Analysis of Hybrid Discontinuous Galerkin Methods for ...€¦ · Analysis of Hybrid Discontinuous Galerkin Methods for Incompressible Flow Problems Christian Waluga1 advised by Prof

Analysis of Hybrid Discontinuous Galerkin Methodsfor Incompressible Flow Problems

Christian Waluga1

advised by Prof. Herbert Egger2 Prof. Wolfgang Dahmen3

1Aachen Institute for Advanced Study in Computational Engineering Science, RWTH Aachen University2M2 – Center of Mathematics, Technische Universitat Munchen

3 Institut fur Geometrie und Praktische Mathematik, RWTH Aachen University

February 3, 2012

Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 1 / 34

Outline

Outline

1 IntroductionMotivationGoverning equationsDiscretizationOverview of the thesis

2 A Hybrid Discontinuous Galerkin MethodPreliminariesPoisson problemStokes problemNavier-Stokes problem

3 A posteriori error estimators and adaptivityError estimationAn adaptive algorithmNumerical results

4 Conclusions


Introduction

Outline

1 IntroductionMotivationGoverning equationsDiscretizationOverview of the thesis

2 A Hybrid Discontinuous Galerkin Method

3 A posteriori error estimators and adaptivity

4 Conclusions


Introduction Motivation

Simulation aims to predict physical phenomena which aredifficult, expensive, or even impossibleto observe in conventional experiments.



Source: Wikimedia Commons



Real-world problems

⇓

Mathematical language

⇓Computable problems

⇓Approximate solutions


Introduction Governing equations

Motion of fluids

Incompressible Navier-Stokes equations:

−ν∆u + u ·∇u + ∇p = fdivu = 0

on Ω ⊂ Rd

velocity: u = [u1, . . . , ud] pressure: p

Boundary conditions:

u = gD on ∂ΩD (Dirichlet)

ν∂nu− pn = gN on ∂ΩN (Neumann)

For simplicity, let us suppose that u = 0 on ∂Ω.

∂ΩN

∂ΩD

Ω

R2

Difficulties? incompressibility, convective terms, nonlinearity



Motion of fluids


−ν∆u + u ·∇u + ∇p = fdivu = 0

on Ω ⊂ Rd






∂ΩN

∂ΩD

Ω

R2




Motion of fluids


−ν∆u + u ·∇u + ∇p = fdivu = 0

on Ω ⊂ Rd






∂ΩN

∂ΩD

Ω

R2



Introduction Discretization

Popular discretization methods

Finite Volume (FV)

• locally conservative

• suitable for convection dominated flow

• extension to higher orders is complicated

Finite Element (FE)

• straightforward extension to higher orders

• not locally conservative

• unstable for dominant convection on coarsemeshes

Discontinuous Galerkin (DG)

• combines advantages of FV and FE methods

• very suitable for adaptivity

• increased number of degrees of freedom

• reduced sparsity in the discrete system

FE

DG



Circumventing the drawbacks of DG...

add additional unknowns at the interfaces (hybridization*).

relax the coupling across interfaces.

eliminate element unknowns (static condensation).

DG HDG* HDG

Literature: Cockburn, Gopalakrishnan and Lazarov. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods forsecond order elliptic problems. 2009.



Hybrid Discontinuous Galerkin (HDG) Methods

DG methods with some (algorithmic) advantages

better sparsity structure (for higher orders)static condensationelement-based assembly

but: Implementation more involved!

Overview: Cockburn, Gopalakrishnan, Lazarov. Unified hybridization of discontinuousGalerkin, mixed and continuous Galerkin methods for second order elliptic problems.


Introduction Overview of the thesis

Overview of the thesis

A priori analysis for different flow model-problems.

Technical results for the hp analysis.

Suitable error estimators to drive adaptive algorithms.

Hybrid mortar methods for domain decomposition.

Numerical experiments.

Analysis of Hybrid Discontinuous Galerkin Methods forIncompressible Flow Problems

Von der Fakultät für Mathematik, Informatik undNaturwissenschaften der RWTH Aachen University zurErlangung des akademischen Grades eines Doktors der

Naturwissenschaften genehmigte

D i s s e r t a t i o n

vorgelegt von

Diplom-Ingenieur

Christian Waluga

aus Würselen

Berichter: Univ.-Prof. Dr. Herbert EggerUniv.-Prof. Dr. Wolfgang Dahmen

Tag der mündlichen Prüfung: 3. Februar 2012

Diese Dissertation ist auf den Internetseitender Hochschulbibliothek online verfügbar.

