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Numerical Methods for Scalar
Convection-Dominated Problems
Volker John
Winter Semester 2013/14
Contents
1 Introduction 2
2 Analysis of Two-Point Boundary Value Problems 32.1 The Model Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Existence and Uniqueness of the Solution of the Model Problem . . . 7
2.2.1 Investigation of the Differential Equation (2.5) . . . . . . . . 82.2.2 Investigation of the Two-Point Boundary Value Problem (2.5),
(2.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Maximum Principle and Stability . . . . . . . . . . . . . . . . . . . . 14
3 Finite Difference Methods 193.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Classical Convergence Theory for Central Difference Schemes . . . . 213.3 Upwind Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Uniformly Convergent Methods . . . . . . . . . . . . . . . . . . . . . 35
3.4.1 Sophisticated Artificial Diffusion . . . . . . . . . . . . . . . . 353.4.2 Layer-Adapted Grids . . . . . . . . . . . . . . . . . . . . . . . 38
4 Weak Solution Theory 424.1 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 Finite Element Methods (FEM) 485.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 The Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . 485.3 Stabilized Finite Element Methods . . . . . . . . . . . . . . . . . . . 51
5.3.1 Petrov–Galerkin Methods and Upwind Methods . . . . . . . 525.3.2 The Streamline-Upwind Petrov–Galerkin (SUPG) Method . . 52
6 Outlook 65
1
Chapter 1
Introduction
Remark 1.1 On the importance of scalar convection-dominated problems and thefundamental problem in their numerical simulation. Scalar convection-dominatedproblems arise in many models of processes from nature and technical applications.This situation will be always happen if a species (dissolved or as particle) or aphysical quantity, e.g., the temperature, is transported by a flow field. Often,further processes besides the flow field are present in real systems, like chemicalreactions.
Solutions of scalar convection-dominated problems possess generally very smallstructures or scales. These scales are important features of the solution. Numericalmethods for the simulation of scalar convection-dominated problems are based ingeneral on a decomposition of the given domain with grids. These grids are oftennot sufficiently fine to resolve the small scales of the solution, i.e., the small scalescannot be even represented on the given grids. In this situation, it turns out thatstandard numerical methods fail completely to produce useful approximations ofthe solutions. Special numerical methods are necessary. In this course, the mostimportant special methods will be introduced and discussed. 2
Remark 1.2 Contents of the course, state of the art of the research. This coursestudies properties of the solution of scalar convection-dominated problems and theirnumerical approximation. It will start with investigations of problems in one di-mension, since
• the main difficulties arise already in one dimension,• the analysis for one-dimensional problems is comparable simple,• one can construct for one-dimensional problems numerical methods which pro-
duce, in a certain sense, perfect numerical solutions.
Higher-dimensional problems will be studied towards the end of the course. Thebasic approach consists in trying to transfer the ideas for one-dimensional problemsto higher dimensions. It turns out that
• one obtains much better solutions compared with the numerical solutions com-puted with standard methods,• however, the solutions are by far not perfect, often they are not even good, e.g.,
see Augustin et al. (2011); John and Schumacher (2014).
Methods for the computation of good numerical solutions for scalar convection-dominated problems in higher dimensions are an active field of research. 2
Remark 1.3 Literature. The best monograph on this topic is Roos et al. (2008).The course follows in large parts this monograph. A shorter survey paper that canbe recommended is Stynes (2005). 2
2
Chapter 2
Analysis of Two-PointBoundary Value Problems
2.1 The Model Problem
Definition 2.1 Linear two-point boundary value problem. A linear two-point boundary value problem has the form
− εu′′ + b(x)u′ + c(x)u = f(x), for x ∈ (d, e), (2.1)
with the boundary conditions
αdu(d)− βdu′(d) = γd,αeu(e)− βeu′(e) = γe.
(2.2)
Here are b, c, f ∈ C([e, d]), 0 < ε ∈ R, and the constants αd, αe, βd, βe, γd, γe aregiven. 2
Remark 2.2 Importance of two-point boundary value problems. The boundaryvalue problem (2.1), (2.2) is the simplest model for processes which have diffusionand transport.
An example from Goering (1977) is as follows. Consider a flow reactor with con-stant temperature in which there is a continuous inflow of a species (reactant) andan outflow of a product. Then, the concentration c(t, x, y, z) = c(t,x), [kmol/m3], ofthe species in the reactor is the solution of the following partial differential equation
∂c
∂t+ div (cu)︸ ︷︷ ︸
convection
−div (D gradc)︸ ︷︷ ︸diffusion
= r(c),
where u(t,x), [m/s], is the velocity, r(c(t,x)), [kmol/(m3s)], is a function whichmodels the reaction, and D(t,x), [m2/s], is the diffusion coefficient. If the reactorworks stationary, i.e., the temporal changes are very slow and they are negligible,if the parameters D and u are constant, and if the concentration changes only inx-direction, then one obtains an ordinary differential equation for c(x)
−Dc′′(x) + uc′(x) = r(c(x)).
Let x ∈ [0, L], where L, [m], is the length of the reactor. With the dimensionlessquantities
ξ :=x
L, γ :=
c
c0,
3
where c0, [kmol/m3], is a reference concentration, one gets a dimensionless ordinarydifferential equation. Using the chain rule gives
dγ(ξ)
dξ=d(c(x)/c0)
dx
dx
dξ= L
c′(x)
c0and
d2γ(ξ)
dξ2= L2 c
′′(x)
c0.
Inserting into the differential equation yields, if u 6= 0,
− 1
Peγ′′(ξ) + γ′(ξ) = ρ(γ(ξ)), ξ ∈ (0, 1), with Pe :=
uL
D, ρ =
L
uc0r.
The dimensionless number Pe is called Peclet number. For completing the problem,appropriate (dimensionless) boundary conditions at ξ ∈ 0, 1 have to be prescribed.
2
Figure 2.1: Jean Claude Eugene Peclet (1793 – 1857).
Remark 2.3 Names for the individual terms in (2.1). Based on the applicationdescribed in Remark 2.2, the terms in (2.1) are called as follows:
• −εu′′ – diffusive term,• b(x)u′ – convective term, advective term, transport term,• c(x)u – reactive term.
The model problem (2.1), (2.2) is called convection–diffusion(–reaction) problem, ifb(x) 6≡ 0.
The Peclet number describes the ratio of convection and diffusion. If this ratiois high, then the numerical solution of (2.1), (2.2) might be difficult. 2
Definition 2.4 Boundary conditions. Let γd, γe ∈ R, αd, αe ∈ R \ 0. Bound-ary conditions of the form:
1.u(d) = γd, u(e) = γe
are called boundary condition of first kind or Dirichlet boundary condition,2.
u′(d) = γd, u′(e) = γe
are called boundary condition of second kind or Neumann boundary condition,
4
3.αdu(d) + u′(d) = γd, αeu(e) + u′(e) = γe
are called boundary condition of third kind or Robin1 boundary condition.
Dirichlet boundary conditions are
• most important in applications,• most difficult from the point of view of the analysis.
In this course, it will be concentrated on Dirichlet boundary conditions. 2
Figure 2.2: Left: Johann Peter Gustav Lejeune Dirichlet (1805 – 1859), right: CarlGottfried Neumann (1832 – 1925).
Remark 2.5 Normalization of a linear two-point boundary value problem.
• One can assume, without loss of generality, that x ∈ (0, 1). The original interval(d, e) is mapped to (0, 1) with the transform
x 7→ x− de− d
.
• One can also assume, without loss of generality, that homogeneous boundaryconditions γd = γe = 0 are prescribed. To this end, one subtracts from u(x) asmooth function ψ(x) which satisfies the original boundary conditions. If, e.g.,the original Dirichlet boundary conditions are given by
u(d) = γd, u(e) = γe,
then one can set
ψ(x) = γdx− ed− e
+ γex− de− d
andu∗(x) = u(x)− ψ(x).
It follows that u∗(x) is the solution of a two-point boundary value problem withhomogeneous Dirichlet boundary conditions.
2
1Victor Gustave Robin (1855 – 1897)
5
Definition 2.6 Model problem. The model problem has the form
Lu := −εu′′ + b(x)u′ + c(x)u = f(x) for x ∈ (0, 1), (2.3)
with the boundary conditions
u(0) = u(1) = 0. (2.4)
For the data of the problem it holds b, c, f ∈ C([0, 1]) and 0 < ε ∈ R. 2
Remark 2.7 Differential operator. In (2.3), L denotes a differential operator. Anoperator is a map between two function spaces. A linear operator is a linear mapA, defined on a linear space X, such that
A(αu+ βv) = αAu+ βAv
for all scalars α, β and all u, v ∈ X. A differential operator is an operator, which,if it is applied to appropriate functions, contains derivatives of these function. Forthe complete definition of an operator, its domain has to be given. 2
Example 2.8 Standard example. The boundary value problem
−εu′′ + u′ = 1 on (0, 1), u(0) = u(1) = 0
has the solution
u(x) = x−exp
(− 1−x
ε
)− exp
(− 1
ε
)1− exp
(− 1
ε
) .
The smaller the parameter ε, the steeper becomes the solution at the right boundary,see Figure 2.3. This part of the solution is called (boundary) layer. Such strongchanges of the solutions in a very small part of the domain lead to difficulties in thenumerical approximation of the solution.
Figure 2.3: Solution of Example 2.8 for ε = 0.1 (left) and ε = 0.0001 (right).
2
Remark 2.9 Transform of the model problem to a symmetric problem. Let b(x)be sufficiently smooth. If one defines
u(x) := u(x) exp
(− 1
2ε
∫ x
0
b(ξ) dξ
), x ∈ [0, 1],
6
then one can transform (2.3), (2.4) to the symmetric problem
−εu′′(x) + c(x)u(x) = f(x), x ∈ (0, 1), u(0) = u(1) = 0,
where
c(x) :=1
4εb2(x)− 1
2b′(x) + c(x), f(x) := f(x) exp
(− 1
2ε
∫ x
0
b(ξ) dξ
),
exercise. The weak or variational formulation of this problem, see Section 4.1,contains only symmetric bilinear forms. 2
Definition 2.10 Reduced problem, reduced solution. The reduced problemis obtain by setting formally ε = 0, yielding
L0u0 := b(x)u′0 + c(x)u0 = f(x), for x ∈ (0, 1).
The Dirichlet boundary condition has to be set at the boundary where the con-vection comes from, i.e., where the inflow is situated. In the case b(x) > 0 for allx ∈ [0, 1], the problem has the boundary condition
u0(0) = 0,
and for b(x) < 0 for all x ∈ [0, 1], the boundary condition is
u0(1) = 0.
