2∂u2

∂y

∂u2

∂z+

∂u3

∂y

∂u1

∂z+

∂u3

∂x

∂u2

∂z+

∂u3

∂y2∂u3

∂z

. (5.12)

Component by component, and using the Maxwell stress functions just presented, we

can write (5.11) as

φyy + ζzz = −p + 2ν∂u1

∂x(5.13)





−φxy = ν

(∂u1

∂y+

∂u2

∂x

), (5.14)

−ζxz = ν

(∂u1

∂z+

∂u3

∂x

), (5.15)

φxx + ξzz = −p + 2ν∂u2

∂y, (5.16)

−ξyz = ν

(∂u2

∂z+

∂u3

∂y

), (5.17)

ζxx + ξyy = −p + 2ν∂u3

∂z. (5.18)

By subtracting (5.18) from (5.13) and (5.18) from (5.16) we eliminate the pressure to

give us a system of five equations, which are

φyy + ζzz − ζxx − ξyy = 2ν

(∂u1

∂x− ∂u3

∂z

), (5.19)

−φxy = ν

(∂u1

∂y+

∂u2

∂x

), (5.20)

−ζxz = ν

(∂u1

∂z+

∂u3

∂x

), (5.21)

φxx + ξzz − ζxx − ξyy = 2ν

(∂u2

∂y− ∂u3

∂z

), (5.22)

−ξyz = ν

(∂u2

∂z+

∂u3

∂y

). (5.23)

We introduce as new variables the gradients of the stress functions

U1 =∂φ

∂x, U2 =

∂φ

∂y, U3 =

∂φ

∂z, (5.24)

U4 =∂ζ

∂x, U5 =

∂ζ

∂y, U6 =

∂ζ

∂z, (5.25)

U7 =∂ξ

∂x, U8 =

∂ξ

∂y, U9 =

∂ξ

∂z. (5.26)

Using the same notation scheme we signify the velocities by

U10 = u1, U11 = u2, U12 = u3. (5.27)

Applying these substitutions, the equations (5.19) to (5.23) become

∂U2

∂y− ∂U4

∂x+

∂U6

∂z− ∂U8

∂y− 2ν

∂U10

∂x+ 2ν

∂U12

∂z= f1, (5.28)

∂U1

∂y+

∂U2

∂x+ 2ν

∂U10

∂y+ 2ν

∂U11

∂x= f2, (5.29)





∂U4

∂z+

∂U6

∂x+ 2ν

∂U10

∂x+ 2ν

∂U12

∂z= f3, (5.30)

∂U1

∂x− ∂U4

∂x− ∂U8

∂y+

∂U9

∂z− 2ν

∂U11

∂y+ 2ν

∂U12

∂z= f4, (5.31)

∂U8

∂z+

∂U9

∂y+ 2ν

∂U11

∂z+ 2ν

∂U12

∂y= f5. (5.32)

The definitions (5.24) to (5.26) give the nine extra equations

∂U1

∂y− ∂U2

∂x= f6, (5.33)

∂U1

∂z− ∂U3

∂x= f7, (5.34)

∂U2

∂z− ∂U3

∂y= f8, (5.35)

∂U4

∂y− ∂U5

∂x= f9, (5.36)

∂U4

∂z− ∂U6

∂x= f10, (5.37)

∂U5

∂z− ∂U6

∂y= f11, (5.38)

∂U7

∂y− ∂U8

∂x= f12, (5.39)

∂U7

∂z− ∂U9

∂x= f13, (5.40)

∂U8

∂z− ∂U9

∂y= f14. (5.41)

The equations (5.9) and (5.10) are rewritten as

∂U1

∂x+

∂U2

∂y+

∂U3

∂z− ∂U4

∂x− ∂U5

∂y− ∂U6

∂z= f15,

∂U1

∂x+

∂U2

∂y+

∂U3

∂z− ∂U7

∂x− ∂U8

∂y− ∂U9

∂z= f16.

Mass conservation gives the additional relation

∂U10

∂x+

∂U11

∂y+

∂U12

∂z= f17. (5.42)

5.3.1 Boundary Conditions

The values taken by the functions on the boundary must be derived from suitable

boundary conditions for the Stokes equations, for instance

u× n = g3 on Γ, u.n = g4 on Γ





or

σ.n = g5 on Γ.

5.3.2 A Reformulation of the Navier-Stokes Equations in Three Di-

mensions in Terms of Stress Functions and Velocities

So as to develop a set of equations which describes non-linear flow in three dimensions

we modify the definition of the stress tensor (5.11), with the addition of the three-

dimensional symmetric Reynolds stress tensor R3

σ3R = −p + 2νd−R3.

Explicitly R3 is

R3 =

u21 u1u2 u1u3

u1u2 u22 u2u3

u1u3 u2u3 u23

.

If we express σ3R using the Maxwell stress functions defined by (5.5) and make the

substitutions (5.24) to (5.26) in place of the stress gradients this gives us a way to write

down the Navier-Stokes equations for three-dimensional flow as a first order system.

Making the substitutions in (5.27) for the velocities the equations (5.28) to (5.32) are

modified to become

∂U2

∂y− ∂U4

∂x+

∂U6

∂z− ∂U8

∂y− 2ν

∂U10

∂x+ 2ν

∂U12

∂z+ U2

10 − U212 = f1,

∂U1

∂y+

∂U2

∂x+ 2ν

∂U10

∂y+ 2ν

∂U11

∂x− 2U10U11 = f2,

∂U4

∂z+

∂U6

∂x+ 2ν

∂U10

∂x+ 2ν

∂U12

∂z− 2U10U12 = f3,

∂U1

∂x− ∂U4

∂x− ∂U8

∂y+

∂U9

∂z− 2ν

∂U11

∂y+ 2ν

∂U12

∂z+ U2

11 − U212 = f4,

∂U8

∂z+

∂U9

∂y+ 2ν

∂U11

∂z+ 2ν

∂U12

∂y− 2U11U12 = f5.

The other equations of the system are unchanged.





5.4 The Three-Dimensional Velocity-Vorticity-Pressure For-

mulation

The first-order reformulation of the Stokes equations utilising the velocities, vorticities

and pressure is set out and analysed in [52]. We refer also to [23]. In a three-dimensional

Cartesian coordinate system the vorticity ~ω = (ω1, ω2, ω3) is given by the curl of the

velocity ~u = (u1, u2, u3). Explicitly

ω1 =∂u3

∂y− ∂u2

∂z,

ω2 =∂u1

∂z− ∂u3

∂x,

ω3 =∂u2

∂x− ∂u1

∂y.

The velocity-vorticity-pressure system of equations for incompressible Stokes flow in

three dimensions is

∇× ~ω +∇p = f(x, y, z), (5.43)

∇.~ω = 0, (5.44)

~ω −∇× ~u = 0, (5.45)

∇.~u = 0. (5.46)

We introduce the notation

U1 = u1, (5.47)

U2 = u2, (5.48)

U3 = u3, (5.49)

U4 = ω1, (5.50)

U5 = ω2, (5.51)

U6 = ω3, (5.52)

U7 = p. (5.53)

Using (5.47) to (5.53) we can write the system as

−∂U5

∂z+

∂U6

∂y+

∂U7

∂x= f1,

5.4. The Three-Dimensional Velocity-Vorticity-Pressure Formulation



∂U4

∂z− ∂U6

∂x+

∂U7

∂y= f2,

−∂U4

∂y+

∂U5

∂x+

∂U7

∂z= f3,

∂U4

∂x+

∂U5

∂y+

∂U6

∂z= f4,

∂U2

∂z− ∂U3

∂y+ U4 = f5,

∂U1

∂z− ∂U3

∂x+ U5 = f6,

∂U1

∂y− ∂U2

∂x+ U6 = f7,

∂U1

∂x+

∂U2

∂y+

∂U3

∂z= f8.

This system consists of seven unknowns in eight equations. It is therefore not square.

In order to establish ellipticity of ADN type [2], it is necessary to introduce what

is called a slack variable. This is a variable which can be set as trivial everywhere

when obtaining actual solutions; see for instance [23], [52], [53] and [143]. The term is

borrowed from linear programming. We refer to [75], [76] and [116]. From [23] we have

that the equation (5.45) is modified by introducing a variable φ such that

~ω +∇φ−∇× ~u = 0. (5.54)

Taking the divergence of (5.54) and applying the identity (5.44) it can be seen that if

we set φ equal to zero on the boundary of our region it will vanish everywhere. Letting

φ = U8 equations can be written as

−∂U5

∂z+

∂U6

∂y+

∂U7

∂x= f1,

∂U4

∂z− ∂U6

∂x+

∂U7

∂y= f2,

−∂U4

∂y+

∂U5

∂x+

∂U7

∂z= f3,

∂U4

∂x+

∂U5

∂y+

∂U6

∂z= f4,

∂U2

∂z− ∂U3

∂y+ U4 +

∂U8

∂x= f5,

∂U1

∂z− ∂U3

∂x+ U5 +

∂U8

∂y= f6,

∂U1

∂y− ∂U2

∂x+ U6 +

∂U8

∂z= f7,




∂U1

∂x+

∂U2

∂y+

∂U3

∂z= f8.

Compatible boundary conditions for this system are given in [81]. If enclosed flow

conditions

~u.n = g1, ~u× n = ~g2 (5.55)

are specified all around the boundary then the pressure must also be fixed at a point.

For this set of boundary conditions the complementing condition of ADN theory is

not satisfied if the equation indices si all take the same value; see [17] and [23]. Ex-

periments we have carried out modelling channel flow through a cubic region using

trilinear interpolation on hexahedral elements confirm that flow is not well conserved

in solutions satisfying these boundary conditions unless the mass conservation term

(5.46) is weighted. On the other hand if the pressure p, tangential vorticities ~ω× n and

normal velocity ~u.n are specified on the boundary then the complementing condition

is satisfied even with equal equation indices; see [23]. There is no loss of flow in the

solutions we have obtained when enforcing these boundary conditions, even with equal

equation weighting. When p, the normal vorticity ~ω.n and the tangential velocities

~u× n are specified on the boundary the system also fails to satisfy the complementing

condition unless the equation indices are unequal; see [23]. This is different from the

case in two dimensions; see [25]. Our investigations suggest that nevertheless mass is

well conserved in solutions satisfying these boundary conditions even if all the equation

terms are weighted equally.

5.4.1 The Navier-Stokes Velocity-Vorticity-Pressure Formulation in

Three Dimensions

In [81] an equation system in terms of the velocity and vorticity and pressure is given

which incorporates the convection term ~u.∇~u. For incompressible flow this system is

ν∇× ~ω + ~u.∇~u +∇p = ~f,

∇.~ω = 0,




~ω −∇× ~u = 0,

∇.~u = 0.