Updated and corrected version: May 23, 2013


Introduction Overview of the thesis

Overview of the thesis

A priori analysis for different flow model-problems.

Technical results for the hp analysis.

Suitable error estimators to drive adaptive algorithms.

Hybrid mortar methods for domain decomposition.

Numerical experiments.

Analysis of Hybrid Discontinuous Galerkin Methods forIncompressible Flow Problems

Von der Fakultät für Mathematik, Informatik undNaturwissenschaften der RWTH Aachen University zurErlangung des akademischen Grades eines Doktors der

Naturwissenschaften genehmigte

D i s s e r t a t i o n

vorgelegt von

Diplom-Ingenieur

Christian Waluga

aus Würselen

Berichter: Univ.-Prof. Dr. Herbert EggerUniv.-Prof. Dr. Wolfgang Dahmen

Tag der mündlichen Prüfung: 3. Februar 2012

Diese Dissertation ist auf den Internetseitender Hochschulbibliothek online verfügbar.

Updated and corrected version: May 23, 2013


HDG for incompressible flow

Outline

1 Introduction

2 A Hybrid Discontinuous Galerkin MethodPreliminariesPoisson problemStokes problemNavier-Stokes problem


4 Conclusions


HDG for incompressible flow Preliminaries

Triangulations of the domain Ω

Hybrid meshes with a bounded level of nonconformity (shape regular, quasi-uniform)

T1 T2 T3

T4 T5

T6

T7

T8T9

T10

T11 T12

E0,8E0,9E0,10

E′0,10

E0,1

E′0,1 E0,2

E0,3

E0,5

E0,6

E′0,6

E1,2 E2,3

E3,4

E4,7

E4,5

E5,6

E6,7E7,8

E7,12

E2,12E1,10 E2,11

E11,12

E8,9

E8,12E9,11

E10,11

E9,10

Collection of elements: Th = T1, T2, . . .

Boundary + interior facets: Eh := E0,1, E0,2, . . . , E1,2, E2,3, . . .


HDG for incompressible flow Preliminaries

Hybrid Discontinuous Galerkin?

Galerkin?

seek a weak solution in a finite dimensional spacesolution can be computed by solving a linear system of equations

AU = F A : stiffness matrix, U : unknowns, F : right hand side

Discontinuous?

find a discrete solution uh ∈ Vh on Thspace Vh consists of piecewise discontinuous polynomial functions.

Vh =vh ∈ L2(Ω) : vh|T ∈ Pk(T ), T ∈ Th

Hybrid?

also approximate the trace uh ∈ Vh on Eh

Vh =vh ∈ L2(Eh) : vh|E ∈ Pk(E), E ∈ Eh, vh = 0 on ∂Ω


HDG for incompressible flow Poisson problem

HDG for Poisson

−∆u = f in Ω and u = 0 on ∂Ω.

Discrete problem

Find (uh, uh) ∈ Vh × Vh, such that

ah(uh, uh; vh, vh) = fh(vh, vh) for all (vh, vh) ∈ Vh × Vh.

where we define

ah(uh, uh; vh, vh) :=∑

T∈Th

(∫T∇uh ·∇vh dx−

∫∂T

∂nuh · (vh − vh) ds

−∫∂T

(uh − uh) · ∂nvh ds+

∫∂T

γTk2T

hT(uh − uh) · (vh − vh) ds

)fh(vh, vh) :=

∑T∈Th

∫Tf · vh dx

Consistency For u ∈ H10 (Ω) ∩H2(Th), there holds

ah(u, u|Eh; vh, vh) = fh(vh, vh) for all (vh, vh) ∈ Vh × Vh.


HDG for incompressible flow Poisson problem

A priori analysis for Poisson

Theorem (Coercivity):

For γT sufficiently large, there holds

ah(vh, vh; vh, vh) ≥ 12‖(vh, vh)‖21,h, ∀ (vh, vh) ∈ Vh × Vh.