The solution of the reduced problem is called reduced solution. 2
Example 2.11 Reduced problem for Example 2.8. The reduced problem for Ex-ample 2.8 has the form
u′0 = 1 in (0, 1), u0(0) = 0.
Its solution is u0(x) = x.It follows that the solution of the non-reduced problem from Example 2.8 is the
sum of the solution of the reduced problem and another term, which is responsiblefor the fulfillment of the second boundary condition. 2
2.2 Existence and Uniqueness of the Solution ofthe Model Problem
Remark 2.12 The model problem. For the investigation of the unique solvabilityof the two-point boundary value problem (2.3), (2.4) is the value of ε > 0 notimportant. After having divided the equation by ε and a renaming of the data, onecan consider the problem
Lu := −u′′(x) + b(x)u′(x) + c(x)u(x) = f(x), for x ∈ (0, 1), (2.5)
with the boundary conditions
u(0) = u(1) = 0. (2.6)
2
Definition 2.13 Classical solution. A function u(x) is called classical solutionof (2.5), (2.6), if
• u ∈ C2(0, 1) ∩ C([0, 1]),• u(x) satisfies (2.5) identically,• u(x) satisfies the boundary conditions (2.6).
2
7
2.2.1 Investigation of the Differential Equation (2.5)
Remark 2.14 Contents of this section. A classical solution of (2.5) has to satisfythe first two properties of Definition 2.13. This topic is studied in this section. Thenecessary tools are known already from the theory of ordinary differential equations,e.g., see Numerical Mathematics I. 2
Definition 2.15 Linearly independent functions. Two functions u1(x) andu2(x), both defined on the interval (a, b), are called linearly independent if from
c1u1(x) + c2u2(x) = 0 for all x ∈ (a, b)
it follows that c1 = c2 = 0. They are called linearly dependent, if they are notlinearly independent. 2
Remark 2.16 Wronski determinant. If two linearly dependent functions, whichare defined on (a, b), are continuously differentiable, then it follows from the condi-tion for the linear dependence also that
c1u′1(x) + c2u
′2(x) = 0 for all x ∈ (a, b).
Hence, with u1(x), u2(x) there are also u′1(x), u′2(x) linearly dependent. It followsthat the homogeneous linear system of equations(
u1(x) u2(x)u′1(x) u′2(x)
)(c1c2
)=
(00
)has a nontrivial solution for all x ∈ (a, b). Thus, the so-called Wronski determinant
W (x) := det
(u1(x) u2(x)u′1(x) u′2(x)
)vanishes for all x ∈ (a, b). Note that the reverse statement is not true: the Wron-ski determinant might vanish for all x ∈ (a, b) but the two functions are linearlyindependent. One can find an example for this case in Emmrich (2004). 2
Lemma 2.17 Linear independence of two solutions of the homogeneousdifferential equation. Let x0 ∈ (0, 1) be arbitrary. Two classical solutions de-fined on (0, 1) of the homogeneous second order linear differential equation withcontinuous coefficients (2.5) are linearly independent if and only if the correspond-ing Wronski determinant does not vanish for x0.
Proof: The proof should be known from the course on the theory of ordinary dif-ferential equations or from Numerical Mathematics I. It is given just for completeness ofpresentation.
i) W (x0) 6= 0 =⇒ linear independence. Let u1(x) and u2(x) be classical solutions of
−u′′(x) + b(x)u′(x) + c(x)u(x) = 0, x ∈ (0, 1),
where b, c ∈ C(0, 1). One obtains for the Wronski determinant, applying the product rule,
W ′(x) =(u1(x)u′2(x)− u′1(x)u2(x)
)′= u′1(x)u′2(x) + u1(x)u′′2 (x)− u′′1 (x)u2(x)− u′1(x)u′2(x)
= u1(x)u′′2 (x)− u′′1 (x)u2(x)
= u1(x)(b(x)u′2x(x) + c(x)u2(x)
)− u2(x)
(b(x)u′1x(x) + c(x)u1(x)
)= b(x)
(u1(x)u′2x(x)− u′1x(x)u2(x)
)+ c(x) (u1(x)u2(x)− u1(x)u2(x))
= b(x)W (x).
8
Figure 2.4: Left: Joseph Marie Wronski (1758 – 1853), right: Joseph Liouville (1809– 1882).
Hence, the Wronski determinant solves the homogeneous first order linear differentialequation
y′(x) = b(x)y(x).
The general solution of this differential equation is given by the Liouville formula. Applyingthis formula to the differential equation for W (x), then one gets for every x0 ∈ (0, 1) that
W (x) = W (x0) exp
(∫ x
x0
b(ξ) dξ
), x ∈ (0, 1).
Since the exponential takes only positive values, it follows that the Wronski determinantis not equal to zero (equal to zero) if and only if it is not equal to zero (equal to zero) atan arbitrary point x0 ∈ (0, 1). In particular, one has with Remark 2.16 that in the caseW (x0) 6= 0 the functions u1(x) and u2(x) are linearly independent.
ii) Linear independence =⇒ W (x0) 6= 0. This statement is proved by contradiction.Assume that u1(x) and u2(x) are two linearly independent solutions and the Wronskideterminant vanishes in x0 ∈ (0, 1). Then, it follows from part i) that it vanishes evenin the whole interval (0, 1). Hence, there is a nontrivial solution of the linear system ofequations (
u1(x) u2(x)u′1(x) u′2(x)
)(c1c2
)=
(00
).
Consider the functionv(x) := c1u1(x) + c2u2(x).
By the assumption, it is v(x0) = v′(x0) = 0. The function v(x) is a solution of thedifferential equation, because this equation is linear. Hence, v(x) solves the followinginitial value problem
−v′′(x) + b(x)v′(x) + c(x)v(x) = 0, x ∈ (x0, 1), v(x0) = v′(x0) = 0
for every x0 ∈ (0, 1). Applying the Theorem of Picard–Lindelof, one shows that this initial
value problem has only the trivial solution. Hence, one gets that v(x) = 0 for all x ∈ (0, 1).
This result contradicts the linear independence of u1(x) and u2(x).
Theorem 2.18 Super position principle. Consider the homogeneous linear dif-ferential equation
−u′′ + b(x)u′ + c(x)u = 0, x ∈ (0, 1),
9
Figure 2.5: Left: Charles Emile Picard (1856 – 1941), right: Ernst Leonard Lindelof(1870 – 1946).
with coefficients b, c ∈ C([0, 1]). Then, there are two linearly independent solutionsin C2([0, 1]) and every classical solution can be represented as a linear combinationof them.
Proof: The proof should be known from the course on ordinary differential equationsor Numerical Mathematics I. It is presented here just for completeness.
i) Existence and uniqueness of the solution of the initial value problem. One can rewritethe second order differential equation (2.5) as an equivalent system of first order differentialequations
d
dx
(u(x)u′(x)
)=
(0 1c(x) b(x)
)(u(x)u′(x)
).
From the Theorem of Picard–Lindelof it follows that for given initial conditions u(x0),u′(x0), x0 ∈ (0, 1), there is a uniquely determined solution (u(x), u′(x)) of the initialvalue problem in [0, 1]. Both components of the solution are in C1([0, 1]), from whichu ∈ C2([0, 1]) follows.
ii) Existence of two linearly independent solutions. Let u1(x) be the solution with theinitial conditions u(x0) = 1, u′(x0) = 0 and u2(x) be the solution with the initial conditionsu(x0) = 0, u′(x0) = 1. One obtains for the Wronski determinant that
W (x0) = u1(x0)u′2(x0)− u′1(x0)u2(x0) = 1.
With Lemma 2.17 it follows that u1(x) and u2(x) are linearly independent.iii) Representation of each classical solution as a linear combination. Each linear
combinationu(x) = c1u1(x) + c2u2(x), c1, c2 ∈ R,
is a solution of the differential equation. Consider a function of the form
u(x) = u(x0)u1(x) + u′(x0)u2(x), x ∈ [a, b], u(x0), u′(x0) ∈ R.
This function satisfies the initial value problem with the initial conditions u(x0), u′(x0),
where u(x0), u′(x0) are two arbitrary real numbers. From the uniqueness of the solution
of the initial value problem, it follows that every solution of the differential equation can
be represented in this form.
Theorem 2.19 Classical solution of the inhomogeneous differential equa-tion. Consider the inhomogeneous linear differential equation
−u′′ + b(x)u′ + c(x)u = f(x), x ∈ (0, 1),
10
with b, c, f ∈ C([0, 1]). Then, there exists a classical solution up(x), the so-calledparticular solution, and every classical solution can be represented in the form
u(x) = c1u1(x) + c2u2(x) + up(x), c1, c2 ∈ R,
where u1(x), u2(x) is a system of two linearly independent solutions (fundamentalsystem) of the corresponding homogeneous differential equation. It is u ∈ C2([0, 1]).
Proof: The proof of this theorem follows from the global (existence and uniqueness)
Theorem of Picard–Lindelof, exercise.
2.2.2 Investigation of the Two-Point Boundary Value Prob-lem (2.5), (2.6)
Example 2.20 Non-uniqueness of the solution of the two-point Dirichlet boundaryvalue problem. Consider the differential equation
−u′′(x)− u(x) = 0.
The general solution of this linear homogeneous differential equation has the form
u(x) = c1 cosx+ c2 sinx, c1, c2 ∈ R.
• Let the boundary conditions be given by
u(0) = u(π/2) = 1,
then there is the unique solution u(x) = cosx+ sinx.• If the boundary conditions are prescribed by
u(0) = u(π) = 1,
then there is no solution of the boundary value problem, since one gets that atthe same time c1 = 1 and c1 = −1 have to hold.
• With the boundary conditions
u(0) = 1, u(π) = −1,
there are infinitely many solution, because from the boundary conditions onegets only that c1 = 1. The value for c2 can be chosen arbitrarily.
This example shows that even in simple cases there might be not a unique solution ofthe two-point boundary value problem (2.5), (2.6). It will be shown in the followingthat the coefficients of this problem have to satisfy certain conditions such that aunique solution is guaranteed. 2
Theorem 2.21 Existence and uniqueness of a solution of the model prob-lem with homogeneous right-hand side. Consider the two-point boundaryvalue problem (2.5), (2.6) with b ∈ C1([0, 1]), c ∈ C([0, 1]), and f(x) ≡ 0. If for allx ∈ (0, 1)
c(x) :=1
4b2(x)− 1
2b′(x) + c(x) ≥ 0, (2.7)
then (2.5), (2.6) has only the trivial solution.