Making the substitutions (5.47) to (5.53)

−∂U5

∂z+

∂U6

∂y+

∂U7

∂x+ U1

∂U1

∂x+ U2

∂U1

∂y+ U3

∂U1

∂z= f1,

∂U4

∂z− ∂U6

∂x+

∂U7

∂y+ U1

∂U2

∂x+ U2

∂U2

∂y+ U3

∂U2

∂z= f2,

∂U5

∂x− ∂U4

∂y+

∂U7

∂z+ U1

∂U3

∂x+ U2

∂U3

∂y+ U3

∂U3

∂z= f3,

∂U4

∂x+

∂U5

∂y+

∂U6

∂z= f4,

U4 +∂U2

∂z− ∂U3

∂y= f5,

U5 +∂U1

∂z− ∂U3

∂x= f6,

U6 +∂U1

∂y− ∂U2

∂x= f7,

∂U1

∂x+

∂U2

∂y+

∂U3

∂z= f8.

Boundary conditions which are appropriate for equations (5.43) to (5.46), the reformu-

lation of the Stokes equations in terms of the velocity, the vorticity and the pressure,

are also appropriate for this system; see [81]. Solutions to the driven cavity problem

obtained using a linearisation of this system are discussed in [84].

5.4.2 The Navier-Stokes Velocity-Vorticity-Head Formulation in Three

Dimensions

The pressure head in three dimensions is defined by the relation

b = p +12

(u2

1 + u22 + u2

3

).

The velocity - vorticity - head reformulation of the Navier-Stokes equations for incom-

pressible flow in three dimensions, as given in [22], [17] and [81], is

ν∇× ~ω + ~ω × ~u +∇b = ~f,




∇.~ω = 0,

~ω −∇× ~u = 0,

∇.~u = 0.

Adopting a Cartesian coordinate system and making the substitutions (5.47) to (5.52)

and U7 = b the full system is

−∂U5

∂z+

∂U6

∂y+

∂U7

∂x+ U3U5 − U2U6 = f1,

∂U4

∂z− ∂U6

∂x+

∂U7

∂y+ U1U6 − U3U4 = f2,

−∂U4

∂y+

∂U5

∂x+

∂U7

∂z+ U2U4 − U1U5 = f3,

∂U4

∂x+

∂U5

∂y+

∂U6

∂z= f4,

∂U2

∂z− ∂U3

∂y+ U4 = f5,

∂U1

∂z− ∂U3

∂x+ U5 = f6,

∂U1

∂y− ∂U2

∂x+ U6 = f7,

∂U1

∂x+

∂U2

∂y+

∂U3

∂z= f8.

The solution of this system is unique given any boundary conditions for which the

solution of the Stokes problem (5.43) to (5.46) is unique; see [81]. Bochev derives

convergence estimates for this system in [17], exploiting results drawn from [33] and

[71].

5.5 The Three-Dimensional Velocity-Velocity Gradient-

Pressure Formulation of the Stokes Equations

The equation system is

U−∇~uT = 0, (5.56)

−ν(∇.U)T +∇p = ~f, (5.57)

5.5. The Three-Dimensional Velocity-Velocity Gradient-Pressure Formulation of the

Stokes Equations



∇.~u = 0, (5.58)

∇×U = 0, (5.59)

∇( trace U) = 0, (5.60)

D−1 trace U = 0. (5.61)

The velocity gradient U is in this case a 3 × 3 tensor with components U = ∂ui∂xj

; see

[25]. We follow practice for planar systems by using the labels G1 for the system of

equations (5.56) to (5.58) and G2 for the system (5.56) to (5.60). The parameter D

which appears in (5.61) is the distance from any given element to the nearest vertex

of the region . We call the full set of equations (5.56) to (5.61) and the corresponding

least-squares functional G3.

We make the substitutions

U1 = u1, (5.62)

U2 = u2, (5.63)

U3 = u3, (5.64)

U4 = U11 =∂u1

∂x, (5.65)

U5 = U21 =∂u2

∂x, (5.66)

U6 = U31 =∂u3

∂x, (5.67)

U7 = U12 =∂u1

∂y, (5.68)

U8 = U22 =∂u2

∂y, (5.69)

U9 = U32 =∂u3

∂y, (5.70)

U10 = U13 =∂u1

∂z, (5.71)

U11 = U23 =∂u2

∂z, (5.72)

U12 = U33 =∂u3

∂z, (5.73)

U13 = p. (5.74)


Stokes Equations



In terms of these variables the G3 system is

−∂U1

∂x+ U4 = f1, (5.75)

−∂U2

∂x+ U5 = f2, (5.76)

−∂U3

∂x+ U6 = f3, (5.77)

−∂U1

∂y+ U7 = f4, (5.78)

−∂U2

∂y+ U8 = f5, (5.79)

−∂U3

∂y+ U9 = f6, (5.80)

−∂U1

∂z+ U10 = f7, (5.81)

−∂U2

∂z+ U11 = f8, (5.82)

−∂U3

∂z+ U12 = f9, (5.83)

−ν∂U4

∂x− ν

∂U7

∂y− ν

∂U10

∂z+

∂U13

∂x= f10, (5.84)

−ν∂U5

∂x− ν

∂U8

∂y− ν

∂U11

∂z+

∂U13

∂y= f11, (5.85)

−ν∂U6

∂x− ν

∂U9

∂y− ν

∂U12

∂z+

∂U13

∂z= f12, (5.86)

∂U1

∂x+

∂U2

∂y+

∂U3

∂z= f13, (5.87)

−∂U7

∂z+

∂U10

∂y= f14, (5.88)

∂U4

∂z− ∂U10

∂x= f15, (5.89)

−∂U4

∂y+

∂U7

∂x= f16, (5.90)

−∂U8

∂z+

∂U11

∂y= f17, (5.91)

∂U5

∂z− ∂U11

∂x= f18, (5.92)

−∂U5

∂y+

∂U8

∂x= f19, (5.93)

−∂U9

∂z+

∂U12

∂y= f20, (5.94)

∂U6

∂z− ∂U12

∂x= f21, (5.95)


Stokes Equations



−∂U6

∂y+

∂U9

∂x= f22, (5.96)

∂U4

∂x+

∂U8

∂x+

∂U12

∂x= f23, (5.97)

∂U4

∂y+

∂U8

∂y+

∂U12

∂y= f24, (5.98)

∂U4

∂z+

∂U8

∂z+

∂U12

∂z= f25, (5.99)

D−1U4 + D−1U8 + D−1U12 = f26. (5.100)

From [25] the enclosed flow boundary conditions on the bounding surface are that

~u = gb(x, y, z) (5.101)

together with the further conditions on the variables of the G2 and G3 formulations

that

U× n = Gb(x, y, z), (5.102)

where n is the outward normal. The pressure must also be fixed at a point in the region.

The least-squares functionals for the G1, G2 and G3 functionals are given in equations

(2.41), (2.43) and (2.46) respectively. The bounds on the estimates are given by the

inequalities (2.47) to (2.52). We have found that mass is not conserved in solutions of

equations (5.75) to (5.100) which satisfy the boundary conditions (5.101) and (5.102)

unless the mass conservation term (5.87) is weighted.

5.6 Navier-Stokes Equations in Velocity-Velocity Gradient-

Pressure Form

We let Ω ∈ <n, where n = 2 or n = 3, and define Γ as the boundary of Ω. In terms

of the variables of the G formulations in two or three dimensions the Navier-Stokes

equations for incompressible flow can be written as

−ν(∇.U)T + UT~u +∇p = ~f in Ω,

∇.~u = 0 in Ω,

U−∇~uT = 0 in Ω;

5.6. Navier-Stokes Equations in Velocity-Velocity Gradient-Pressure Form



see [20]. From [20] appropriate boundary conditions are

~u = 0 on Γ,∫

Ωp dΩ = 0.

Let

L20 (Ω) =

p ∈ L2 (Ω) |

∫

Ωp dΩ = 0

.

Solutions of this system can be found by minimising the functional

GN,−1 =| −(∇.U)T +∇p +1ν

(UT~u− f

) |2−1 + ‖ ∇.~u ‖20 + ‖ U−∇~uT ‖2

0

in the space

X =

(U, ~u, p) ∈ L2 (Ω)n2 ×H10 (Ω)n × L2

0 (Ω) | ~u = 0 on Γ

;

see [20], [21] and [18]. An augmented system is

−ν(∇.U)T + UT~u +∇p = ~f in Ω, (5.103)

∇.~u = 0 in Ω, (5.104)

U−∇~uT = 0 in Ω, (5.105)

∇ ( trace U) = 0 in Ω, (5.106)

∇×U = 0 in Ω. (5.107)

The boundary conditions for homogeneous enclosed flow for this system are

~u = 0 on Γ,∫

Ωp dΩ = 0,

U× n = 0 on Γ;

see [21]. We let ~f ∈ [L2(Ω)]n and (~u0, p0) be the solution of the Stokes problem

−∇2~u +∇p =1ν

~f in Ω,

∇.~u = 0 in Ω,

~u = 0 on Γ,∫

Ωp dΩ = 0.




Given that

U0 = ∇~uT0

then we can rewrite equation (5.103) as

− (∇.U)T +1ν

(U + U0)T (~u + ~u0) +∇p = 0

The solution of the system is the minimum of the functional

GN = ‖ − (∇.U)T +1ν

(U + U0)T (~u + ~u0) +∇p ‖2

0 + ‖ ∇.~u ‖20 + ‖ U−∇~uT ‖2

0

+ ‖ ∇ ( trace U) ‖20 + ‖ ∇ ×U ‖2

0

in the space

X =

(U, ~u, p) ∈ H1 (Ω)n2 ×H1 (Ω)n ×H1 (Ω) ∩ L20 (Ω) | ~u = 0, U× n = 0 on Γ

.


Chapter 6

Concluding Remarks

We have presented and discussed the solutions by least-squares methods of a number of

first-order formulations of the Stokes and Navier-Stokes systems. We have used for the

most part enclosed flow boundary conditions, which are canonical ones for the primitive

formulations. It emerges that with these boundary conditions, mass conservation is not

enforced very well. We also examined solutions of the S formulation satisfying what we

termed downstream stress conditions, which can be derived from appropriate boundary

conditions for the primitive Stokes formulation. Again we found a great deal of mass

was lost in the solutions. In general errors tend to be large. The observed convergence

rates sometimes agree with theoretical estimates. However convergence in most cases

seems to be a long way from optimal unless the grids used are highly refined.

Much less mass is lost with all the formulations when large weights are applied to

appropriate terms. It seems from our experience with linear elements that provided

these weights are not too large, the solutions can be of acceptable accuracy in all the

variables. In particular, the errors in the velocities reduce considerably and these are

usually the variables of most practical interest. Observed convergence rates seem in

general to be in line with theory and in many cases it appears that convergence rate are

close to optimal between solutions on the grids studied. Weighting particular terms in

175

Chapter 6. Concluding Remarks 176

the least-squares functional is a reasonable way of obtaining a reliable solution whilst

not increasing the complexity of the solution process or the time required to obtain a

solution.

We have shown that mass is conserved in solutions of the S formulation which

satisfy certain classes of boundary conditions. These are however inappropriate for most

problems in fluid mechanics, though in some circumstances these boundary conditions

may be equivalent to ones which are physically meaningful for the biharmonic problem.