Energy norm:

‖(vh, vh)‖21,h :=∑

T∈Th

(‖∇vh‖2T + γT

k2T

hT|vh − vh|2∂T

)1/2.

Remarks:

Optimal γT is explicitly given by sharp trace inverse estimates.

Existence, uniqueness and stability bounds of discrete solution (Lax-Milgram).

Standard arguments and approximation results yield order-optimal error estimates.

Literature: Arnold et. al. Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. 2002.Warburton, Hesthaven. On the constants in hp-finite element trace inverse inequalities, 2003.Burman, Ern. Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations, 2007.Egger. A class of hybrid mortar finite element methods for interface problems with non-matching meshes, 2009.


HDG for incompressible flow Stokes problem

HDG for Stokes

System of equations for u = [u1, . . . , ud] with incompressibility constraint.

−∆u + ∇p = f and divu = 0 in Ω and u = 0 on ∂Ω.

Finite dimensional spaces

discrete velocity (uh, uh): Vh := Vdh (on Th), Vh := Vd

h (on Eh)

discrete pressure ph: Qh :=qh ∈ L2

0(Ω) : qh|T ∈ Pk−1(T )

Discrete (saddle-point) problem

Find (uh, uh) ∈ Vh × Vh and ph ∈ Qh, such that

ah(uh, uh;vh, vh) + bh(vh, vh; ph) = fh(vh, vh) for all (vh, vh) ∈ Vh × Vh,

bh(uh, uh; qh) = 0 for all qh ∈ Qh.

The bilinear form associated with the incompressibility constraint is defined as

bh(uh, uh; qh) :=∑

T∈Th

(−∫T

divuh · qh dx−∫∂T

(uh − uh) · n · qh ds)



A priori analysis for Stokes

The crucial part of the analysis is the following stability condition.

Theorem (discrete inf-sup stability):

There exists a constant β independent of the mesh and the polynomial degree k, such that

sup(vh,vh)∈Vh×Vh

bh(vh, vh; qh)

‖(vh, vh)‖1,h≥ βk−1/2‖qh‖0,h, ∀ qh ∈ Qh.

Remarks:

The proof is based on an argument due to Fortin.

We employ new hp-estimates for the L2-orthogonal projections:

ΠT : H1(T )→ Pk(T ) ΠE : H1(T )→ Pk(E).

The analysis applies to hybrid meshes (e.g. tri/quad or tet/hex) with hanging nodes.

Error estimates of such mixed methods depend on the approximation properties of the finiteelement spaces and the discrete inf-sup estimate.

Hence, the k-dependence is important for high order discretizations.

Literature: Fortin. Analysis of the Convergence of Mixed Finite Element Methods, 1977.Brezzi, Fortin. Mixed and Hybrid Finite Element Methods. Springer, 1991.Boffi et. al. Mixed finite elements, compatibility conditions, and applications, Springer, 2008.Egger, W. hp-Analysis of a Hybrid DG Method for Stokes Flow, 2011.



Comparison with related work

The discrete inf-sup constant is usually of the order k−s (optimal method: s = 0).

supvh∈Vh

bh(vh, qh)

‖vh‖Vh

≥ βk−s ‖qh‖Qh∀qh ∈ Qh,

Type Reference Suboptimality s Element types Balanced approx.Spectral [BM:99] 0 quad, hex yes

CG [AC:02] 0 quad yesCG [SS:96] ε quad, hex yesCG [S:98] 1/2 quad no

HDG [EW:11] 1/2 tri, quad, tet, hex yes

DG [T:02] d−12

quad, hex noDG [SST:03] 1 quad, hex yesCG [S:98] ( 3 ) tri no

Literature: [SS:96] Stenberg, Suri. Mixed finite element methods for problems in elasticity and Stokes flow, 1996.[S:98] Schwab. p- and hp- finite element methods: theory and applications in solid and fluid mechanics, 1998.[BM:99] Bernardi, Maday. Uniform inf-sup conditions for the spectral discretization of the Stokes problem, 1999.[AC:02] Ainsworth, Coggins. A uniformly stable family of mixed hp-finite elements with continuous pressures for incompressible flow.[T:02] Toselli. hp discontinuous Galerkin approximations for the Stokes problem, 2002.[SST:03] Schotzau, Schwab, Toselli. Mixed hp-DGFEM for incompressible flows, 2003.[EW:11] Egger, W. hp-Analysis of a Hybrid DG Method for Stokes Flow, 2011.