Proof: It is obvious that u(x) ≡ 0 is a solution of the considered problem.Proof by contradiction. Assume that u(x) 6≡ 0 is another classical solution. From
Theorem 2.19 it follows that u ∈ C2([0, 1]). Applying the transform from Remark 2.9, onegets the problem
−u′′(x) + c(x)u(x) = 0, x ∈ (0, 1), u(0) = u(1) = 0.
11
One solution of this problem is u(x) ≡ 0. Let u(x) be another solution. Multiplying thisequation with the second solution and applying integration by parts yields
0 =
∫ 1
0
(−u′′(x)u(x) + c(x)u2(x)
)dx
= −u′′(1)u(1) + u′′(0)u(0) +
∫ 1
0
((u′(x)
)2+ c(x)u2(x)
)dx
=
∫ 1
0
((u′(x)
)2+ c(x)u2(x)
)dx,
since u(x) vanishes at the boundary. Since c(x) ≥ 0, the term in the integral is nonnegative.Hence, this term must vanish. It follows that (u′(x))
2 ≡ 0, i.e., u′(x) ≡ 0, from what onegets that u(x) has to be a constant. The continuity of u(x) gives in combination with theboundary conditions that u(x) ≡ 0. Hence
u(x) = u(x) exp
(1
2
∫ x
0
b(ξ) dξ
)≡ 0,
in contradiction to the assumption.
Remark 2.22 Constant coefficients. In the special case of constant coefficients,condition (2.7) reduces to
D :=b2
4+ c ≥ 0.
It is possible to describe the solution of the two-point boundary value problem alsofor the case D < 0, exercise. 2
Remark 2.23 Different criterion for the uniqueness of the solution of the fullyhomogeneous problem. Consider the two-point boundary value problem (2.5), (2.6)with homogeneous right-hand side. Let u1(x), u2(x) be two linearly independentsolutions of the equation and denote
R := det
(u1(0) u2(0)u1(1) u2(1)
).
The general solution of the homogeneous differential equation has the form
u(x) = c1u1(x) + c2u2(x).
The parameters has to be determined from the boundary conditions
0 = c1u1(0) + c2u2(0), 0 = c1u1(1) + c2u2(1),
which is equivalent to the solution of the linear system of equations(u1(0) u2(0)u1(1) u2(1)
)(c1c2
)=
(00
).
The solution of this system is unique (c1 = c2 = 0) if and only if R 6= 0. Exactlyin this case there is only the trivial solution of the fully homogeneous two-pointboundary value problem. 2
Remark 2.24 The inhomogeneous two-point boundary value problem. Considerthe two-point boundary value problem (2.5), (2.6) with inhomogeneous right-handside. Let u1(x), u2(x) be two linearly independent solutions of the correspondinghomogeneous differential equation and
A(x) := det
(u1(0) u2(0)u1(x) u2(x)
), B(x) := det
(u1(x) u2(x)u1(1) u2(1)
).
For the investigation of the boundary value problem with inhomogeneous right-hand side, Green’s function will be used. 2
12
Definition 2.25 Green2’s function. The function Γ(x, ξ) is called Green’s func-tion for the homogeneous two-point boundary value problem Lu = 0, u(0) = u(1) =0, if:
1. Γ(x, ξ) is continuous on the square Q := (x, ξ) : x, ξ ∈ [0, 1].2. In both of the triangles
Q1 := (x, ξ) : 0 ≤ ξ ≤ x ≤ 1, Q2 := (x, ξ) : 0 ≤ x ≤ ξ ≤ 1
there exist continuous partial derivatives ∂xΓ(x, ξ) and ∂xxΓ(x, ξ).3. For fixed ξ ∈ I = (0, 1) is Γ(x, ξ), as a function of x, a solution of LΓ = 0 forx 6= ξ, x ∈ I.
4. On the diagonal x = ξ, the first derivative has a jump of the form
∂xΓ(x+ 0, x)− ∂xΓ(x− 0, x) = −1, 0 < x < 1.
5. Γ(0, ξ) = Γ(1, ξ) = 0 for all ξ ∈ (0, 1).
2
Theorem 2.26 Existence and uniqueness of the solution of the modelproblem with inhomogeneous right-hand side. Consider the model problem(2.5), (2.6) with b, c, f ∈ C([0, 1]). If the corresponding fully homogeneous two-point boundary value problem has only the trivial solution, then there is exactly oneclassical solution of the two-point boundary value problem (2.5), (2.6). This solutionhas the form
u(x) =
∫ 1
0
Γ(x, ξ)f(ξ) dξ
with Green’s function
Γ(x, ξ) =1
R W (ξ)
A(ξ)B(x) for 0 ≤ ξ ≤ x ≤ 1,A(x)B(ξ) for 0 ≤ x ≤ ξ ≤ 1.
Proof: Sketch. Direct calculations show that Γ(x, ξ) is a Green’s function. Similarly,
a direct calculation shows that u(x) is a solution of the two-point boundary value problem,
exercise. Hence, there exists a solution. The uniqueness follows by assuming that there is a
second solution. The difference of both solutions satisfies the fully homogeneous problem.
But by the assumption, this problem has only the trivial solution. Hence, the second
solution is the same solution as the solution given in the theorem.
Theorem 2.27 Existence and uniqueness of the solution of the modelproblem with homogeneous right-hand side, reverse statement of Theo-rem 2.26. If the inhomogeneous two-point boundary value problem (2.5), (2.6) hasfor a right-hand side f(x) ∈ C([0, 1]) exactly one classical solution, then there isonly the trivial solution for the corresponding fully homogeneous two-point boundaryvalue problem.
Proof: Let u(x) be the unique classical solution of the inhomogeneous two-point
boundary value problem for f(x) and let uhom(x) be a nontrivial solution of the fully
homogeneous two-point boundary value problem. From the linearity of the problem, it
follows that u(x)+uhom(x) is a classical solution of the two-point boundary value problem
for the same right-hand side f(x), which is a contradiction to the assumed uniqueness of
this solution.
2Georg Green (1793 – 1841)
13
Corollary 2.28 Existence and uniqueness of the solution of the modelproblem with arbitrary Dirichlet boundary conditions. Consider the modelproblem (2.5) with b ∈ C1([0, 1]), c, f ∈ C([0, 1]), and with the Dirichlet boundaryconditions u(0) = a, u(1) = b with a, b ∈ R. If for all x ∈ (0, 1) condition (2.7) issatisfied, then there is exactly one classical solution of the two-point boundary valueproblem.
Proof: Inhomogeneous Dirichlet boundary conditions can be transferred to the right-
hand side, see Remark 2.5. This transform is two times continuously differentiable and it
can be performed in such a way that the new right-hand side is continuous in [0, 1]. For
this transformed two-point boundary value problem with homogeneous Dirichlet boundary
conditions, one can apply the previous statements. It follows from Theorem 2.21 that the
fully homogeneous problem has only the trivial solution. With Theorem 2.26, one gets
that there is exactly one classical solution. Since the back transform is also two times
continuously differentiable, one obtains that there is exactly one solution of the two-point
boundary value problem with inhomogeneous Dirichlet boundary conditions.
Remark 2.29 Another sufficient condition for unique solvability of the fully homo-geneous problem. There are also other sufficient conditions than (2.7) for the fullyhomogeneous problem to possess only the trivial solution, e.g., see Corollary 2.37below. 2
2.3 Maximum Principle and Stability
Remark 2.30 Differential operator. In this section, L defined by
(Lu)(x) := −u′′(x) + b(x)u′(x) + c(x)u(x), x ∈ (0, 1),
denotes a linear differential operator. For b, c ∈ C([0, 1]), this operator maps obvi-ously from C2(0, 1) into C(0, 1).
This section proves a maximum principle for this operator. With this principle,the stability of the solution can be shown. Finally, a strong maximum principle willbe proved. 2
Lemma 2.31 First form of the maximum principle. Let b ∈ C([0, 1]) andc(x) = 0 for all x ∈ [0, 1]. Then it holds for each u ∈ C2(0, 1) ∩ C([0, 1]):
i) from (Lu)(x) ≤ 0 for all x ∈ (0, 1), it follows that u(x) ≤ maxu(0), u(1) forx ∈ [0, 1],
ii) from (Lu)(x) ≥ 0 for all x ∈ (0, 1), it follows that u(x) ≥ minu(0), u(1) forx ∈ [0, 1].
Proof: It is sufficient to prove statement i). Statement ii) follows by replacing u(x)with −u(x).
In the first step, it will be proved that the statement follows with the stronger assump-tion (Lu)(x) < 0 on (0, 1). Assume that the function u(x) takes its maximum not on theboundary but in the interior of the interval. Then, there exists an argument x0 ∈ (0, 1)with u′(x0) = 0 (local extremum) and u′′(x0) ≤ 0 (local maximum). It follows that
(Lu)(x0) = −u′′(x0) + b(x0)u′(x0) = −u′′(x0) ≥ 0,
which contradicts the assumption.In the next step, statement i) will be proved. Denote for δ, λ > 0
w(x) = δeλx, x ∈ [0, 1].
If λ is sufficiently large, more precisely if λ > maxx∈[0,1] b(x), then it follows for all x ∈ (0, 1)that
(Lw)(x) = −λ2w(x) + b(x)λw(x) = −λ (λ− b(x))w(x) < 0.
14
Using the linearity of the differential operator, one gets
(L(u+ w)) (x) = (Lu)(x) + (Lw)(x) < 0.
Applying the first part of the proof gives
u(x) + w(x) ≤ maxu(0) + w(0), u(1) + w(1).
Statement i) follows now for δ → 0.
Theorem 2.32 Maximum principle. Let b, c ∈ C([0, 1]) and c(x) in [0, 1] non-negative. Then, it holds for each u ∈ C2(0, 1) ∩ C([0, 1]) that:
i) from (Lu)(x) ≤ 0 for all x ∈ (0, 1), it follows that u(x) ≤ max0, u(0), u(1) forx ∈ [0, 1],
ii) from (Lu)(x) ≥ 0 for all x ∈ (0, 1), it follows that u(x) ≥ min0, u(0), u(1) forx ∈ [0, 1].
Proof: Again, statement ii) follows from statement i) by replacing u(x) with −u(x).Since u(x) is continuous in [0, 1], the set
M+ := x ∈ (0, 1) : u(x) > 0
is either empty or a union of open subintervals from (0, 1), see the course Calculus I. LetM+ = ∅, i.e., u(x) is in (0, 1) nonpositive. Then, statement i) is trivially satisfied.