We have written down a set of first-order equations in terms of the gradients of the

stream and stress functions which is equivalent to the planar Navier-Stokes equations.

It appears from our results that the solutions of this system are reasonable, again

provided appropriate terms are weighted. We have also set out how the Stokes and

Navier-Stokes systems in three dimensions may be reformulated as first-order systems

using the gradients of stress functions together with the velocities. It remains to be

determined what forms of boundary conditions should be specified so that problems in

terms of these reformulations are well-posed and how such boundary conditions can be

related to appropriate boundary conditions for the primitive second-order systems.

Appendix A

ADN Ellipticity Analysis of the

Stress and Stream System

The H1 ellipticity of (2.14) to (2.17) is established in [130] using the theory presented

in [133]. Here we examine the system of equations (2.14) to (2.17) and appropriate

accompanying boundary conditions using the methods of analysis presented by Agmon,

Douglis and Nirenberg in [1] and [2]. ADN ellipticity theory is more general than that

of [133] which is only appropriate for equations in two independent variables. Both

methods have been heavily exploited in the analysis of first-order systems. We refer to

[8], [17], [23], [24] and [51] for examples using ADN theory and to [51], [52], [136], [138]

and [139] which cite [133]. In particular, in [23] Bochev and Gunzburger investigate the

H1 coercivity of the velocity-vorticity-pressure reformulation of the Stokes equation,

which we have elsewhere referred to as the J formulation; see (2.24) to (2.27). Bochev

goes on to examine the two- and three-dimensional Stokes and Navier-Stokes equations

rephrased in terms of the same variables in [17]. Having established ellipticity and

hence certain coercivity estimates it can thence be shown that least-squares functionals

derived from these systems are equivalent or close to norms in the approximation spaces,

usually H1, so that least-squares finite element solutions should be convergent.

177

Appendix A. ADN Ellipticity Analysis of the Stress and Stream System 178

Further discussion of ADN ellipticity appears in [8], [99], [117] and [129]. In par-

ticular in [8] the application of the theory to least-squares finite element analysis is

discussed in some detail.

A.1 ADN Ellipticity Theory

A method for establishing the ellipticity of a single equation in the ADN sense is

presented in [1] and this is extended in [2] so that the ellipticity of a general equation

system defined over a region Ω can be investigated; see also [64]. We let this system

be given by

LU = F in Ω (A.1)

with accompanying boundary conditions on the boundary Γ of Ω. We may write these

as

BkU = fk for k = 1, . . . , m, (A.2)

which taken together give

BU = f. (A.3)

In the above, U is a vector of dependent variables or unknowns

U = (u1, u2, . . . , uN )T .

The number of independent variables is n+1. These are labelled (x1, x2, . . . , xn, xn+1)

or (x1, x2, . . . , xn, t), so Ω ⊂ <n+1 or Ω ⊂ <n × T . Finally, l consists of a system of

partial differential operators which are polynomials in the partial derivative operators

on the unknowns, taken with respect to the independent variables. Given the shorthand

notation

∂i =∂

∂xi

we can define a multi-index of differential operators in these variables

(∂β11 , ∂β2

2 , . . . , ∂βn+1

n+1 ). (A.4)

A.1. ADN Ellipticity Theory


The order of differentiation p of an operator with this vector is∑n+1

i=1 βi = p.

The terms in L may also have coefficients depending on the independent variables.

This dependence is expressed by use of the notation L(P ), where P is a point in Ω.

The components Lij of L, act on Uj . These components are sums of terms of the

form (A.4); the degree of Lij , denoted deg Lij , is the maximum order of any of these

individual terms in Lij . The definitions of the components Bij of the matrix operator

B on the boundary and of their respective degrees follow in an analogous way.

The authors of [2] assign an integer to each equation in the system. These are called

the equation indices. They use the notation s1, s2, . . . , si, . . . , sN to denote equation

indices for a set of N equations represented by the action of the operator L. They also

associate an integer with each dependent variable of the system. These are called the

unknown indices and are denoted t1, t2, . . . , tj , . . . , tN .

The degree of differentiation of a particular unknown in a given equation is bounded

by the sum of the corresponding equation and unknown indices.

deg Lij ≤ si + tj . (A.5)

By convention if a given equation contains no term in a particular unknown then the

corresponding degree is negative.

The choice of these indices is not unique but they are related. In [2] the set of

equation indices is normalised so that

si ≤ 0 for i = 1, . . . , N, (A.6)

0 ≤ tj for j = 1, . . . , N. (A.7)

A.1. ADN Ellipticity Theory


A.2 Conditions on an ADN Elliptic System of Equations

An operator L′ is defined, derived from L and called the principal part of L. Terms of

L only appear in L′ if they are of order si + tj . The system is elliptic in the classical

sense if the symbol L(P, θ), defined as

L(P, θ) = detL′(θ),

is non-zero for all non-zero θ ∈ <n+1. This is also a condition for a system to be

ADN elliptic, which we shall term Condition 1. A second condition that we call simply

Condition 2 is that the polynomial L in θ must also be of order 2m, where m is the

number of boundary conditions. Since m is an integer L will be of even order.

Let ~n be a vector which is orthogonal to θ. For instance θ might be a tangent

to the boundary and ~n a normal. Condition 3 for a system to be ADN elliptic is

that L(P, θ + τ~n) has m roots in τ with positive imaginary parts. The polynomial

L(P, θ + τ~n) for an elliptic operator will have 2m complex roots in τ consisting of m

conjugate pairs. We represent a general root with positive imaginary part as

τj = τjr + iτji, τjr ∈ <, τji ∈ <+,

where j = 1, 2, . . . , m. Making use of this notation the roots can be written as

(τ1r + τ1ii, τ2r + τ2ii, . . . , τjr + τjii, . . . , τmr + τmii) (A.8)

and

(τ1r − τ1ii, τ2r − τ2ii, . . . , τjr − τjii, . . . , τmr − τmii).

In the work to follow, and without loss of generality, we will choose the vector θ to

be θ, the unit vector tangent to the boundary at a point P [2]; in component form

θ = (θ1, θ2, . . . , θn+1). In a similar way, ~n will be chosen to be n = (n1, n2, . . . , nn+1),

the unit normal.

Taken together, the Conditions 2 and 3 are referred to in [2] as the Supplementary

Condition on L . The Supplementary Condition is satisfied by all elliptic systems for

which n > 1. Where n = 1 it must be verified on a case-by-case basis.

A.2. Conditions on an ADN Elliptic System of Equations


A.3 The Boundary Equations

Recalling the boundary conditions (A.3), each boundary equation is assigned an integer

label in the same way that the system equations are. These are called the boundary

indices and are labelled r1, r2, . . . , rk, . . . , rm. Using the same unknown indices as

above, terms in the boundary operator must satisfy the inequality

deg Bkj ≤ rk + tj . (A.9)

Also defined are a set of boundary equations B′k, the principal parts of the equations

Bk. The equation B′k is derived from Bj in the same way as l′ is derived from l. A

given term in Bk in an unknown j appears in B′k if and only if

deg Bkj = rk + tj . (A.10)

The boundary conditions and system of equations taken together must also satisfy the

Complementing Condition.

A.3.1 The Complementing Condition

Let Ljk denote the adjoint matrix to L′(P, θ + τn). We also introduce the polynomial

M+ with roots given by (A.8) as

M+ = (τ − τ1)(τ − τ2) . . . (τ − τm). (A.11)

Then the system satisfies the Complementing Condition if

m∑

h=1

ChB′hjL

jk(P, θ + τn) = 0 mod M+ (A.12)

which is to say that the rows of this matrix are linearly independent modM+.

Once a system of the form (A.1) and (A.2), with u = (u1, u2, . . . , uN )T , f =

(f1, f2, . . . , fN )T and g = (g1, g2, . . . , gm)T , is shown to be ADN elliptic, then the

A.3. The Boundary Equations


following inequality (see [2], Theorem 2.1 of [8] and [17]) will hold

N∑

j=1

‖ uj ‖l+tj≤ C

N∑

i=1

‖ fi ‖l−si+

m∑

i=1

| gk |l−rk− 12

+N∑

j=1

‖ uj ‖0

. (A.13)

Here C > 0 and l ≥ 0. In addition uj must fall in the space H l+tj (Ω), fi ∈ H l−si and

gk ∈ H l−rk− 12 . In cases where the solution of the system which satisfies the specified

boundary conditions is unique, the bound continues to hold when all of the L2 norms

of uj on the right-hand side are weighted by zero. Conditions on the boundary Γ are

given in [2] and [17]. Specifically, the boundary must be of class Cr+t; the variable r

is the greater of either the maximum value for rk + 1 or the parameter l whilst t is the

maximum value for tj . Relations of this form are termed Schauder estimates in [2].

REMARK

Both the methodology of [2] and that of [133] fell out of favour to an extent because

strictly they only apply to regions with smooth boundaries. This analysis is thus

inadequate for many of the actual problems solved using finite element methods. For

example, in the plane this restriction on the boundary even excludes convex polygons.

Nevertheless, as pointed out in [17], this analysis is still useful for regions where the

boundary does meet the necessary smoothness requirements. Other researchers have

developed alternative means to establish convergence. The coercivity of a number of

first-order systems is established in [80] and [81]. The approach used there makes use

of Poincare-Friedrichs inverse inequalities together with the bounded inverse theorem

to establish equivalence between norms and hence ellipticity. Theory relying on inverse

inequalities and the Lax-Milgram theorem [81] has also been developed by Manteuffel,

McCormick and others, for instance in [35], [36], [39], [41], [43], [91], [96], [114], [115],

[135] and [137]. The work in [131] exploits the trace inequality

‖ u ‖k, Ω≥ C ‖ u ‖k− 12, Γ, k = 0, 1 (A.14)

in which the trace norm ‖ u ‖k− 12, Γ is obtained from the trace norms of the NE

A.3. The Boundary Equations


individual elements with sides Γi, i = 1, . . . , NE

‖ u ‖k− 12,Γ=

√√√√NE∑

i=1

‖ u ‖k− 12,Γi

.

The inequality (A.14) holds in polygonal domains.

More recently interest has been rekindled in ADN theory; see for example [25], [60],

[63], [74], [92] and [140].

A.4 Illustrations

Two systems of equations are considered, the second-order Stokes planar formulation

in the primitive variables and the S formulation.

A.4.1 The Primitive Second-Order Stokes Equations

Various systems of equations are examined in [8]. One example is the system of steady-

state incompressible Stokes equations in two dimensions introduced earlier in the main

body of the thesis and which we express here as

−∇2U +∇p = f, (A.15)

∇.U = 0. (A.16)

The dependent variables are the velocity U = (u, v) and the pressure p.

Component by component (A.15) and (A.16) can be written in the form

−uxx − uyy + px = f1, (A.17)

−vxx − vyy + py = f2, (A.18)

ux + vy = f3. (A.19)

Here U = (u, v) and f = (f1, f2). We note that f3 is zero in the incompressible case,

but is allowed to be non-zero for the purposes of the analysis. The subscripts x and y

A.4. Illustrations


indicate partial differentiation with respect to the appropriate variable, with multiple

subscripts indicating multiple differentiation.