Convergence rates for Stokes

We can prove order-optimal convergence rates; i.e, for (u, p) ∈Hm+1(Th)×Hm(Th):

‖(u− uh,u|Eh− uh)‖1,h + 1√

k‖p− ph‖0,h

hm

km−1‖u‖m+1,Ω +

hm

km−1/2‖p‖m,Ω

k level velocity error rate pressure error rate

1

0 2.2052 · 101 − 9.9237 · 100 −1 1.1569 · 101 0.93 4.9250 · 100 1.012 5.8722 · 100 0.98 2.4036 · 100 1.033 2.9487 · 100 0.99 1.1837 · 100 1.02

2

0 3.3991 · 100 − 1.8315 · 100 −1 8.6146 · 10−1 1.98 4.6010 · 10−1 1.99

2 2.1494 · 10−1 2.00 1.1302 · 10−1 2.03

3 5.3539 · 10−2 2.01 2.7859 · 10−2 2.02

3

0 2.5261 · 10−1 − 1.7153 · 10−1 −1 3.1045 · 10−2 3.02 2.0734 · 10−2 3.05

2 3.8526 · 10−3 3.01 2.5606 · 10−3 3.02

3 4.8003 · 10−4 3.00 3.1885 · 10−4 3.01


HDG for incompressible flow Navier-Stokes problem

HDG for Navier-Stokes

Add nonlinear convective terms...

−ν∆u + u ·∇u + ∇p = f and divu = 0 in Ω and u = 0 on ∂Ω.

Discrete (nonlinear) problem


νah(uh, uh;vh, vh) + ch(uh, uh;uh, uh;vh, vh) + bh(vh, vh; ph) = fh(vh, vh),

bh(uh, uh; qh) = 0.

for all (vh, vh) ∈ Vh × Vh and all qh ∈ Qh.

The form associated with the convective terms is defined as

ch(wh, wh;uh, uh;vh, vh)

:=∑

T∈Th

(−∫Tuh · (wh ·∇vh) dx+

∫∂T

(wh · n) uh/uh · (vh − vh) ds

+ 12bh(wh, wh;uh · vh).

where uh/uh = f(wh, wh;uh, uh) denotes an upwind value.



What is upwinding?

Numerical scheme adapts to the direction of propagation of information in the flow field.

uh/uh :=

uh if wh · n ≤ 0,

uh if wh · n > 0.

wh

uh/uh = uh

uh/uh = uh

Literature: Reed and Hill. Triangular mesh methods for the neutron transport equation. 1973.Egger and Schoberl. A mixed-hybrid-discontinuous Galerkin finite element method for convection-diffusion problems. 2010.



Dealing with the nonlinearity...

The discrete Navier-Stokes problem is equivalent to a fixed point problem

(uh, uh) = Φh(uh, uh)

The operator Φh : (wh, wh)→ (uh, uh) is defined by the following discrete Oseen problem


νah(uh, uh;vh, vh) + ch(wh, wh;uh, uh;vh, vh) + bh(ph;vh, vh) = fh(vh, vh),

bh(qh;uh, uh) = 0.

for all (vh, vh) ∈ Vh × Vh and all qh ∈ Qh.

Remarks:

The fixed-point operator is well-defined.

Existence of fixed points (thus discrete solutions) by Leray-Schauder principle.

For small ‖f‖Ω, we also obtain

Uniqueness of a discrete solution (Banach)Order-optimal convergence rates.



Lid-driven cavity flow



Lid-driven cavity flow (ν = 1/100)

Comparison of a third order solution with reference data by Ghia et. al.

0 0.5 10

0.2

0.4

0.6

0.8

1

u1(x = 0.5)

y

Literature: Ghia et. al. High-Re Solutions for Incompressible Flow using the Navier-Stokes Equations and a Multigrid Method, 1982.



Lid-driven cavity flow (ν = 1/1000)

Comparison of a third order solution with reference data by Ghia et. al.

−0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

u1(x = 0.5)

y

Literature: Ghia et. al. High-Re Solutions for Incompressible Flow using the Navier-Stokes Equations and a Multigrid Method, 1982.