Let M+ = (0, 1). Then it is for x ∈ (0, 1)
−u′′(x) + b(x)u′(x) ≤ −u′′(x) + b(x)u′(x) + c(x)u(x) = (Lu)(x) ≤ 0.
From Lemma 2.31 it follows that
u(x) ≤ maxu(0), u(1),
which proves statement i) also in this case.Consider now ∅ 6=M+ 6= (0, 1). It will be shown that M+ has to be arbitrarily close
to 0 or 1. Let (a0, b0) ⊆M+. If a0 6= 0 and u(a0) > 0, then it is, because of the continuityof u(x) that either u(0) > 0 or there exists a value 0 ≤ a1 < a0 with u(a1) = 0. An analogstatement holds true for b0. Thus, one can assume that a0 = 0 or u(a0) = 0 as well asb0 = 1 or u(b0) = 0. That means, (a0, b0) is chosen as large as possible. Altogether, thereare four cases to consider. From the assumption, it follows that for all x ∈ (a0, b0)
(Lu)(x) ≤ 0 =⇒ −u′′(x) + b(x)u′(x) ≤ −c(x)u(x) ≤ 0.
Now, one can apply again Lemma 2.31. One obtains for all x ∈ (a0, b0)
0 < u(x) ≤ maxu(a0), u(b0). (2.8)
Obviously, it is not possible that u(a0) = u(b0) = 0, because there would be a contradictionto (2.8). The case a0 = 0, b0 = 1 was already considered. Hence, there remain the casesa0 = 0 and u(b0) = 0 as well as u(a0) = 0 and b0 = 1.
So far, the following is proved: If the set M+ is not empty, then there are numbersa, b ∈ [0, 1] with a ≤ b, such that
M+ = (0, a) ∪ (b, 1),
where u(a) = 0 if a 6= 0, and u(b) = 0, if b 6= 1. Using in the next step that u(x) isbounded in [a, b] by zero from above and applying in the following step (2.8), one gets forx ∈ (0, 1)
u(x) ≤ max
maxx∈(0,a)
u(x), maxx∈(b,1)
u(x), 0
≤ max
maxu(0), u (a),maxu(b), u(1), 0
= max0, u(0), u(1).
15
Remark 2.33 Physical interpretation. The model problem (2.5), (2.6) can be writ-ten in the form
(Lu)(x) = f(x), u(0) = u0, u(1) = u1.
If (Lu)(x) ≤ 0 for all x ∈ (0, 1), i.e., f(x) ≤ 0 for all x ∈ (0, 1), then there areno sources of u(x) in (0, 1). The maximum principle says that if at least one ofthe boundary values is positive, u(x) takes its largest value at the boundary. Forinstance, if u(x) is a concentration and there are no sources of the concentration inthe domain, then there does not exist a local maximum of the concentration in thedomain which is larger than the concentration at the boundary. 2
Corollary 2.34 Inverse monotonicity, comparison principle. Let the as-sumption of Theorem 2.32 be satisfied and let u, v ∈ C2(0, 1)∩C([0, 1]) with u(0) ≤v(0) and u(1) ≤ v(1). If (Lu)(x) ≤ (Lv)(x) for all x ∈ (0, 1), then it follows thatu(x) ≤ v(x) for x ∈ [0, 1].
Proof: This statement follows by applying Theorem 2.32 to the difference (u− v)(x).
Theorem 2.35 Stability of the solution, continuous dependence on thedata. Consider the two-point boundary value problem (2.5), (2.6), with b, c, f ∈C([0, 1]). If c(x) is nonnegative in [0, 1], then it holds for each classical solutionu(x) the following estimate
‖u‖C([0,1]) ≤ Λ ‖f‖C([0,1]) ,
where the constant Λ > 0 depends on b(x), c(x), but not on f(x).
Proof: Set for λ > 0
w(x) := Beλx −A, x ∈ [0, 1],
withA := ΛB, B := ‖f‖C([0,1]) , Λ := eλ − 1 > 0,
where λ will be specified later. Then, using c(x) ≥ 0, it is for x ∈ (0, 1)
(Lw)(x) = −(λ2 − λb(x)− c(x)
)Beλx −Ac(x)
≤ −(λ2 − λb(x)− c(x)
)Beλx.
Now, λ is chosen such that(λ2 − λb(x)− c(x)
)eλx ≥ 1. This relation is satisfied if λ is
sufficiently large, e.g., if
λ ≥ maxx∈[0,1]
(b(x)
2+
√b2(x)
4+ c(x) + e−λx
),
One gets for all x ∈ (0, 1) that
(Lw)(x) ≤ −B = −‖f‖C([0,1]) .
It follows for all x ∈ (0, 1), using the definition of the norm in C([0, 1]), that
(L(±u+ w))(x) = ±f(x) + (Lw)(x) ≤ |f(x)| − ‖f‖C([0,1]) ≤ 0.
The application of the maximum principle gives
±u(x) + w(x) ≤ max0,±u(0) + w(0),±u(1) + w(1) = max0, w(0), w(1).
Hence, it is for all x ∈ (0, 1)
±u(x) ≤ max0, w(0), w(1) − w(x).
16
From eλx ≥ 1 it follows that
w(x) ≥ B −A = w(0), w(1) = Beλ −A,
i.e.,
|u(x)| ≤ max0, w(0), w(1) − w(x) ≤ max0, B −A,Beλ −A+A−B= maxA−B, 0, B(eλ − 1) = maxA−B, 0,ΛB = maxA−B, 0, A = A.
This inequality is the statement of the theorem.
Remark 2.36 Non-normalized problem. For the non-normalized two-point bound-ary value problem (2.1), (2.2) with Dirichlet boundary conditions u(d) = α, u(e) =β, one obtains analogously
‖u‖C([e,d]) ≤ Λ ‖f‖C([e,d]) + max |α| , |β| ,
where Λ depends also on e−d, but not on α, β, (Emmrich, 2004, Satz 2.5.4), exercise.One can see that this estimate is in fact a stability estimate, if one applies it
to the difference u(x) − u(x). Here, u(x) is the solution of the exact problem andu(x) is the solution of a problem with perturbed right-hand side f or perturbedboundary conditions α, β. It follows from the linearity of the problem that
‖u− u‖C([e,d]) ≤ Λ∥∥∥f − f∥∥∥
C([e,d])+ max
|α− α| ,
∣∣∣β − β∣∣∣ .Hence, changes in the solution depend continuously, in the norm of C([e, d]), on thedata of the problem. 2
Corollary 2.37 Uniqueness of the solution of the fully homogeneous prob-lem. Consider the two-point boundary value problem (2.5), (2.6) with b, c ∈ C([0, 1])and f(x) ≡ 0. If c(x) is nonnegative in [0, 1], then the problem has only the trivialsolution u(x) ≡ 0.
Proof: This statement follows directly from the estimate from Theorem 2.35, since
‖f‖C([0,1]) = 0.
Remark 2.38 Different proof of Corollary 2.37. The statement of Corollary 2.37follows also from the maximum principle, Theorem 2.32, because for a fully homo-geneous problem, both statements i) and ii) of this theorem can be applied and itis u(0) = u(1) = 0. 2
Corollary 2.39 Uniqueness of the solution of the inhomogeneous prob-lem. Consider the two-point boundary value problem (2.5), (2.6) with b, c, f ∈C([0, 1]) and let c(x) be nonnegative in [0, 1]. Then there is exactly one classicalsolution of the boundary value problem.
Proof: The statement of this corollary follows directly from Corollary 2.37 and The-
orem 2.26.
Lemma 2.40 Another maximum principle. Let b, c ∈ C([0, 1]), let c(x) benonnegative in [0, 1], and let u ∈ C2(0, 1)∩C([0, 1]). If (Lu)(x) < 0 for all x ∈ (0, 1),then there is no local nonnegative maximum of u(x) in (0, 1).
Proof: Assume that there is a nonnegative local maximum x0 ∈ (0, 1). From standardcalculus, one obtains that u(x0) ≥ 0, u′(x0) = 0, and u′′(x0) ≤ 0. It follows that
(Lu)(x0) = −u′′(x0) + b(x0)u′(x0) + c(x0)u(x0) ≥ 0,
which is a contradiction to the assumption.
17
Theorem 2.41 Strong maximum principle. Let b, c ∈ C([0, 1]) and let c(x)nonnegative in [0, 1]. If u ∈ C2(0, 1) ∩ C([0, 1]) has in (0, 1) a nonnegative localmaximum and it is (Lu)(x) ≤ 0 for all x ∈ (0, 1), then u(x) is constant.
Proof: Assume that the function u(x) takes in x0 a nonnegative local maximum, i.e.,it is in particular u(x0) ≥ 0.
Assume that u(x) is not constant. Then, there is a point x2 ∈ (0, 1) with u(x2) < u(x0).Without loss of generality, let x2 > x0. The case x2 < x0 can be proved analogously. Now,one chooses x2 and x1 ∈ [0, x0) such that u(x) takes in x0 the largest value with respectto the interval [x1, x2]. This maximum is taken in a closed interval.
Define for δ, λ > 0
w(x) := δ(eλ(x−x0) − 1
), x ∈ [x1, x2].
Then, one has obviously
w(x)
< 0 for x < x0,= 0 for x = x0,> 0 for x > x0.
Now, one chooses λ sufficiently large, e.g., satisfying the inequality
λ > maxx∈[x1,x2]
(b(x)
2+
√b2(x)
4+ c(x)
).
Then, it follows with c(x) ≥ 0 for all x ∈ (x1, x2) that
(Lw)(x) = −(λ2 − λb(x)− c(x)
)δeλ(x−x0) − c(x)δ < 0.
One has from the assumption also that
(L(u+ w))(x) = (Lu)(x) + (Lw)(x) < 0, x ∈ (x1, x2).
Now, δ is chosen to be sufficiently small, such that
u(x2) + w(x2) < u(x0).
Hence,
u(x) + w(x) < u(x) ≤ u(x0), for x ∈ (x1, x0),
u(x0) + w(x0) = u(x0) ≥ 0,
u(x2) + w(x2) < u(x0),
from what follows that the function (u + w)(x) has a nonnegative maximum in (x1, x2).
This property contradicts the statement of Lemma 2.40, since (L(u+w))(x) < 0. It follows
that the assumption, u(x) being not constant, is wrong.