In this case N , the number of unknowns, is three. The number of equations is

also three. Hence the matrix corresponding to the system operator is square and the

analysis of [2] can be applied. Also we see that n = 1, since the number of independent

variables is defined as n + 1. Two boundary conditions (i.e. m = 2) are specified on

the boundary Γ of Ω; these are

B1U = u = g1, (A.20)

B2U = v = g2. (A.21)

In matrix form the boundary conditions are

BU = g,

where

B =

1 0 0

0 1 0

, (A.22)

and

gT = (g1, g2), UT = (u, v, p).

The equation and unknown indices given in [8] are

(s1, s2, s3) = (0, 0, −1),

(t1, t2, t3) = (2, 2, 1).

The accompanying boundary condition indices are

r = (−2, −2).

With these choices for the equation and unknown indices, the principal part of the

system (A.17) to (A.19) is identical to the system itself, as all terms are of the order

si + tj . Likewise, all terms appearing in the boundary equations (A.20) and (A.21) are

of order rh + tj , and so the principal part B′ of B is equal to B itself.

A.4. Illustrations


This gives an expression for the symbol L of the system

L(P, θ) = det

−(θ21 + θ2

2) 0 θ1

0 −(θ21 + θ2

2) θ2

θ1 θ2 0

, (A.23)

= (θ21 + θ2

2)θ22 + (θ2

1 + θ22)θ

21, (A.24)

= (θ21 + θ2

2)2, (A.25)

= θ4. (A.26)

This expression is non-zero for non-zero θ. It is also of degree four in θ, twice the

number of boundary conditions. Let τ be a parameter and n be a unit vector which is

orthogonal to the unit vector θ. In particular θ is a tangent to the boundary Γ and n a

normal. It can be seen that the Supplementary Condition is satisfied as L(P, θ + τ n)

has two pairs of conjugate roots. The polynomial M+ from (A.11) is given by

M+ = (τ − i) (τ − i) .

We also have to verify the Complementing Condition. We define

ξ = θ + τ n,

= (ξ1, ξ2).

We let L′(P, θ) refer to the matrix appearing in (A.23). Let Ljk(P, ξ) be the matrix

which is adjoint to L′(P, ξ). This is

Ljk =

−ξ22 ξ1ξ2 ξ1(ξ2

1 + ξ22)

ξ1ξ2 −ξ21 ξ2(ξ2

1 + ξ22)

ξ1(ξ21 + ξ2

2) ξ2(ξ21 + ξ2

2) (ξ21 + ξ2

2)2

. (A.27)

We note that both L′ and the matrix Ljk are symmetric.

Multiplying (A.27) and (A.22) as in (A.12) gives

B′hjL

jk =

−ξ2

2 ξ1ξ2 ξ1(ξ21 + ξ2

2)

ξ1ξ2 −ξ21 ξ2(ξ2

1 + ξ22)

. (A.28)

A.4. Illustrations


We adopt the reasoning followed in [52]. A row matrix formed from a non-trivial linear

combination of the two rows of the above matrix has two elements in the first two

columns which are quadratic in τ with real coefficients. These equations therefore will

have at most one root in τ with positive imaginary part; these elements cannot be non-

zero integer multiples of M+, which by its definition has two such roots. Explicitly,

we let a general element of matrix (A.28) be denoted by Fhk(τ). Then if a linear

combination of the rows of (A.28) add up to integer multiples nk of M+ we have

c11F11 + c21F21 = n1M+, (A.29)

c12F12 + c22F22 = n2M+. (A.30)

But as the left hand-side and right hand-side have unequal roots, n1 = n2 = 0.

The other column has elements which are cubic in τ , and a linear combination can

also not be a non-zero integer multiple of M+. We also see that these rows are linearly

independent, so that (n1, n2) 6= 0 for any non-trivial cij , i, j = 1, 2. Therefore the

Complementing Condition is satisfied. Thus the Stokes system (A.17) to (A.19) with

the boundary conditions (A.20) and (A.21) is ADN elliptic given an appropriate choice

of indices. Let V = (V1, V2, V3)T = (u, v, p)T . Then the Schauder estimate (A.13)

takes the form2∑

j=1

‖ Vj ‖l+2 + ‖ V3 ‖l+1≤ C

(2∑

i=1

‖ fi ‖l + ‖ f3 ‖l+1 +2∑

k=1

‖ gk ‖l− 52

);

see [8]. The estimate holds for l ≥ 0, provided that V ∈ [H l+2(Ω)]2 × H l+1(Ω), f ∈[H l(Ω)]2 ×H l+1(Ω) and gk ∈ [H l− 5

2 (Γ)]2; the boundary must be of class C l+2. So in

this case there are high continuity requirements on the estimate. This is reasonable, as

the initial equation system contains second-order terms.

A.4.2 The System of Equations of the S Formulation

We reproduce the equations (2.14) to (2.17) of the S formulation in dimensionless form

−∂U1

∂x+

∂U2

∂y− ∂U3

∂y− ∂U4

∂x= f1, (A.31)

A.4. Illustrations


∂U1

∂y+

∂U2

∂x− ∂U3

∂x+

∂U4

∂y= f2, (A.32)

∂U1

∂y− ∂U2

∂x= f3, (A.33)

∂U3

∂y− ∂U4

∂x= f4. (A.34)

This is a planar system and so n = 1. From [131] two boundary conditions are required

for this problem, so that m = 2.

An acceptable set of equation indices for this system according to the theory of [2]

is

(s1, s2, s3, s4) = (0, 0, 0, 0).

The accompanying unknown indices are

(t1, t2, t3, t4) = (1, 1, 1, 1).

This is verified as follows. As all terms in (A.31) to (A.34) are first-order derivatives

deg Lij ≤ 1 ∀ i, j = 1, 2, 3, 4.

In addition it is easily seen that

si + tj = 1 ∀ i, j = 1, 2, 3, 4.

Then

deg Lij ≤ si + tj (A.35)

and the system with these indices obeys the bound (A.5).

Furthermore, by (A.35), the corresponding principal part L′ of L is equal to L itself.

The symbol L of L′ is given by

L(P, θ) = det

−θ1 θ2 −θ2 −θ1

θ2 θ1 −θ1 θ2

θ2 −θ1 0 0

0 0 θ2 −θ1

,

= − (θ21 + θ2

2

)2,

= −θ4

A.4. Illustrations


This is non-zero for θ 6= 0 and is of order 2m = 4. In addition we have that

L(P, θ + τ n) = −((θ1 + τn1)2 + (θ2 + τn2)2)2,

= −(1 + τ2)2,

= − (τ − i)2 (τ + i)2 .

Hence the Supplementary Condition on L is satisfied and

M+ = (τ − i)2.

Now we consider the two boundary conditions (2.19) to (2.20). Firstly, consider the

case (2.19) where U1 and U2 are fixed with the other two unknowns free

U1 = b1, (A.36)

U2 = b2. (A.37)

As there are no derivatives appearing in this system

deg Bh, j = 0 for h = 1, j = 1,

deg Bh, j = 0 for h = 2, j = 2

with Bi, j being negative for all other combinations of h = 1, 2 and j = 1, 2, 3, 4.

A suitable choice of boundary equation indices is

(r1, r2) = (−1, −1). (A.38)

Working with the unknown indices previously established it is easily seen that

deg Bh, j ≤ rh + tj for h = 1, 2 and j = 1, 2, 3, 4.

In fact the principal part B′ of B is simply B itself. Consider the Complementing

Condition(A.12). We again make use of the notation

ξi = θi + τ ni.

A.4. Illustrations


Then the matrix adjoint to L′ is given by

Ljk(P, ξ) =

ξ1(ξ21 − ξ2

2) −2ξ21ξ2 −ξ2(ξ2

1 + ξ22) −ξ1(ξ2

1 + ξ22)

ξ2(ξ21 − ξ2

2) −2ξ1ξ22 ξ1(ξ2

1 + ξ22) −ξ2(ξ2

1 + ξ22)

2ξ21ξ2 ξ1(ξ2

1 − ξ22) ξ1(ξ2

1 + ξ22) −ξ2(ξ2

1 + ξ22)

2ξ1ξ22 ξ2(ξ2

1 − ξ22) ξ2(ξ2

1 + ξ22) ξ1(ξ2

1 + ξ22)

.

The matrix expression for B′(P, θ + τ n) = B′(P, ξ) is 1 0 0 0

0 1 0 0

.

And so B′hj(P, ξ)Ljk(P, ξ) is

ξ1(ξ2

1 − ξ22) −2ξ2

1ξ2 −ξ2(ξ21 + ξ2

2) −ξ1(ξ21 + ξ2

2)

ξ2(ξ21 − ξ2

2) −2ξ1ξ22 ξ1(ξ2

1 + ξ22) −ξ2(ξ2

1 + ξ22)

.

A linear combination of these rows has elements which are cubic in τ , and these cannot

be integer multiples of M+. We refer back to (A.29) and (A.30) and the accompany-

ing discussion. These rows are also linearly independent, and so the Complementing

Condition is satisfied.

A similar case is the one in which

U3 = b1 on Γ, (A.39)

U4 = b2 on Γ, (A.40)

with U1 and U2 free on the boundary. The same choice of boundary equation indices

holds. It is simple to show that the Complementing Condition also holds in this case,

following the technique outlined above.

This means that the system (A.31) to (A.34) is ADN elliptic with boundary condi-

tions (A.36) and (A.37) or (A.39) and (A.40).

The Schauder estimate (A.13) with parameter l ≥ 0 takes the form

4∑

j=1

‖ Uj ‖l+1≤ C

(4∑

i=1

‖ fi ‖l +2∑

k=1

‖ bk ‖l− 32

).

A.4. Illustrations


In this instance Uj ∈ H l+tj (Ω) = H l+1(Ω), fi ∈ H l−si(Ω) = H l(Ω) and bk ∈H l−rk− 1

2 = H l− 32 . The boundary is of class C l+1.

A.5 The Application of ADN Theory to the Development

of Mesh Dependent Weights in Least-Squares Func-

tionals

Where we have weighted equations in our least-squares functionals the weights have

been independent of the grid size. In [8] mesh dependent weights are suggested and

justified by referring to ADN theory. Given a system of n equations we let Ri denote

the residual term corresponding to system equation i in the least-squares functional.

According to [8], the appropriate least-squares sum to be minimised is notn∑

i=1

‖Ri‖20

but is insteadn∑

i=1

‖Ri‖2−si

(A.41)

where si is the index of equation i. We recall that the equation indices are bounded

above by zero

si ≤ 0, i = 1, . . . , n

and it would appear that solving a functional of the form (A.41) would mean that

approximate solutions would in general be restricted to spaces with high degrees of

continuity. However the inverse inequality

| V |1≤ Ch−1 ‖ V ‖0 (A.42)

applies for functions V located in finite dimensional subspaces of H1; see [81]. More

generally

| V |s≤ Ch−s ‖ V ‖0 (A.43)

A.5. The Application of ADN Theory to the Development of Mesh Dependent

Weights in Least-Squares Functionals


This relation can be extended to instances where the left-hand side norm is one over

spaces which have non-integer differentiability, for instance H12 , as pointed out in [25].