A posteriori error estimators and adaptivity

Outline

1 Introduction


3 A posteriori error estimators and adaptivityError estimationAn adaptive algorithmNumerical results

4 Conclusions


A posteriori error estimators and adaptivity Error estimation

Error estimation

Exact solution of relevant problems is usually unknown.

The jump error estimator ηJ is given by a sum of local contributions

ηJ :=(∑

T∈Th η2T

)1/2where η2

T := γTk2T

hT

∣∣uh − uh

∣∣2L2(∂T )

Estimator bounds the error from below (efficiency) and above (reliability)

ηJ ≤ ‖(u− uh,u− uh)‖1,h + ‖p− ph‖0,h k ηJ + osc

Reliability proved for HDG Methods for Poisson and Stokes.

ηT can be used as error indicator to drive adaptive refinement strategies.

Literature: Egger, W. hp-Analysis of a Hybrid DG Method for Stokes Flow, 2011.


A posteriori error estimators and adaptivity An adaptive algorithm

The adaptive algorithm

Given an initial mesh T 0h of Ω, we invoke the algorithm: For i = 0, 1, . . .

SOLVE→ ESTIMATE→ MARK→ REFINE,

(SOLVE) The discrete problem on T ih is solved by the HDG Method.

(ESTIMATE) For each T ∈ T ih , we compute the local error indicator ηT .

(MARK) Dorfler: obtain minimal M(Th) ⊆ Th, such that∑T∈M(Th)

η2T ≥ θ2

∑T∈Th

η2T , here: θ = 0.5

(REFINE) Refine T ih by subdividing all marked triangles in M(Th) into four

similar ones. Ensure that the maximal difference of the refinementlevels between two neighboring elements is one (1-irregular).

Literature: Dorfler. A convergent adaptive algorithm for Poissons equation, 1996.


A posteriori error estimators and adaptivity Numerical results

Stokes in L-shape domain

Exact solution due to Verfurth exhibits corner singularity.

Initial mesh and adaptively refined meshes after 20 refinement steps.

initial k = 1

k = 2 k = 3

Literature: Verfurth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Teubner, 1996.Egger, W. hp-Analysis of a Hybrid DG Method for Stokes Flow, 2011.


A posteriori error estimators and adaptivity Numerical results

Convergence rates and effectivity index

102 103 104

10−1

100

number of elements (k=1)

energy error

ηJ estimate

102 103

10−2

10−1

100


energy error

ηJ estimate

102 103

10−2

100


energy error

ηJ estimate

102 103 1040

2

4

6

8

10


effectivity index

102 1030

2

4

6

8

10


effectivity index

102 1030

2

4

6

8

10


effectivity index


Conclusions

Outline

1 Introduction



4 Conclusions


Conclusions

Conclusions

We derived analyzed HDG methods for a class of incompressible flow problems.

Some technical results may also be useful for related work.

HDG methods are promising for high order simulations.

Reliable, efficient and simple error estimators.

Outlook: HDG methods for other interesting physical models.


Financial support from theDeutsche Forschungsgemeinschaft

through grant GSC 111is gratefully acknowledged

Conclusions

Selected references

Discontinuous Galerkin (DG)

Arnold et. al. Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. 2002.

Di Pietro, Ern. Mathematical Aspects of Discontinuous Galerkin Methods. 2011.

Cockburn et. al. A locally conservative LDG method for the incompressible Navier-Stokes equations. 2005.

Girault et. al. A discontinuous Galerkin method with non-overlapping domain decomposition for theStokes and Navier-Stokes problems. 2005.

Hybrid Discontinuous Galerkin (HDG)

Cockburn et. al. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methodsfor second order elliptic problems. 2009.

Egger and Schoberl. A mixed-hybrid-discontinuous Galerkin finite element method for convection-diffusionproblems. 2010.

Nguyen et. al. An implicit high-order hybridizable discontinuous Galerkin method for the incompressibleNavier-Stokes equations. 2010.

Cockburn et. al. Analysis of HDG methods for Stokes flow. 2011.

Egger and W. hp-analysis of a hybrid DG method for Stokes flow. 2011.


Documents

Analysis of Hybrid Discontinuous Galerkin Methods for ...€¦ · Analysis of Hybrid Discontinuous Galerkin Methods for Incompressible Flow Problems Christian Waluga1 advised by Prof