Remark 2.42 Minimum principles. One obtains corresponding minimum princi-ples by replacing u(x) with −u(x). 2
18
Chapter 3
Finite Difference Methods
3.1 Notations
Remark 3.1 Idea. The basic idea of finite difference methods consists in approxi-mating the derivatives of a differential equation with appropriate finite differences.Consider a decomposition of the interval [0, 1], which is at the moment assumed tobe equidistant:
xi = ih, i = 0, . . . , N, h = 1/N,
ωh = xi : i = 0, . . . , N – grid, mesh.
2
Definition 3.2 Grid function. A vector uh = (u0, . . . , uN )T ∈ RN+1, whichassigns to each grid point (node) a value is called grid function. The restriction ofa function u ∈ C([0, 1]) to a grid function is denoted by Rhu, i.e.,
Rhu := (u(x0), u(x1), . . . , u(xN ))T.
2
Example 3.3 Grid function. Consider a grid with the nodes 0, 0.25, 0.5, 0.75, 1.Then, the grid function of u(x) = x2 is
Rhu =
(0,
1
16,
1
4,
9
16, 1
)T
.
Different functions might have for a given grid the same grid function. Consider,e.g., u(x) = sin(4πx) on the same grid as used above. The corresponding gridfunction is
Rhu = (0, 0, 0, 0, 0)T.
This grid function is obviously also the grid function of u(x) = 0. The consideredgrid is too coarse to represent the u(x) = sin(4πx) in a reasonable way. 2
Definition 3.4 Finite difference operators. Let v(x) be a sufficiently smoothfunction and denote vi = v(xi), where xi are the nodes of the grid. The followingdifference quotients (finite differences) are called
D+v(xi) = vx,i =vi+1 − vi
h– forward difference,
D−v(xi) = vx,i =vi − vi−1
h– backward difference,
D0v(xi) = vx,i =vi+1 − vi−1
2h– central difference,
D+D−(v)(xi) = vxx,i =vi+1 − 2vi + vi−1
h2– second order difference,
19
see Figure 3.1. 2
Figure 3.1: Finite differences.
Remark 3.5 To the finite differences. The formula for D+D−(v)(xi) can bechecked with a direct calculation. In addition, it is
D0v(xi) =1
2
((D+v(xi) +D−v(xi)
).
2
Definition 3.6 Consistency of a finite difference operator, discrete max-imum norm. Let L be a differential operator. The finite difference operatorLh : RN+1 → RN+1 is said to be consistent with L of order k if
max0≤i≤N
|(Lu)(xi)− (Lhuh)i| =: ‖Lu− Lhuh‖∞,d = O(hk).
Here, ‖·‖∞,d is the discrete maximum norm in the space of grid functions. 2
Example 3.7 Orders of consistency for standard finite difference operators. Theconsistency is a measure of the approximation property of Lh. Applying a Taylorseries expansion for v(x) at the node xi yields
D+v(xi) = v′(xi) +O(h),
D−v(xi) = v′(xi) +O(h),
D0v(xi) = v′(xi) +O(h2),
D+D−(v)(xi) = v′′(xi) +O(h2).
The finite difference operators D+v(xi), D−v(xi), D
0v(xi) are consistent to L =ddx of first, first, and second order, respectively. The operator D+D−(v)(xi) is
consistent of second order to L = d2
dx2 . 2
20
Figure 3.2: Brook Taylor (1685 – 1731).
3.2 Classical Convergence Theory for Central Dif-ference Schemes
Remark 3.8 Contents of this section. This section considers the two-point bound-ary value problem
Lu := −u′′ + b(x)u′ + c(x)u = f(x), for x ∈ (0, 1), u(0) = u(1) = 0, (3.1)
i.e., the model problem with ε = 1. The classical convergence theory will be pre-sented. It will be assumed that the parameter functions b, c, f are sufficiently smoothand that c(x) ≥ 0 for all x ∈ [0, 1]. 2
Definition 3.9 Central difference scheme. The central difference scheme for(3.1) has the form
(Lhuh)i := −D+D−ui + biD0ui + ciui = fi, for i = 1, . . . , N − 1,
u0 = uN = 0. (3.2)
2
Remark 3.10 To the central difference scheme.
• The central difference scheme leads to a tridiagonal system of linear equations
riui−1 + siui + tiui+1 = fi, i = 1, . . . , N − 1, u0 = uN = 0,
with
ri = − 1
h2− 1
2hbi, si = ci +
2
h2, ti = − 1
h2+
1
2hbi.
• The following questions have to be answered:
Which properties has the discrete problem (3.2)? What can be said about the error ‖u− uh‖∞,d?
For answering these questions, the concepts of consistency and stability will beused.
2
21
Definition 3.11 Consistency of a difference scheme and order of consis-tency. Consider a difference scheme of the form Lhuh := Rh(Lu). The boundaryconditions should be integrated into this scheme such that the first and last row of Lh
are identical to the first and last row of the identity matrix and it is Rh(Lu)0 = u0,Rh(Lu)N = uN . The scheme is called consistent of order k in the discrete maximumnorm, if
‖LhRhu−Rh(Lu)‖∞,d ≤ chk,
where the positive constants c and k are independent of h. 2
Lemma 3.12 Consistency order of the central difference scheme. Assumethat u ∈ C4([0, 1]), then the central difference scheme (3.2) has consistency order 2.
Proof: The proof uses a Taylor series expansion, exercise.
Definition 3.13 Stability of a difference scheme. A difference scheme Lhuh =fh is called stable in the discrete maximum norm, if there is a stability constant cS ,which is independent of h, with
‖uh‖∞,d ≤ cS ‖Lhuh‖∞,d = cS ‖fh‖∞,d
for all grid functions uh. 2
Definition 3.14 Convergence of a difference scheme and order of con-vergence. A difference scheme for (3.1) is convergent of order k in the discretemaximum norm, if there are positive constants c and k, which are independent ofh, such that
‖uh −Rhu‖∞,d ≤ chk.
2
Theorem 3.15 Consistency + stability =⇒ convergence. A consistent andstable difference scheme is convergent. The orders of consistency and convergenceare the same.
Proof: It is
‖uh −Rhu‖∞,dstab.
≤ cS ‖Lh (uh −Rhu)‖∞,dlin.= cS ‖Lhuh − LhRhu‖∞,d
= cS ‖fh − LhRhu‖∞,d = cS ‖Rhf − LhRhu‖∞,d
= cS ‖RhLu− LhRhu‖∞,dcons.
≤ Khk,
where the constant K is the product of the constants from the stability and consistency
condition.
Remark 3.16 To consistency and stability. One has to prove consistency andstability.
• Consistency proofs are based often on Taylor series expansions and they areperformed in a standard way.• Stability proofs are not performed only at functions but they are performed at
matrices and functions, see Definition 3.13. They are generally not simple andthey require the introduction of some new notations.
2
22
Definition 3.17 Natural order of vectors and matrices, inverse-monotonematrices. Let x,y ∈ Rn. Then one writes x ≤ y if and only if xi ≤ yi forall i = 1, . . . , n. The notation x ≥ 1 means that xi ≥ 1 for all i = 1, . . . , n.Analogously, the notation A ≥ 0 means for a matrix A = (aij) ∈ Rn×n that aij ≥ 0for all i, j = 1, . . . , n.
A matrix A, for which the inverse A−1 exists with A−1 ≥ 0 is called inverse-monotone matrix. 2
Lemma 3.18 Discrete comparison principle. Let A ∈ Rn×n be inverse-mono-tone. If Av ≤ Aw for v,w ∈ Rn, then it follows that v ≤ w.
Proof: From the assumption, it follows that
A(v −w) := b ≤ 0.
Multiplication with A−1 givesv −w = A−1b ≤ 0.
The last inequality follows from the property that A is an inverse-monotone matrix. Non-
negative matrix entries are multiplied with nonpositive vector entries of b. The result is
a vector with nonpositive components.
Remark 3.19 To Lemma 3.18. Lemma 3.18 is a discrete analog to the comparisonprinciple from Corollary 2.34. 2
Definition 3.20 M-matrix. A matrix A = (aij) ∈ Rn×n is called an M-matrix,if:
1. aij ≤ 0 for i 6= j,2. A−1 exists with A−1 ≥ 0.
2
Lemma 3.21 M-matrices have positive diagonal entries. Let A = (aij) ∈Rn×n be an M-matrix, then it is aii > 0, i = 1, . . . , n.
Proof: Exercise.
Remark 3.22 On M-matrices. M-matrices are an important subclass of inverse-monotone matrices, which will become very useful in the analysis. However, inpractice, the second condition of their definition is hard to check. But there arecharacterizations of M-matrices which are easier to check.
Then name refers to Hermann Minkowski. 2
Theorem 3.23 M-matrix criterion. Let A = (aij) ∈ Rn×n with aij ≤ 0 fori 6= j. Then A is an M-matrix if and only if there exists a vector e ∈ Rn, e > 0,such that Ae > 0. In this case, one obtains for the row sum norm
∥∥A−1∥∥∞ ≤
‖e‖∞,d
mink (Ae)k. (3.3)
The vector e is called majorizing element.
Proof: See the literature, e.g., Bohl (1981); Axelsson and Kolotilina (1990).
Remark 3.24 Concerning the M-matrix criterion.
23
• The following approach is often successful for the construction of a majorizingelement.
Find a function e(x) > 0 such that (Le)(x) > 0 for x ∈ (0, 1). This functionis a majorizing element of the differential operator L. Restrict e(x) to its corresponding grid function eh.
If the first step of this approach is possible and the discretization Lh of L isconsistent, then this approach generally works, at least if the mesh width issufficiently small. The matrix A is the matrix representation of Lh.
• The constant cS in the definition of the stability can be estimated with (3.3)
‖uh‖∞,d =∥∥A−1fh
∥∥∞,d≤∥∥A−1
∥∥∞ ‖fh‖∞,d =
∥∥A−1∥∥∞ ‖Lhuh‖∞,d .
Hence, it holds for this constant
cS ≤‖e‖∞,d
mink (Ae)k.
Thus, from the M-matrix criterion, which is equivalent to the M-matrix property,it follows stability.• If Dirichlet boundary conditions are prescribed, then the variables u0 and uN
should be eliminated before Theorem 3.23 is applied.• For the central finite difference scheme, the first requirement of the M-matrix
criterion, aij ≤ 0 for i 6= j, is satisfied if h is sufficiently small, see Remark 3.10for the coefficients of the matrix.