We can write down an expression in cases where the left hand-side semi-norm is defined

over a space with negative differentiability, although it is generally better to approxi-

mate negative norms using the technique outlined here in the first chapter. Again see

[25].

By invoking (A.42) or (A.43) together with a Poincare-Friedrichs inequality [32] for

sets of functions which satisfy homogeneous conditions on at least a portion of a given

boundary, we see that the terms in (A.41) for which si 6= 0 can be replaced by squares

of norms in L2, multiplied by h−2si . We give a particular example from [23] and [63].

Given the enclosed flow boundary conditions (2.30) for the J system of equations (2.24)

to (2.27) then the appropriate modification of (2.28) is

J =∫

Ω

((ν

∂ω

∂y+

∂p

∂x− f1

)2

+(−ν

∂ω

∂x+

∂p

∂y− f2

)2

+1h2

(ω +

∂u1

∂y− ∂u2

∂x− f3

)2

+1h2

(∂u1

∂x+

∂u2

∂y− f4

)2)

dΩ.

The error bounds

‖ ω − ωh ‖0 + ‖ p− ph ‖0 + ‖ ~uh − ~u ‖1≤ Chk (‖ ω ‖k + ‖ p ‖k + ‖ ~u ‖k+1)

hold for the approximation (ωh, ph, ~uh) to (ω, p, ~u). The estimate holds for k ≥ 2

if for example the velocities are approximated using quadratic interpolation, with the

other two variables approximated linearly; see [25]. Also, from [8], [66] and [25], it is

suggested that a standard least-squares functional may be augmented with a boundary

integral term, giving a functional to be minimised of the form

F =‖ Lu− f ‖20 + ‖ Ru− g ‖2

12, Γ

.

A mesh-dependent rescaling of the boundary term can be performed so that the integral

takes the form

F =‖ Lu− f ‖20 +h−1 ‖ Ru− g ‖2

0, Γ .




It may be advisable to use a different factor depending on the order of differentiabil-

ity required of the boundary conditions and the concomitant values of the boundary

indices; see [8].



Appendix B

Preconditioning Matrices for

Least-Squares Solutions of the

Stokes Equations

We have found that the number of iterations of the basic conjugate gradient algorithm

[72] required to obtain a solution of linear systems arising from the application of

least-squares finite element methods is very high. Therefore we have tested various

preconditioners.

A simple form of preconditioning is diagonal scaling, for which the matrix takes the

form

Mij = Aij , i = j,

Mij = 0, i 6= j.

We recall that rows and columns of the stiffness matrix are associated with a particular

vertex in blocks of size nf , where nf is the number of degrees of freedom at each of

the NV vertices. This suggests another choice of conditioner. We construct the sets

SI , I = 1, . . . , NV , where SI consists of the indices of the rows and columns which

193

Appendix B. Preconditioning Matrices for Least-Squares Solutions of the Stokes

Equations 194

are connected with vertex I. We have that

SI = (I − 1)nf + l, l = 1, 2, . . . , nf .

For example in the S formulation nf = 4 and we have that

S1 = 1, 2, 3, 4 ,

S2 = 5, 6, 7, 8

and so on up to

SNV= (NV ) (nf )− 3, (NV ) (nf )− 2, (NV ) (nf )− 1, (NV ) (nf ) .

Then we can develop another form of preconditioner, the block diagonal matrix

Mij = Aij if i ∈ SI and j ∈ SI , I = 1, . . . , NV ,

Mij = 0 otherwise.

Both diagonal conditioning and block vertex conditioning have the advantage that

storage requirements are low.

Given nf degrees of freedom at each node we can also develop sets sJ , J = 1, . . . , nf

with elements which are the indices of the rows and columns associated with variable

UJ . That is

sJ = (l − 1)nf + J, l = 1, . . . , NV .

In the S formulation

s1 = 1, 5, . . . , (NV ) (nf )− 3 ,

s2 = 2, 6, . . . , (NV ) (nf )− 2 ,

s3 = 3, 7, . . . , (NV ) (nf )− 1 ,

s4 = 4, 8, . . . , (NV ) (nf ) .

Our third form of preconditioning matrix is obtained by implementing the scheme

Mij = Aij if i ∈ sJ and j ∈ sJ , J = 1, . . . , nf ,

Mij = 0 otherwise.


Equations 195

Using a similar approach we can generate the following preconditioning matrix for a

stiffness matrix A obtained using the S formulation

Mij = Aij if (i ∈ s1 or i ∈ s2) and (j ∈ s1 or j ∈ s2),

Mij = Aij if (i ∈ s3 or i ∈ s4) and (j ∈ s3 or j ∈ s4),

Mij = 0 otherwise.

To indicate the relative effectiveness of these conditioning schemes we looked at the

number of iterations required for the solution to converge for a particular problem.

Specifically, we looked at the simulation of Poiseuille flow in a square region. We

obtained solutions of the S, J and G3 formulations with enclosed flow boundary condi-

tions and solutions of the S formulation with normal velocities and tangential stresses

specified on the boundary. The tables below list the matrix sizes and the number of

iterations required with the different forms of conditioning for unweighted and weighted

functionals respectively. The first scheme is diagonal scaling, the second block vertex

conditioning, the third banded conditioning using rows and columns associated with

a given unknown and the fourth a modification of the third for the S formulation as

presented above. The number of iterations required to obtain a converged solution with

an unconditioned matrix in each case is also shown.

B.1 Enclosed Flow Boundary Conditions

Matrix Iterations for Scheme

nx × ny Size N 1 2 3 4 Unconditioned

4× 4 100 68 52 32 12 83

8× 8 324 161 111 41 20 240

16× 16 1156 362 240 49 22 562

Table B.1: Comparison of conditioning rules for S functional with equal weights

B.1. Enclosed Flow Boundary Conditions


Equations 196



4× 4 100 117 117 31 7 148

8× 8 324 490 487 74 8 811

16× 16 1156 1609 1613 152 9 3074

Table B.2: Comparison of conditioning rules for S functional with weights of

1, 1, 103, 103

Matrix Seconds Taken to Converge


4× 4 100 0.28 0.28 0.34 0.14 0.16

8× 8 324 1.28 1.50 0.88 3.79 1.94

16× 16 1156 10.44 11.97 25.08 241.34 13.33

Table B.3: Comparison of solution times for S functional with equal weights



4× 4 100 0.22 0.43 0.33 0.11 0.63

8× 8 324 2.11 5.59 1.28 1.66 3.42

16× 16 1156 40.58 70.10 32.41 102.70 66.89

Table B.4: Comparison of solution times for S functional with weights of 1, 1, 103, 103



Equations 197


nx × ny Size N 1 2 3 Unconditioned

4× 4 100 46 46 14 54

8× 8 324 99 99 22 115

16× 16 1156 233 233 36 268

Table B.5: Comparison of conditioning rules for J functional with equal weights



4× 4 100 40 40 14 101

8× 8 324 149 149 32 625

16× 16 1156 539 539 149 2535

Table B.6: Comparison of conditioning rules for J functional with weight of 103 on

mass conservation term



4× 4 100 0.19 0.30 0.16 0.38

8× 8 324 1.72 2.78 0.67 1.84

16× 16 1156 15.67 25.56 16.55 16.19

Table B.7: Comparison of solution times for J functional with equal weights



Equations 198



4× 4 100 0.24 0.30 0.14 0.55

8× 8 324 2.38 3.97 0.84 8.77

16× 16 1156 34.13 59.11 25.83 139.17

Table B.8: Comparison of solution times for J functional with weight of 103 on mass

conservation term



4× 4 175 60 52 21 74

8× 8 567 145 113 41 197

16× 16 2023 339 253 50 465

Table B.9: Comparison of conditioning rules for G3 functional with equal weights



4× 4 175 62 55 21 153

8× 8 567 249 209 63 979

16× 16 2023 803 618 180 3448

Table B.10: Comparison of conditioning rules for G3 functional with weight of 103 on

mass conservation term



Equations 199



4× 4 175 3.41 3.03 1.28 3.75

8× 8 567 29.63 23.28 9.14 39.77

16× 16 2023 271.01 202.53 68.70 362.42

Table B.11: Comparison of solution times for G3 functional with equal weights



4× 4 175 3.53 2.95 1.20 7.39

8× 8 567 50.73 42.81 13.71 176.09

16× 16 2023 625.20 470.48 178.58 2570.84

Table B.12: Comparison of solution times for G3 functional with weight of 103 on mass

conservation term

We solved these problems using programs written in FORTRAN and compiled by

FORTRAN 95 Version 2.0 from NA Software. We ran the executable code under

Windows XP on a PC with a single 1.7 GHz Intel Pentium 4 processor and 768Mb of

RAM.

In obtaining solutions which minimise the weighted functionals without condition-

ing, more than N iterations were needed; see Tables B.2, B.6 and B.10. From Table

B.2 we see that as far as the weighted S functional is concerned the number of itera-

tions required was still greater than N even after preconditioning with a diagonal or

block vertex conditioning. Furthermore the number of iterations required appears to

rise approximately at order h2 or greater. The number of iterations required to obtain

a solution of an unconditioned matrix obtained in solving for an unweighted functional



Equations 200

arising from any of the three formulations does not increase at this rate, though the

rate is still greater than order h; see Tables B.1, B.5 and B.9. In obtaining a solution

of the weighted S formulation diagonal scaling is generally to be preferred to block

vertex conditioning. The shorter time spent on each iteration compensates for the

slightly greater number of iterations required; see Tables B.2 and Table B.4. With the

unweighted S formulation the advantages of diagonal scaling over block vertex condi-

tioning are less clear; see Table B.1 and Table B.3. In solving for the J formulation

the number of iterations required after either diagonal or vertex conditioning is the

same, so diagonal scaling is obviously to be preferred; see Tables B.5, B.6, B.7 and B.8.

In solving the system arising from the G3 functional solutions to these problems can

be obtained more quickly using block vertex conditioning than diagonal scaling; see

Tables B.5, B.6, B.7 and B.6. Though the number of iterations may be reduced using

the third or fourth conditioning schemes the sizes of the arrays required to store non-

trivial values are comparable to those needed for solution of the full matrix problem by

Gaussian elimination. The number of variables in any given test problem considered

in this appendix is relatively small, but these schemes are less suitable for the solution

of problems on grids with a great many nodes. A further disadvantage is that invert-

ing these matrices can take a large amount of computer time, as can be seen even in

obtaining solutions on these coarse grids; see Tables B.3 and B.4. The third scheme

does have its advantages in that when solving a system arising from an S functional or

the unweighted J or G functionals the number of iterations required does not increase

anywhere nearly as rapidly with the size of the matrix as it does with diagonal scaling

or block vertex conditioning; see Tables B.1, B.2, B.5 and B.9. In using the fourth

scheme to solve a system of equations arising from an S functional it appears that

the number of iterations hardly rises with the size of the matrix; see Tables B.1 and

B.2. Furthermore fewer iterations are required in obtaining a weighted solution than

in obtaining an unweighted one.