2
Example 3.25 M-matrix criterion. Consider (3.1) with b(x) ≡ 0, i.e.,
Lu(x) = −u′′(x) + c(x)u(x), u(0) = u(1) = 0, c(x) ≥ 0 in [0, 1].
Choose e(x) := 12x(1− x), then it follows that
Le(x) = 1 + c(x)e(x) ≥ 1.
Setting eh := Rhe gives for all nodes xi
(Lheh)i = −D+D−eh,i + cieh,i = 1 + cieh,i ≥ 1,
because the second order finite difference discretizes the second derivative of aquadratic function in the interior nodes exactly, see Example 3.7. One gets
Lheh ≥ (1, . . . , 1)T ⇐⇒ Ae ≥ 1.
As bound for the stability constant, one obtains
cS ≤‖e‖∞,d
mink (Ae)k≤ eh(1/2)
1=
1/8
1=
1
8.
This example shows that in the case b(x) ≡ 0 the M-matrix property holdswithout restrictions on the fineness of the grid. 2
Lemma 3.26 Stability of the central finite difference scheme for suffi-ciently fine grids. If the mesh width h is sufficiently small, then the central finitedifference scheme (3.2) for the two-point boundary value problem (3.1) is stable inthe discrete maximum norm. The stiffness matrix is an M-matrix.
24
Proof: A majorizing element will be constructed. To this end, let e(x) be the solutionof the two-point boundary value problem
−e′′ + b(x)e′ = 1, e(0) = e(1) = 0.
From the maximum principle, Lemma 2.31, it follows that e(x) ≥ 0 for x ∈ (0, 1). Inaddition, e(x) has no local minima in (0, 1) since in this case, one obtains from the equationthat −e′′(x) = 1 for the local minima, which is a contradiction. Since e(x) 6≡ 0, it ise(x) > 0 for x ∈ (0, 1). Since c(x) ≡ 0, one gets with Corollary 2.39 that the givenproblem possesses a unique solution and that e ∈ C([0, 1]). Hence, e(x) is bounded. Byconstruction, it is
Le(x) = −e′′(x) + b(x)e′(x) + c(x)e(x) = 1 + c(x)e(x).
Let eh be the grid function of e(x). For interior nodes, one obtains with c(x) ≥ 0
(Lheh)i = (RhLe)i + (Lheh −RhLe)i= (Rh (1 + c(x)e(x)))i +
(−D+D−eh + biD
0eh + cieh − 1− cieh)i
≥ 1 +(−D+D−eh + biD
0eh − 1)i
=(−D+D−eh + biD
0eh)i.
Since eh is the grid function that corresponds to e(x), the expression in the last lineapproximates −e′′(xi) + b(xi)e
′(xi)(= 1) sufficiently well, if h sufficiently small, see theconsistency estimate in Example 3.7. In particular, there is a H > 0 such that for allh ∈ (0, H] it is
(Lheh)i ≥1
2.
Now, the proof of the theorem is finished with the application of the M-matrix criterion
and using the property that aij ≤ 0 for sufficiently fine meshes.
Corollary 3.27 Second order convergence of the central difference scheme.If u ∈ C4([0, 1]), then the central difference scheme (3.2) is convergent with secondorder.
Proof: The statement follows with Theorem 3.15 by combining Lemma 3.12 and
Lemma 3.26.
Example 3.28 Second order convergence of the central difference scheme for suf-ficiently fine meshes. Consider the two-point boundary value problem
−u′′(x) + 2u′(x) + 3u(x) = 1 in (0, 1), u(0) = u(1) = 0.
The solution of this problem is
u(x) =1
3
(1 +
1− e−1
e−1 − e3e3x +
e3 − 1
e−1 − e3e−x
).
One obtains the following errors for different mesh widths
no. of intervals N ‖u− uh‖∞,d
4 4.2388e-48 9.8811e-5
16 2.4529e-532 6.1537e-664 1.5368e-6
128 3.8440e-7256 9.6093e-8512 2.4023e-8
1024 6.0058e-9
It can be observed that the error is reduced by the factor four if the mesh width isreduced by the factor two. This behavior is second order convergence. 2
25
3.3 Upwind Schemes
Remark 3.29 Singularly perturbed two-point boundary value problem. From nowon, finite difference schemes will be studied for the two-point boundary value prob-lem
Lu := −εu′′ + b(x)u′ + c(x)u = f(x), for x ∈ (0, 1), (3.4)
with the boundary conditions
u(0) = u(1) = 0, (3.5)
and the assumptions
ε > 0,
b(x) > 0 for all x ∈ [0, 1],
c(x) ≥ 0 for all x ∈ [0, 1],
with sufficiently smooth functions b(x), c(x), and f(x). This problem is called sin-gularly perturbed if ε ‖b‖L∞(Ω). The parameter ε is called singular perturbation
parameter. With respect to the convection, it is only important that b(x) 6= 0 forall x ∈ [0, 1]. If b(x) < 0 in [0, 1], then one obtains a problem of form (3.4) byapplying the variable transform x 7→ 1− x.
If ε is small, then the solution of (3.4), (3.5) has in general a boundary layer atx = 1, see Example 2.8. This layer influences both the stability and the consistencyof the numerical method. If the boundary values are chosen such that there is noboundary layer, then the consistency improves but the stability of the method mightbe still a problem. 2
Example 3.30 Application of the central difference scheme to a simplified singu-larly perturbed problem. Consider the problem
−εu′′ + u′ = 0 in (0, 1), u(0) = 0, u(1) = 1.
The solution of this problem is
u(x) =e−(1−x)/ε − e−1/ε
1− e−1/ε.
Applying the transform u(x) := x + v(x), one gets a problem with homogeneousboundary conditions. But one can apply the difference scheme directly to the prob-lem with inhomogeneous boundary conditions. This approach is used in practice.The discrete problem has the form
−εD+D−ui +D0ui = 0, u0 = 0, uN = 1
and it has the solution (exercise)
ui =ri − 1
rN − 1with r =
2ε+ h
2ε− h.
In particular, it is |r| > 1, since the numerator is the sum of two positive numbers.The absolute value of this sum is always larger than the absolute value of thedifference of these numbers.
If h 2ε then it is r ≈ −1 and it follows that
ui ≈(−1)i − 1
(−1)N − 1.
26
Figure 3.3: Solution (left) and discrete solution with the central difference scheme(right) for ε = 10−6 and h = 1/32.
If N even, then one divides by a very small positive number, since |r| > 1. For ieven, the numerator is also small and it is positive. Hence, the quotient is positive,too. For i odd, the numerator is negative and of order −2. In this case, the quotientis negative and its absolute value is large. The discrete solution is highly oscillating,see Figure 3.3.
If h < 2ε, then one obtains with the central difference scheme a useful approxi-mation of the solution. However, in applications it is often ε ≤ 10−6, i.e., one needsvery fine grids in order to apply the central difference scheme. The use of such gridsis possible in one dimension, but not in two or three dimensions. 2
Remark 3.31 Application of the central difference scheme to the general singularlyperturbed problem. Consider now the singularly perturbed problem (3.4), (3.5) andwrite the difference scheme in the form presented in Remark 3.10
riui−1 + siui + tiui+1 = fi, i = 1, . . . , N − 1, u0 = uN = 0,
with
ri = − ε
h2− 1
2hbi, si = ci +
2ε
h2, ti = − ε
h2+
1
2hbi, bi > 0.
One obtains an M-matrix, and hence stability, if one assumes that
ti ≤ 0 =⇒ h ≤ h0(ε) =2ε
‖b‖∞.
This assumption generalizes the observation from Example 3.30. Note that h0(ε)→0 for ε→ 0. 2
Remark 3.32 Motivation for upwind schemes. Another heuristic explanation forthe failure of the central difference scheme in the case ε h is as follows. For smallε, the method applied to Example 3.30 has essentially the form
D0ui = 0, ⇐⇒ ui+1 − ui−1
2h= 0.
It follows for i = N − 1 that uN−2 ≈ uN = 1, which is a very bad approximation ofthe exact value u(xN−2) ≈ 0.
This observations leads to the idea that for the approximation of u′(xN−1) it isbetter not to use the value uN . The simplest candidate which has this feature isthe backward difference
u′(xi) ≈ui − ui−1
h.
27
If one has the goal to modify the matrix entries which are obtained with the centraldifference scheme in such a way that one gets an M-matrix, i.e., ti ≤ 0, then onecan also motivate backward difference approximation of the first derivative, becausethis condition is satisfied at any rate if for the discretization of the convective terma contribution from the node xi+1 is not used. 2
Definition 3.33 Simple upwind scheme. The simple upwind scheme for thesingularly perturbed two-point boundary value problem (3.4), (3.5) has the form
− εD+D−ui + biDNui + ciui = fi, for i = 1, . . . , N − 1,
u0 = uN = 0, (3.6)
with
DN :=
D+ for b < 0,D− for b > 0.
2
Remark 3.34 Concerning the simple upwind scheme.
• In the upwind scheme, the finite difference approximation of the convective termis computed with values from the upwind direction. For convection-dominatedproblems, the transport of information occurs in the direction of convection.Hence, the upwind direction is the direction from which information is coming.
• Using the simple upwind scheme, one obtains a much better numerical solutionfor Example 3.30, see Figure 3.4.
Figure 3.4: Numerical solution of Example 3.30 with the simple upwind scheme forε = 10−6 and h = 1/32.
• In the simple upwind scheme, the second order approximation D0 is replaced bythe first order approximation D+ or D−. One will see this reduced order in theaccuracy of the numerical results.• Let Lh be the matrix of the simple upwind scheme after having eliminated the
boundary values u0 and uN . In the form of Remark 3.10, this matrix has theform
ri = − ε
h2− 1
hmax0, bi, si = ci +
2ε
h2+
1
h|bi| , ti = − ε
h2+
1
hmin0, bi.
One can see that all non-diagonal entries are negative, independently of the sizeof ε and h.
2
Theorem 3.35 Stability of the simple upwind scheme. Under the assump-tions from Remark 3.29, the matrix Lh of the simple upwind scheme (3.6) is an
28
M-matrix. The simple upwind scheme is uniformly stable with respect to the param-eter ε, i.e., it is
‖uh‖∞,d ≤ cS ‖Lhuh‖∞,d ,
where the stability constant cS > 0 is independent of ε and h.
Proof: Consider only the case b(x) ≥ β > 0. The goal consists in constructing anappropriate majorizing element. To this end, choose e(x) = x, which gives
Le(x) = −εe′′(x) + b(x)e′(x) + c(x)e(x) = b(x) + xc(x) ≥ β.
For the simple upwind scheme and the corresponding grid function eh, one obtains
(Lheh)i = rixi−1 + sixi + tixi+1
=
(− ε
h2− 1
hbi
)(xi − h) +
(ci +
2ε
h2+
1
hbi
)xi −
ε
h2(xi + h)
=
(− ε
h2− 1
hbi + ci +
2ε
h2+
1
hbi −
ε
h2
)xi +
(ε
h2+
1
hbi −
ε
h2
)h
= cixi + bi ≥ β.
It follows from Theorem 3.23 that Lh is an M-matrix. Using the estimate of the stabilityconstant from Remark 3.24, one gets
cS ≤‖eh‖∞,d
mink (Lheh)k=
1
β.
Lemma 3.36 Estimates of the norm of derivatives of the solution. Letb(x) ≥ β > 0 and b(x), c(x), f(x) sufficiently smooth. Then, it is for the solutionu(x) of (3.4), (3.5)∣∣∣u(i)(x)
∣∣∣ ≤ C [1 + ε−i exp
(−β 1− x
ε
)], i = 1, 2, . . . , q,
for x ∈ [0, 1]. The maximal order q depends on the smoothness of the data.
Proof: The proof was performed in Kellogg and Tsan (1978), see also (Roos et al.,
2008, p. 21).
Theorem 3.37 Consistency of the simple upwind scheme. Under the as-sumptions from Remark 3.29 with b(x) ≥ β > 0, there is a positive constantβ∗, which depends only on β, such that the error committed by the simple upwindscheme (3.6) in the inner nodes xi : i = 1, . . . , N − 1 can be bounded as follows
|u(xi)− ui| ≤
Ch
[1 + ε−1 exp
(−β∗ 1− xi
ε
)]if h < ε,
Ch+ C exp
(−β∗ 1− xi+1
ε
)if h ≥ ε.
(3.7)
Proof: The proof was performed in Kellogg and Tsan (1978). Here, only the inter-esting case h ≥ ε will be considered and also for this case, not the complete proof will bepresented. The complete proof can be found in (Roos et al., 2008, pp. 49).
In the case h ≥ ε, one decomposes the solution of (3.4), (3.5) into
u(x) = −u0(1) exp
(− b(1)(1− x)
ε
)+ z(x) =: v(x) + z(x),
29
where u0(x) is the reduced solution. To the part z(x), those properties of u(x) are trans-ferred which v(x) does not possess. One finds, analogously to Lemma 3.36, that∣∣∣z(k)(x)
∣∣∣ ≤ C [1 + ε1−k exp
(− b(1)(1− x)
ε
)], k = 1, 2, 3. (3.8)
It isLhuh = fh = Rh(f) = Rh(Lu) = Rh (L(v + z)) = Rh(Lv) +Rh(Lz).
In this way, a decomposition uh = vh+zh is obtained, where the grid functions are definedas the solutions of the following discrete problems
Lhvh = Rh(Lv) and Lhzh = Rh(Lz).
The functions vh and zh coincide with v(x) and z(x), respectively, in x0 and xN . Applyingthe triangle inequality gives
|u(xi)− ui| = |v(xi) + z(xi)− (vi + zi)| ≤ |v(xi)− vi|+ |z(xi)− zi| .
The error for z(x) is estimated with the consistency error, using the stability estimatefrom Theorem 3.35. Applying the Taylor series expansion of z(x) in the node xi, oneobtains for the first step of the consistency error estimate, exercise,
|τi| := |Lhz(xi)−Rh(Lz(xi))| ≤ C∫ xi+1
xi−1
(ε∣∣∣z(3)(t)
∣∣∣+∣∣∣z(2)(t)
∣∣∣) dt.
The right-hand side comes from the remainder in the expansion. Using the estimates (3.8)for the derivatives of z(x) yields
|τi| ≤ C
∫ xi+1
xi−1
(ε+ ε−1 exp
(−b(1)
1− tε
)+ 1 + ε−1 exp
(−b(1)
1− tε
))dt
≤ C
∫ xi+1
xi−1
(ε+ 1) dt+ Cε−1
∫ xi+1
xi−1
(exp
(−β 1− t
ε
)+ exp
(−β 1− t
ε
))dt
≤ Ch+ Cε−1
∫ xi+1
xi−1
exp
(−β 1− t
ε
)dt
= Ch+ Cε−1
(ε
βexp
(−β 1− t
ε
)∣∣∣∣xi+hxi−h
)
= Ch+ C
[exp
(−β 1− xi − h
ε
)− exp
(−β 1− xi + h
ε
)]= Ch+ C exp
(−β 1− xi
ε
)[exp
(βh
ε
)− exp
(−βhε
)]= Ch+ C sinh
(βh
ε
)exp
(−β 1− xi
ε
).
It is
sinh(t) =et − e−t
2≤ et
2= Cet.
Hence, it follows that
|τi| ≤ Ch+ C exp
(−β 1− xi
ε+βh
ε
)= Ch+ C exp
(−β 1− xi+1
ε
).
An estimate for the error of the other part v(x) can be derived with the discretecomparison principle, Lemma 3.18,
|v(xi)− vi| ≤ C exp
(−β 1− xi+1
ε
).
Combining both estimates gives the final estimate.
30
Corollary 3.38 Convergence of the simple upwind scheme away from lay-ers. Under the assumptions of Theorem 3.35 and Theorem 3.37, the simple upwindscheme converges in an interval [0, 1 − δ], where δ > 0 is fixed, of first order. Theconstant in the convergence estimate is independent of ε.
Proof: This statement follows with Theorem 3.15, using Theorems 3.35 and 3.37.
Remark 3.39 Behavior in the layer based on estimate (3.7). Let ε < h, then oneobtains in the node xN−2 the estimate
|u(xN−2)− uN−2| ≤ Ch+ C exp
(−β∗ 1− xN−1
ε
)= Ch+ C exp
(−β∗h
ε
)≤ Ch+ Ch = O (h) ,
since exp(−x) < x if x is sufficiently large. However, for xN−1 one gets
|u(xN−1)− uN−1| ≤ Ch+ C exp
(−β∗ 1− xN
ε
)= Ch+ C = O (1) ,
since xN = 1. If follows that the estimate of the error in xN−1 does not convergetoward zero. 2
Example 3.40 Behavior in the layer. The observation in the previous remark isnot a problem of the estimate. Consider
−εu′′(x)− u′(x) = 0, u(0) = 0, u(1) = 1.
The solution of this problem has a layer at x = 0. One obtains with the simpleupwind scheme, exercise,
ui =1− ri
1− rN, with r =
ε
ε+ h.
For h = ε, one gets
u1 =1− r
1− rN=
1− 1/2
1− (1/2)N=
1/2
1− (1/2)N≈ 1
2.
However, for the solution it is for x1 = h = ε
u(x1) =1− e−1
1− e−1/ε≈ 0.63
for small ε. Hence, the error is O (1) for small ε. Consequently, one cannot expectto improve substantially the estimate from Theorem 3.37. 2
Remark 3.41 Typical behavior in the layer in numerical simulations. Considerconstant ε and variable h. If h is sufficiently large, then all grid points are outsidethe layer. If h decreases, then the error increases since the last node moves into thelayer, see Figure 3.5. If h becomes sufficiently small, then the error decreases. Inthis case the first estimate of Theorem 3.37 becomes effective. 2
Remark 3.42 Interpretation of the simple upwind scheme as artificial diffusion.The difficulties in the numerical solution of singularly perturbed problems originatefrom the different magnitudes of diffusion and convection, and caused by this issue ofthe appearance of sharp (thin) layers. It is clear that the numerical solution becomes
31
101
102
103
10−3
10−2
10−1
100
number of nodeserr
or
in the d
iscre
te m
axim
um
norm
Figure 3.5: Error of the simple upwind scheme for Example 2.8, ε = 1e − 3, anddifferent number of nodes.
the simpler, the larger the diffusion, compared with the convection, become and thelayers become wider.
Consider b > 0, then it is
biDNui = biD
−ui =ui − ui−1
h= bi
ui+1 − ui−1
2h+ bi−ui+1 + 2ui − ui−1
2h
= biD0ui −
bih
2D+D−ui.
Hence, the simple upwind scheme (3.6) can be written in the form
−(ε+
bih
2
)D+D−ui + biD
0ui + ciui = fi, for i = 1, . . . , N − 1,
u0 = uN = 0. (3.9)
One can see that the diffusion coefficient is artificially increased and it hasthe magnitude O (h). Consequently, the simple upwind scheme is nothing elsethen a central difference scheme applied to a problem with sufficiently large, O (h),diffusion. It has been observed already in Example 3.30 that the central differencescheme gives useful results if the diffusion is of order O (h).
One can define methods with artificial diffusion also directly. 2
Definition 3.43 Scheme with artificial diffusion, fitted upwind scheme. Afinite difference scheme with artificial diffusion is defined by
− εσ (q(xi))D+D−ui + biD
0ui + ciui = fi, for i = 1, . . . , N − 1,
u0 = uN = 0, (3.10)
q(x) :=b(x)h
2ε,
and σ(q) is an appropriate function. It is also called fitted upwind scheme. 2
Remark 3.44 Fitted upwind schemes.
• The simple upwind scheme (3.6) is obtained for σ(q) = 1 + q, see (3.9).• The introduction of artificial diffusion changes the original problem significantly.
Consider, e.g.,
−εu′′ + u′ = 1 in (0, 1), u(0) = u(1) = 0,
with the solution
u(x) = x−exp
(− 1−x
ε
)− exp
(− 1
ε
)1− exp
(− 1
ε
) , (3.11)
32
see Example 2.8. The second term is responsible for the satisfaction of theboundary condition at x = 1. It is basically different from zero only in the inter-val [1− ε, 1], see Figure 3.6. Introducing artificial diffusion leads to a perturbedsolution and the term, which is responsible for the satisfaction of the boundarycondition, is in the interval [1−εσ(q(xN−1)), 1] considerably different from zero.That means, the layer is by far less steep. This effect is called smearing of thelayer.
Figure 3.6: Second term of the solution (3.11) for ε = 10−6.
For the simple upwind scheme, it is
εσ(q(xN−1)) = ε+ εbN−1h
2ε= ε+
bN−1h
2.
This expression is in realistic situations, i.e., for ε bN−1 and ε h, larger byorders of magnitude than ε.