Equations 201

B.2 Normal Velocities and Tangential Stresses



4× 4 100 16 15 11 9 28

8× 8 324 56 42 15 11 101

16× 16 1156 136 88 17 13 244

Table B.13: Comparison of conditioning rules for S functional with equal weights



4× 4 100 20 20 8 6 41

8× 8 324 90 85 22 6 307

16× 16 1156 373 356 74 6 1245

Table B.14: Comparison of conditioning rules for S functional with weights of

1, 1, 103, 103



4× 4 100 0.08 0.13 0.08 0.13 0.11

8× 8 324 0.56 0.64 0.52 2.33 0.77

16× 16 1156 4.95 4.87 14.63 146.81 5.94

Table B.15: Comparison of solution times with equal weights

B.2. Normal Velocities and Tangential Stresses


Equations 202



4× 4 175 0.09 0.14 0.06 0.09 0.13

8× 8 567 0.84 1.13 0.56 1.47 1.81

16× 16 2023 9.74 16.17 18.33 70.24 26.67

Table B.16: Comparison of solution times for S functional with weights of 1, 1, 103, 103

Tables B.13 and B.14 show the number of iterations required to obtain a convergent

solution of the S formulation with normal velocities and tangential stresses specified at

every point on the boundary. In all cases the number of iterations required is less than

is the case with enclosed flow boundary conditions, indicating that the conditioning of

the system is better; see Tables B.1 and B.2.

We remark that with these boundary conditions the number of iterations required

to solve the unconditioned matrix arising from the weighted functional is less than N

at

both ny = 4 and ny = 8, though not at ny = 16; see Table B.14. With diagonal

or block vertex scaling the number of iterations needed to obtain a solution which

minimises either the weighted or unweighted functional is less than the N for all sizes

of grid considered. This contrasts with the data for enclosed flow boundary conditions,

where the number needed to obtain a solution of a linear system arising from the

weighted functional and conditioned by either of these two schemes is greater than N

at all grid levels. The time required to obtain a solution by any of the conditioning

schemes or without conditioning is less than that required in obtaining a solution of the

enclosed flow problem; compare Table B.3 with Table B.15 and Table B.4 with Table

B.16.

The necessary number of iterations of the algorithm on a matrix arising from the

B.2. Normal Velocities and Tangential Stresses


Equations 203

unweighted functional and conditioned with the third or fourth scheme increase only

slightly as the grid is refined. The number of iterations required to obtain a weighted

solution with the fourth scheme is constant with respect to the size of the grid and is less

than the number needed to obtain a solution minimising the unweighted functional. We

recall that with enclosed flow boundary conditions the number required is also roughly

constant; see Tables B.2 and B.14. By contrast the number of iterations required to

arrive at a solution which minimises the weighted functional without conditioning or

with conditioning by any of the other three schemes increases at a rate of approximately

h2.

B.3 Conditioning of Linearised Systems arising from the

SN Formulation

We have also solved linear systems arising from the linearisation of the SN reformulation

of the Navier-Stokes equations. We have applied both diagonal scaling and block vertex

conditioning to decrease the condition number of the system. With either conditioning

scheme the number of conjugate gradient iterations required to find a solution of the

linear systems arising in each of the first few iterations of the Newton linearisation is

greater than that required to obtain a solution of an equivalent Stokes problem in the

S formulation1.

1On later iterations of the Newton linearisation algorithm, we can use the solution obtainedon the previous step to give a very accurate estimate of the new solution. The number ofconjugate gradient iterations then required to obtain a new solution is very greatly reduced.

B.3. Conditioning of Linearised Systems arising from the SN Formulation

Appendix C

A Stress and Stream Formulation

of the Unsteady Planar Stokes

Equations

The equations of motion for unsteady incompressible Stokes flow in the plane are

∂~u

∂t− ν∇2~u +∇p = ~f, (C.1)

∇.~u = 0 (C.2)

where t is the time, ~f = (fx, fy) is the body force, p is the pressure and ~u = (u1, u2)

is the velocity. The equations for unsteady incompressible Navier-Stokes flow are

∂~u

∂t− ν∇2~u + ~u.∇~u +∇p = ~f, (C.3)

∇.~u = 0. (C.4)

For incompressible flow the density ρ(x, y, t) = K, a constant. From Newton’s second

law of motion force is proportional to rate of change of momentum, and when no

external body forces actD~u

Dt= ∇.σ. (C.5)

204

Appendix C. A Stress and Stream Formulation of the Unsteady Planar Stokes

Equations 205

The term on the left is called the material or substantial [3] derivative

D~u

Dt=

∂~u

∂t+ u1

∂~u

∂x+ u2

∂~u

∂y. (C.6)

Where the non-linear space derivatives for a particular flow field are non-trivial we have

a Navier-Stokes problem. For brevity we shall consider a reformulation of (C.1) and

(C.2). We introduce new variables U5 and U6 defined as

∂U5

∂x= u1 = ψy = U4,

∂U6

∂y= u2 = −ψx = −U3

and augment the stress tensor σ to generate a new tensor σT written as

σT =

σxx − ∂U5

∂tσxy

σxy σyy − ∂U6

∂t

. (C.7)

Taking the divergence of (C.7) gives us

∇.σT =

−∂u1

∂t+

∂σxx

∂x+

∂σxy

∂y

−∂u2

∂t+

∂σxy

∂x+

∂σyy

∂y

so that

∇.σT = −∂~u

∂t+∇.σ.

Setting ∇.σT equal to zero gives us (C.5) for the case where the non-linear components

of (C.6) vanish.

In terms of the stress function φ, tensor (C.7) is

σT =

φyy − ∂U5

∂t−φxy

−φxy φxx − ∂U6

∂t

.

As in previous chapters, we make the substitutions

U1 = φx, U2 = φy

Appendix C. A Stress and Stream Formulation of the Unsteady Planar Stokes

Equations 206

and introduce as variables the derivatives of ψ, the stream function for the velocity

U3 = ψx = −u2, U4 = ψy = u1.

We introduce the variable U7 = U6 − U5. Rewriting using the variables U1 through to

U7 the system is

∂U7

∂t− ∂U1

∂x+

∂U2

∂y− 2ν

∂U3

∂y− 2ν

∂U4

∂x= f1,

−∂U1

∂y+

∂U2

∂x− 2ν

∂U3

∂x+ 2ν

∂U4

∂y= f2,

∂U1

∂y− ∂U2

∂x= f3,

2ν∂U3

∂y− 2ν

∂U4

∂x= f4,

∂U5

∂x− U4 = f5,

∂U6

∂y+ U3 = f6,

U5 − U6 + U7 = f7

where we allow non-zero right hand-sides.

Applying this approach to reformulate (C.3) and (C.4) is straightforward by incor-

porating the Reynolds stress tensor into the definition of the time-dependent stress

tensor σT . It can also be observed that using the method outlined here we can develop

first-order formulations in terms of stress and stream functions or stress functions and

velocities which are equivalent to the unsteady Stokes and Navier-Stokes equations in

three dimensions.

References

[1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solu-

tions of elliptic partial differential equations I, Comm. Pure and Appl. Math. 12,

pp. 623-727, 1959.

[2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solu-

tions of elliptic partial differential equations II, Comm. Pure and Appl. Math.

17, pp. 35-92, 1964.

[3] J. D. Anderson. Jr., Computational Fluid Dynamics, The Basics with Applica-

tions, McGraw-Hill International Editions, Mechanical Engineering Series, 1995.

[4] D. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equa-

tions, Calcolo 21, pp. 337-344, 1984.

[5] I. O. Arushanian and G. M. Kobelkov, Implementation of a least-squares finite

element method for solving the Stokes problem with a parameter, Numer. Linear

Algebra Appl. 6, pp. 587-597, 1999.

[6] R. B. Ash, Measure, Integration and Functional Analysis, Academic Press, 1972.

[7] T. M. Austin, Advances on a Scaled Least-Squares Method for the 3-D Linear

Boltzmann Equation, Ph. D. Thesis, University of Colorado, 2001.

[8] A. Aziz, R. Kellog and A. Stephens, Least-squares methods for elliptic systems,

Math. Comp. 10, pp. 53-70, 1985.

207

References 208

[9] G. Bachman and L. Narici, Functional Analysis, Academic Press International

Edition, 1966.

[10] G. Bao, Y. Cao and H. Yang, Numerical Solution of Diffraction Problems by a

Least-squares Finite Element Method, Math. Meth. Appl. Sci. 23, pp. 1073-1092,

2000.

[11] G. Bao and H. Yang, A least-squares finite element analysis for diffraction prob-

lems, SIAM J. Numer. Anal. 37, pp. 665-682, 2000.

[12] G. K. Batchelor, An Introduction to Fluid Mechanics, Cambridge University

Press, 1967.

[13] D. M. Bedivan, Error estimates for least squares finite element methods, Comput.

Math. Appl. 43, pp. 1003-1020, 2002.

[14] E. Bertoti, Indeterminacy of first order stress functions and the stress- and

rotation-based formulation of linear elasticity, Comput. Mech. 14, pp. 249-265,

1994.

[15] E. Bertoti, Dual-mixed hp finite element methods using first-order stress functions

and rotations, Comput. Mech. 26, pp. 39-51, 2000.

[16] V. Blokh, Stress functions in the theory of elasticity, Prikladnaia Matematika I

Mekhanika 14, pp. 415-422, 1950 (in Russian).

[17] P. Bochev, Analysis of least-squares finite element methods for the Navier-Stokes

equations, SIAM J. Numer. Anal. 34, pp. 1817-1844, 1997.

[18] P. Bochev, Experiences with negative norm least-square methods for the Navier-

Stokes equations, ETNA 6, pp. 44-62, 1997.

[19] P. Bochev, Negative norm least-squares methods for the velocity-vorticity-

pressure Navier-Stokes equations, Numer. Methods Partial Differential Eq. 15,

pp. 237-256, 1999.

References

References 209

[20] P. Bochev, Z. Cai T. A. Manteuffel and S. F. McCormick, Analysis of Velocity-

Flux Least Squares Principles for the Navier-Stokes Equations Part I, SIAM J.

Numer. Anal. 35, pp. 990-1009, 1998.

[21] P. Bochev, T. A. Manteuffel and S. F. McCormick, Analysis of Velocity-Flux

Least Squares Principles for the Navier-Stokes Equations Part II, SIAM J. Numer.

Anal. 36, pp. 1125-1144, 1999.

[22] P. Bochev and M. D. Gunzburger, Accuracy of Least-Squares Methods for the

Navier-Stokes Equations, Comput. & Fluids 22, pp. 549-563, 1993.

[23] P. Bochev and M. D. Gunzburger, Analysis of Least-Squares Methods for the

Stokes Equations, Math. Comp. 63, No. 208, pp. 479-506, 1994.

[24] P. Bochev and M. D. Gunzburger, Least-squares methods for the velocity-

pressure-stress formulation of the Stokes equations, Comput. Methods Appl.

Mech. Engrg. 126, pp. 267-287, 1995

[25] P. Bochev and M. D. Gunzburger, Finite element methods of least squares type,

SIAM Review 40, pp. 789-837, 1998.

[26] P. Bolton, J. Stratakis and R. W. Thatcher, Mass conservation in least squares

methods for Stokes flow, UMIST internal report, 2001.

[27] A. Bose and G. F. Carey, Least-squares p− r finite element methods for incom-

pressible non-Newtonian flows, Comput. Methods Appl. Mech. Engrg. 180, pp.

431-458, 1999.

[28] J. H. Bramble, R. Lazarov and J. E. Pasciak, A least squares approach based on

a discrete minus one inner product for first order systems, Technical Report 94-

32, Mathematical Science Institute, Cornell University, 1994, published in Math.

Comp. 66, pp. 935-955, 1997.

[29] J. H. Bramble, R. Lazarov and J. E. Pasciak, Least-squares for second-order

elliptic problems, Comput. Methods Appl. Mech. Engrg. 152, pp. 195-210, 1998.

References

References 210

[30] J. H. Bramble, R. Lazarov and J. E. Pasciak, Least-squares methods for linear

elasticity based on a discrete minus one inner product, Comput. Methods Appl.

Mech. Engrg. 191, pp. 727-744, 2001.

[31] J. H. Bramble and J. E. Pasciak, Least-squares methods for Stokes equations

based on a discrete minus one inner product, J. Comp. Appl. Math. 74, pp.

549-563, 1996.

[32] D. Braess, Finite Elements, Cambridge University Press, 1997.

[33] F. Brezzi, J. Rappaz and P.-A. Raviart, Finite-dimensional approximation of

nonlinear problems, Part I: Branches of nonsingular solutions, Numer. Math. 36,

pp. 1-25, 1980

[34] W. L. Briggs, Van Emden Henson, S. F. McCormick, A Multigrid Tutorial, Sec-

ond Edition, SIAM, 2000.

[35] Z. Cai, Least-squares for the perturbed Stokes equations and the Reissner-Mindlin

plate, SIAM J. Numer. Anal. 38, pp. 1561-1581, 2000.

[36] Z. Cai, R. D. Lazarov, T. A. Manteuffel and S. F. McCormick, First-order system

least squares for second-order partial differential equations: Part I, SIAM J.

Numer. Anal. 31, pp. 1785-1799, 1994.

[37] Z. Cai, C.-O. Lee, T. A. Manteuffel and S. F. McCormick, First order system least

squares for the Stokes and linear elasticity equations: Further results, SIAM J.

Sci. Comput. 21, pp. 1728-1739, 2000.

[38] Z. Cai, T. A. Manteuffel and S. F. McCormick, First-order system least squares

for velocity-vorticity-pressure form of the Stokes equations with applications to

linear elasticity, ETNA 3, pp. 150-159, 1995.


for second-order partial differential equations: Part II, SIAM J. Numer. Anal.

34, pp. 425-454, 1997.

References

References 211


for the Stokes equations, with applications to linear elasticity, SIAM J. Numer.

Anal. 34, pp. 1727-1741, 1997.

[41] Z. Cai, T. A. Manteuffel, S. F. McCormick and S. V. Parter, First Order System

Least Squares(FOSLS) for planar linear elasticity: pure traction problem, SIAM

J. Numer. Anal. 35, pp. 320-335, 1998.

[42] Z. Cai, T. A. Manteuffel, S. F. McCormick and J. Ruge, First-order system LL∗

scalar elliptic partial differential equations, SIAM J. Numer. Anal. 39, pp. 1418-

1445, 2001.

[43] Z. Cai and B. C. Shin, The discrete first-order system least squares: the second-

order elliptic boundary value problem, SIAM J. Numer. Anal. 40, pp. 307-318,

2002.

[44] Z. Cai, X. Ye, A Least-Squares Finite Element Approximation to the Compress-

ible Stokes Equations, Numer. Methods Partial Differential Eq. 16, pp. 62-70,

2000.

[45] Z. Cai, X. Ye and H. Zhang, Least-squares Finite Element Approximations for

the Reissner-Mindlin Plate, Numer. Linear Algebra Appl. 6, pp. 479-496, 1999.

[46] G. F. Carey and B.-N. Jiang, Nonlinear preconditioned conjugate gradient and

least-squares finite elements, Comput. Methods Appl. Mech. Engrg. 62, pp. 145-

154, 1987.

[47] G. F. Carey, A. I. Pehlivanov, Y. Shen, A. Bose and K. C. Wang, Least-squares

finite element methods for fluid flow and transport, Int. J. Numer. Methods Fluids

27, pp. 97-107, 1998.

[48] M. Cassidy, A spectral element method for viscoelastic extrudate swell, PhD

Thesis, Univ of Wales, Aberystwyth, Wales, 1996.

[49] C.-L. Chang, A least squares finite element method for the Helmholtz equation,

Comput. Methods Appl. Mech. Engrg. 83, pp. 1-7, 1990.

References

References 212

[50] C.-L. Chang, An Acceleration Pressure Formulation, Appl. Math. Comput. 36,

pp. 135-146, 1990.

[51] C.-L. Chang, Finite Element Approximation for Grad-Div Type Systems in the

Plane, SIAM J. Numer. Anal 29, pp. 452-461, 1992.

[52] C.-L. Chang, An Error Estimate of the Least Square Finite Element Method for

the Stokes Problem in Three Dimensions, Math. Comp. 63, pp. 41-50, 1994.

[53] C.-L. Chang and M. D. Gunzburger, A finite element method for first order

elliptic systems in three dimensions, Appl. Math. Comp. 23, pp. 171-184, 1987.

[54] C.-L. Chang and B.-N. Jiang, An error analysis of least-squares finite element

method of velocity-pressure-vorticity formulation for the Stokes problem, Com-

put. Methods Appl. Mech. Engrg. 84, pp. 247-255, 1990.

[55] C.-L. Chang and J. J. Nelson, Least-squares finite element method for the Stokes

problem with zero residual of mass conservation, SIAM J. Numer. Anal. 34, pp.

480-489, 1997.

[56] C.-L. Chang, S. Y. Yang, Analysis of the L2 least-squares finite element method

for the velocity-vorticity-pressure Stokes equations with velocity boundary con-

ditions, Appl. Math. Comp. 130, pp. 121-144, 2002.

[57] C.-L. Chang, S. Y. Yang and C. H. Hsu, A least squares finite element method for

incompressible flow in stress-velocity-pressure version, Comput. Methods Appl.

Mech. Engrg. 128, pp. 1-9, 1995.

[58] A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechan-

ics, Third Edition, Springer Verlag, 1992.

[59] T. J. Chung, Finite Element Analysis in Fluid Dynamics, McGraw Hill Interna-

tional Advanced Book Program, 1978.

[60] A. L. Codd, Elasticity-Fluid Coupled Systems and Elliptic Grid Generation

(EGG) based on First-Order Systems Least Squares (FOSLS), Ph. D. Thesis,

University of Colorado, 2001.

References

References 213

[61] R. Courant, Variational methods for the solution of problems of equilibrium and

vibration, Bull. Am. Math. Soc., 49, pp. 1-23, 1943.

[62] A. J. Davies, The Finite Element Method: A First Approach, Oxford Applied

Mathematics and Computing Science Series, 1980

[63] J. M. Deang and M. D. Gunzburger, Issues related to least-squares finite element

methods for the Stokes equations, SIAM J. Sci. Comput. 20, pp. 878-906, 1998.

[64] A. Douglis and L. Nirenberg, Interior estimates for elliptic equations, Comm.

Pure and Appl. Math. 8, pp. 503-538, 1955.

[65] F. Dubois, Vorticity-velocity-pressure formulation for the Stokes problem, Math.

Meth. Appl. Sci. 25, pp. 1091-1119, 2002.

[66] E. D. Eason, A review of least-squares methods for solving partial differential

equations, Int. J. Numer. Meth. Engng. 10, pp. 1021-1046, 1976

[67] M. J. Fagan, Finite Element Analysis: Theory and practice, Addison Wesley

Longman Limited, 1992.

[68] J. M. Fiard, T. A. Manteuffel and S. F. McCormick, First-order system least

squares (FOSLS) for convection-diffusion problems: Numerical results, SIAM J.

Sci. Comput. 19, pp. 1958-1979, 1998.

[69] G. J. Fix, M. D. Gunzburger and R. A. Nicolaides, On finite element methods of

the least squares type, Comput. Math. Appl. 5, pp. 87-98, 1979.

[70] M. Fortin, Old and new finite elements for incompressible flow, Int. J. Numer.

Methods Fluids 1, pp. 347-364, 1981.

[71] V. Girault and P. Raviart, Finite Element Methods for Navier-Stokes Equations,

Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986.

[72] G. H. Golub and C. F. Van Loan, Matrix Computation, Third Edition, The John

Hopkins University Press, 1996.

References

References 214

[73] H. Gu and X. Wu, Least-squares mixed finite element methods for the incom-

pressible Navier-Stokes equations, Numer. Methods Partial Differential Eq. 18,

pp. 441-453, 2002.

[74] M. D. Gunzburger and H.-C. Lee, Analysis and approximation of optimal control

problems for first-order elliptic systems in three dimensions, Appl. Math. Comp.

100, pp. 49-70, 1999.

[75] G. Hadley, Linear Programming, Addison Wesley Series in Industrial Manage-

ment, 1962.

[76] W. M. Harper, Operational Research, M and E Handbooks, 1975.

[77] P. Hood and C. Taylor, A numerical solution of the Navier-Stokes equations using

the finite element technique, Comput. & Fluids 1, pp. 73-100, 1973.

[78] W. J. Ibbetson, An elementary treatise on the mathematical theory of perfectly

elastic solids, with a short account of viscous fluids, Macmillan, 1887.

[79] B.-N. Jiang, A least-squares finite element method for incompressible Navier-

Stokes equations, Int. J. Numer. Methods Fluids 14, pp. 843-859, 1992.

[80] B.-N. Jiang, On the least-squares method, Comput. Methods Appl. Mech. Engrg.

152, pp. 239-257, 1998.

[81] B.-N. Jiang, The Least-Squares Finite Element Method, Theory and Applications

in Computational Fluid Dynamics and Electromagnetics, Scientific Computation

Series, Springer, 1998.

[82] B.-N. Jiang, The least-squares finite element method in elasticity. Part II: Bend-

ing of thin plates, Int. J. Numer. Meth. Engng. 54, pp. 1459-1475, 2002.

[83] B.-N. Jiang and G. F. Carey, Adaptive refinement for least-squares finite elements

with element-by-element conjugate gradient solution, Int. J. Numer. Meth. En-

gng. 24, pp. 569-580, 1987.

References

References 215

[84] B.-N. Jiang, T. L. Lin and L. A. Povinelli, Large-scale computation of incom-

pressible viscous flows by least-squares finite element method, Comput. Methods

Appl. Mech. Engrg. 114, pp. 213-231, 1994.

[85] B.-N. Jiang and L. A. Povinelli, Least-squares finite element method for fluid

dynamics, Comput. Methods Appl. Mech. Engrg. 81, pp. 13-37, 1990.

[86] B.-N. Jiang and L. A. Povinelli, Optimal least-squares finite element method for

elliptic problems, Comput. Methods. Appl. Mech. Engrg. 102, pp. 199-212, 1993.

[87] B.-N. Jiang and J. Wu, The least squares finite element method in elasticity-Part

I: Plane stress or strain with drilling degrees of freedom, Int. J. Numer. Meth.

Engng. 53, pp. 621-636, 2002

[88] C. Johnson, Numerical Solution of Partial Differential Equations by the Finite

Element Method, Cambridge University Press, 1987.

[89] J. Jou and S.-Y. Yang, Least-squares finite element approximations to the Tim-

oshenko beam problem, Appl. Math. Comp. 115, pp. 63-65, 2000.

[90] S. D. Kim and E. Lee, An analysis for compressible Stokes equations by first-

order system of least-squares finite element method, Numer. Methods Partial

Differential Eq. 17, pp. 689-699, 2001.

[91] S. D. Kim and E. Lee, Least-squares mixed method for second-order elliptic

problems, Appl. Math. Comp. 115, pp. 89-100, 2000.

[92] S. D. Kim, T. A. Manteuffel and S. F. McCormick, First-Order System Least

Squares (FOSLS) for Spatial Linear Elasticity: Pure Traction, SIAM J. Numer.

Anal. 35., pp. 1454-1482, 2000.

[93] I. Kozak, Remarks and contributions to the variational principles of the linearized

theory of elasticity in terms of the stress functions, Acta Technica Acad. Scient.

Hung. 92, pp. 45-65, 1981.

References

References 216

[94] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,

Translated from Russian by Richard A. Silverman, Gordon and Breach Science

Publishers, 1964.

[95] C. Lanczos, Linear Differential Operators, D. Van Nostrand, 1961.

[96] B. Lee, First-order system least-squares for elliptic problems with Robin bound-

ary conditions, SIAM J. Numer. Anal. 37, pp. 70-104.

[97] B. Lee, T. A. Manteuffel, S. F. McCormick and J. Ruge, First-Order System

Least-Squares (FOSLS) for the Helmholtz Equation, SIAM J. Sci. Comput. 21,

pp. 1927-1949, 2000.

[98] J. S. Li, Z. Y. Yu, X. Q. Xiang, W. P. Ni and C.-L. Chang, Least-squares finite

element method for electromagnetic fields in 2D, Appl. Math. Comp. 58, pp.

143-167, 1999.

[99] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems I,

Springer-Verlag New York, 1972.

[100] J.-L. Liu, Exact a posteriori error analysis of the least squares finite element

method, Appl. Math. Comp. 116, pp. 297-305, 2000.

[101] A. E. H. Love, A Treatise on the Mathematical Theory of Linear Elasticity, Third

Edition, Cambridge University Press, 1920.

[102] P. Lynn and S. Arya, Use of least-squares criterion in the finite element method,

Int. J. Numer. Meth. Engng. 6, pp. 75-88, 1973.

[103] H. R. Macmillan, First-Order System Least Squares and Electrical Impedance

Tomography, University of Colorado, Ph. D. Thesis, 2001.

[104] T. Manteuffel, S. F. McCormick and C. Pflaum, Improved Discretisation Error

Estimates for First-Order System Least Squares (FOSLS), available by electronic

transfer from ftp://amath.colorado.edu/pub/fosls as nitsche.ps.

References

References 217

[105] T. A. Manteuffel and K. J. Ressel, Least-squares finite element solution of the

neuton transport equation in diffusive regimes, SIAM J. Numer. Anal. 35, pp.

806-835, 1998.

[106] T. A. Manteuffel, K. J. Ressel and Gerhard Starke, A boundary functional for

the least-squares finite-element solution of Neutron Transport Problems, SIAM

J. Numer. Anal. 37, pp. 556-586, 2000.

[107] S. F. McCormick, FOSLoSophy, Informal disussion of some advantages and dis-

advantages of First-Order System Least Squares (FOSLS), available by electronic

transfer from ftp://amath.colorado.edu/pub/fosls as FOSLoSophy.ps.

[108] R. von Mises and K. O. Friedrichs, Fluid Mechanics, Springer Verlag, 1971.

[109] K. Nafa and R. W. Thatcher, Low-Order Macroelements for Two- and Three-

Dimensional Stokes Flow, Numer. Methods Partial Differential Eq. 9, pp. 579-591,

1993.

[110] P. Neittaanmaki and J. Saranen, Finite element approximation of electromagnetic

fields in three dimensional space, Numer. Funct. Anal. and Optimiz. 2, pp. 487-

506, 1980.

[111] S. Norburn and D. Silvester, Stabilised vs. stable mixed methods for incompress-

ible flow, Comput. Methods Appl. Mech. Engrg. 166 , pp. 131-141, 1998.

[112] S. H. Park and S.-K. Youn, The least-squares meshfree method, Int. J. Numer.

Meth. Engng. 52, pp. 997-1012, 2001

[113] C. E. Pearson, Theoretical Elasticity, Harvard University Press, 1959.

[114] A. I. Pehlivanov, G. F. Carey and R. D. Lazarov, Least-squares mixed finite

elements for second-order elliptic problems, SIAM J. Numer. Anal. 31, pp. 1368-

1377, 1994.

[115] A. I. Pehlivanov, G. F. Carey, R. D. Lazarov and Y. Shen, Convergence Analysis

of Least-Squares Mixed Finite Elements, Computing 51, pp. 111-123, 1993.

References

References 218

[116] W. H. Press, S. A. Teukolsky, W. T. Vettering and B. P. Flannery, Numerical

Recipes in Fortran, The Art of Scientific Computing, Second Edition, Cambridge

University Press, 1992.

[117] M. Renardy and R. Rogers, Introduction to Partial Differential Equations,

Springer-Verlag, Berlin, 1993.

[118] H. M. Schey, Div, Grad, Curl and All That, An Informal Text on Vector Calculus,

Second Edition, W. W. Norton, 1972.

[119] V. V. Shaidurov, Multigrid Methods for Finite Elements, Mathematics and its

Applications, Boston: Kluwer Academic Publishers, 1995.

[120] D. G. Shepherd, Elements of Fluid Mechanics, Harcourt, Brace and World Inc.,

1965.

[121] I. S. Sokolnikoff, Mathematical Theory of Elasticity, Second Edition, McGraw-

Hill, 1956.

[122] R. V. Southwell, An Introduction to the Theory of Elasticity for Engineers and

Physicists, Oxford University Press, 1941.

[123] J. Stratakis, Least Squares Methods for Stokes Problem, M. Sc. Thesis, UMIST,

1999.

[124] L. Q. Tang, T. Cheng and T. T. H. Tsang, Transient solutions for three-

dimensional lid-driven cavity flows by a least-squares finite-element method, Int.

J. Numer. Methods Fluids 21, pp. 413-432, 1995.

[125] L. Q. Tang and T. T. H. Tsang, A least-squares finite element method for time-

dependent incompressible flows with thermal convection, Int. J. Numer. Methods

Fluids 17, pp. 271-289, 1993.

[126] L. Q. Tang and T. T. H. Tsang, Temporal, spatial and thermal features of 3-D

Rayleigh-Benard convection by a least-squares finite element method, Comput.

Methods Appl. Mech. Engrg. 140, pp. 201-219, 1997.

References

References 219

[127] L. Q. Tang, J. L. Wright and T. T. H. Tsang, Simulations of 2D and 3D thermo-

capillary flows by a least-squares finite element method, Int. J. Numer. Methods

Fluids 28, pp. 983-1007, 1998.

[128] F. Taghaddosi, W. D. Habashi, G. Guevremont and D. Ait-Ali-Yahia, An adap-

tive least-squares method for the compressible Euler Equations, Int. J. Numer.

Methods Fluids 31, pp. 1121-1139, 1999.

[129] R. Temam, The Navier-Stokes Equations, Studies in Mathematics and its Appli-

cations, Volume 2, North Holland, 1979.

[130] R. W. Thatcher, A least squares method for Stokes flow based on stress and

stream functions, Manchester Centre for Computational Mathematics Report

330, 1998.

[131] R. W. Thatcher, A least squares method for biharmonic problems, SIAM J.

Numer. Anal. 38, pp. 1523-1539, 2000.

[132] B. M. Fraejis de Veubeke, Stress function Approach, Proc. World Cong. on Finite

Element Methods in Structural Mechanics, pp. J.1-J.51, Bournemouth, UK, 1975.

[133] W. L. Wendland, Elliptic systems in the plane, Pitman, London, 1979.

[134] J. Wu, S.-T. Yu, B.-N. Jiang, Simulation of two-fluid flows by the least-squares

finite element method using a continuum surface tension model, Int. J. Numer.

Meth. Engng. 42, pp. 583-600, 1998.

[135] S.-Y. Yang, On the convergence and stability of the standard least squares finite

element method for first-order elliptic systems, Appl. Math. Comp. 93, pp. 51-62,

1998

[136] S.-Y. Yang, Error analysis of a weighted least-squares finite element method for 2-

D incompressible flows in velocity-stress-pressure formulation, Math. Meth. Appl.

Sci. 21, pp. 1637-1654, 1998.

[137] S.-Y. Yang, Analysis of a least-squares finite element method for the circular arch

problem, Appl. Math. Comp. 114, pp. 263-278, 2000.

References

References 220

[138] S.-Y. Yang and C.-L. Chang, A Two-Stage Least-Squares Finite Element Method

for the Stress-Pressure-Displacement Elasticity Equations, Numer. Methods Par-

tial Differential Eq. 14, pp. 297-315, 1998.

[139] S.-Y. Yang and C.-L. Chang, Analysis of a Two-stage Least-Squares Finite Ele-

ment Method for the Planar Elasticity Problem, Math. Meth. Appl. Sci. 22, pp.

713-732, 1999.

[140] S.-Y. Yang and J.-L. Liu, A unified analysis of a weighted least squares method

for first-order systems, Appl. Math. Comp. 92, pp. 9-27, 1998.

[141] X. Ye, Domain decomposition for least-squares finite element methods for the

Stokes equations, Appl. Math. Comp. 97, pp. 45-53, 1998.

[142] K. Yoshida, Functional Analysis, Springer-Verlag, Fifth Edition, 1978.

[143] S.-T. Yu, B.-N. Jiang, J. Wu and N.-S. Liu, A div-curl-grad formulation for

compressible buoyant flows solved by the least-squares finite element method,

Comput. Methods Appl. Mech. Engrg. 137, pp. 59-88, 1996.

[144] O. C. Zienkiewicz and Y. K. Cheung, The Finite Element Method in Structural

and Continuum Mechanics, McGraw-Hill, 1967.

References

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A LEAST-SQUARES FINITE ELEMENT METHOD FOR …...A LEAST-SQUARES FINITE ELEMENT METHOD FOR THE STOKES AND NAVIER-STOKES EQUATIONS A thesis submitted to The University of Manchester