2
Remark 3.45 Key point of stabilized methods. Appropriate discretizations forconvection-dominated problems are called stabilized methods. The introductionof artificial diffusion is the key issue of stabilized methods. The difficulty in theconstruction of stabilized methods is that one needs to apply the right amount ofartificial diffusion, at the right locations, and in the correct directions (in multipledimensions).
A question is if one can construct stable methods which lead to less smearing ofthe layers than the simple upwind scheme. 2
Theorem 3.46 Stability of schemes with artificial diffusion. Let b(x) >β > 0, c(x) ≥ 0, and σ(q) > q. Then, the matrix of the scheme with artificialdiffusion (3.10) is an M-matrix and the method is stable in the discrete maximumnorm. The stability constant does not depend on ε.
Proof: The proof is very similar to the proof of Theorem 3.35, exercise.
Theorem 3.47 Consistency of schemes with artificial diffusion. Let theassumptions of Theorem 3.46 be satisfied, let u ∈ C4([0, 1]), and let
|σ(q)− 1| ≤ minq,Mq2, q ≥ 0,
with a constant M > 0. Then, for fixed ε, the consistency error of the scheme withartificial diffusion (3.10) is of second order.
33
Proof: The consistency error in the node xi is
|τi| =∣∣[− εσ (qi)D
+D−u(xi) + biD0u(xi) + ciu(xi)
]−[− εu′′(xi) + biu
′(xi) + ciu(xi)]∣∣
=∣∣εσ(qi)
(u′′(xi)−D+D−ui
)+ ε (1− σ(qi))u
′′(xi) + bi(D0u(xi)− u′(xi)
)∣∣ .From the consistency error estimate in Example 3.7 it follows that
|τi| ≤ C(εσ(qi)h
2∥∥∥u(4)
∥∥∥∞
+ ε |1− σ(qi)|∥∥u′′∥∥∞ + h2
∥∥∥u(3)∥∥∥∞
).
Using the assumptions of the theorem and the definition of q(x) gives
σ(qi) ≤ |σ(qi)− 1|+ 1 ≤ minqi,Mq2
i
+ 1 ≤ qi + 1 ≤ C h
ε+ 1,
|1− σ(qi)| ≤ Mq2i ≤ C
h2
ε2.
Inserting this estimate yields
|τi| ≤ C
((h
ε+ 1
)εh2
∥∥∥u(4)∥∥∥∞
+ εh2
ε2
∥∥u′′∥∥∞ + h2∥∥∥u(3)
∥∥∥∞
)(3.12)
≤ C(ε)h2.
Remark 3.48 The consistency of the scheme with artificial diffusion.
• Examples of functions σ(q) that satisfy the assumptions of Theorem 3.47 are(exercise)
σ(q) =√
1 + q2, σ(q) = 1 +q2
1 + q.
The second variant is called Samarskii upwind scheme.• The consistency is of second order only for constant ε. The factor C(ε) diverges
to infinity for ε→ 0, see the middle term in (3.12). In addition, also derivativesof the solution might depend in a bad way on ε and they might explode forε → 0. It follows that these methods become worse and worse for ε → 0. Onecan show that the consistency independent of ε away from the layer is only offirst order. The typical behavior in the layer is analogously as for the simpleupwind scheme, see Remark 3.41.
2
Remark 3.49 Summary.
• The central difference scheme is not suited for singularly perturbed problems.• The simple upwind scheme is stable, but too inaccurate (of first order consistent).
Layers are heavily smeared.• Upwind methods can be interpreted as methods with artificial diffusion.• Fitted upwind methods might be of second order consistent, but only for fixedε. This property is not uniform in ε.
The upwind schemes that have been introduced so far are not satisfactory sincethey are too inaccurate for small ε and the convergence in the layer depends on ε.
2
34
Figure 3.7: Alexander Andreewitsch Samarskii (1919 – 2008).
3.4 Uniformly Convergent Methods
Remark 3.50 Motivation. The goal consists in the development of discretizationswhich converge uniformly in the whole interval [0, 1], in particular in the layer. Twoways for achieving this goal will be presented:
• a scheme which is obtained with an appropriate choice of artificial diffusion σ(q)in (3.10),
• schemes, which are defined by an appropriate choice of the grid.
In practice, one often has very small diffusion. Therefore it is important to constructnumerical methods which provide accurate results in this case. The constructionof such methods is not trivial. This might become clear already from the fact thatthe limit ε→ 0 is in a certain sense discontinuous, since the order of the differentialequation changes. With this change of order, also other important features change,e.g., the needed boundary conditions to define a well-posed problem. Also theproperties of the solutions of differential equations with different order are different,e.g., the smoothness properties. A uniformly convergent method has to deal withthis limit without deterioration of the quality of the computed solution.
This section follows in part Großmann and Roos (2005). 2
Definition 3.51 Uniform convergence. A scheme for the solution of (3.4), (3.5)is called uniformly convergent of order p with respect to the singularly perturbationparameter ε in the discrete maximum norm if an estimate of the form
‖u− uh‖∞,d ≤ Chp, p > 0,
holds with a constant C that does not depend on ε. 2
3.4.1 Sophisticated Artificial Diffusion
Remark 3.52 Idea. The choice of a sophisticated artificial diffusion σ(q) can bemotivated with a study of the solution of (3.4), (3.5) for ε→ 0. 2
Lemma 3.53 Convergence to the reduced solution. Let u(x, ε) be the solu-tion of (3.4), (3.5) with b(x) ≥ β > 0, c(x) ≥ 0, and let u0(x) be the solution of thereduced problem. Then it holds for all x ∈ [0, x0) with x0 < 1 that
limε→0
u(x, ε) = u0(x).
35
Proof: The proof is based on the comparison principle, Corollary 2.34. Set
v1(x) := γ exp(βx), γ > 0,
then it follows that
(Lv1)(x) = γ(−εβ2 + b(x)β + c(x)) exp(βx) ≥ γβ2(1− ε) exp(βx) ≥ 1
for sufficiently large γ. Now, let
v2(x) := exp
(−β 1− x
ε
),
then it holds
(Lv2)(x) =
(−εβ
2
ε2+ b(x)
β
ε+ c(x)
)exp
(−β 1− x
ε
)≥ β
ε(−β + b(x)) exp
(−β 1− x
ε
)≥ 0.
Consider nowv(x) := M1εv1(x) +M2v2(x),
then one obtains, for appropriately chosen constants M1 and M2 and using the estimatesfor (Lv1)(x) and (Lv2)(x),
(Lv)(x) = M1ε(Lv1)(x) +M2(Lv2)(x) ≥M1ε(Lv1)(x) ≥M1ε ≥ ε∣∣u′′0 (x)
∣∣= |L(u− u0)(x)| ,
v(0) = M1εv1(0) +M2v2(0) = M1εγ +M2 exp(−β/ε) ≥ 0 = |(u− u0)(0)| ,v(1) = M1εv1(1) +M2v2(1) = M1εγ exp(β) +M2 ≥M2 ≥ |(u− u0)(1)| = |u0(1)| .
The constants have to be sufficiently large and they only depend on u0(x). There is nocoefficient ε in the reduced problem such that u0(x) cannot depend on ε and hence theconstants do not depend on ε. With the comparison principle it follows that
|(u− u0)(x)| ≤ v(x) = M1εγ exp(βx) +M2 exp
(−β 1− x
ε
).
Hence, one gets for x < 1limε→0|(u− u0)(x)| = 0.
Lemma 3.54 Estimate to the reduced solution plus a correction term.Under the assumptions of Lemma 3.53, there is a constant C, which is independentof x and ε, such that for the solution of (3.4), (3.5) it holds∣∣∣∣u(x, ε)−
[u0(x)− u0(1) exp
(−b(1)
1− xε
)]∣∣∣∣ ≤ Cε, x ∈ [0, 1].
Proof: The proof is similar to the proof of Lemma 3.53.
Remark 3.55 Necessary condition for a sophisticated artificial diffusion σ(q). Letρ∗ := h/ε be fixed, i.e., for h→ 0 it holds also ε→ 0. The goal consists in finding forthis case a condition for an appropriate function σ(q) in the fitted upwind scheme.
Let a node i be fixed. Since ε→ 0 for h→ 0 it follows from Lemma 3.54 that
limh→0
u(1− ih) = limh→0
(u0(1− ih)− u0(1) exp
(−b(1)
1− (1− ih)
ε
))= u0(1)− u0(1) lim
h→0exp
(−b(1)
ih
ε
)= u0(1)− u0(1) exp (−ib(1)ρ∗)
= u0(1) (1− exp(−2iq(1)) , (3.13)
36
where the definition of q(x) has been used. The fitted upwind scheme has the form
−εσ (q(bi))ui+1 − 2ui + ui−1
h2+ bi
ui+1 − ui−1
2h= fi − ciui,
or after multiplication with h2/ε
−σ (q(bi)) (ui+1 − 2ui + ui−1) + q(bi) (ui+1 − ui−1) =h2
ε(fi − ciui)
= hρ∗ (fi − ciui) .
For the right boundary, i.e., i = N − 1, it is in particular
limh→0
(−σ (qN−1) (uN − 2uN−1 + uN−2) + qN−1 (uN − uN−2))
= limh→0
hρ∗ (fN−1 − cN−1uN−1) = 0,
since f(x), c(x) are bounded and ρ∗ is a constant. Inserting of (3.13) gives, whereon has to take in (3.13) the indices i ∈ 0, 1, 2 and one assumes without loss ofgenerality u0(1) 6= 0,
0 = −σ (q(1))(− exp(0) + 2 exp(−2q(1))− exp(−4q(1))
)+q(1)
(− exp(0) + exp(−4q(1))
).
It is, using the binomial theorem,
−1 + e−4x
−1 + 2e−2x − e−4x=
(e−2x − 1)(e−2x + 1)
−(e−2x − 1)2=
1 + e−2x
1− e−2x=ex + e−x
ex − e−x= coth(x).
Hence, one gets as necessary condition
σ (q(1)) = q(1)1− exp(−4q(1))
1− 2 exp(−2q(1)) + exp(−4q(1))= q(1) coth (q(1)) .
One choice which satisfies this limit is
σ(q) = q coth(q).
This function satisfies also the conditions for the consistency of the scheme withartificial diffusion from Theorem 3.47, see Figure 3.8.
Figure 3.8: coth(x) and comparison with the conditions of Theorem 3.47.
2
37
Definition 3.56 Iljin scheme, Iljin–Allen–Southwell scheme. The scheme
−h2bi coth
(h
2εbi
)D+D−ui + biD
0ui + ciui = fi, for i = 1, . . . , N − 1,
u0 = uN = 0,
is called Iljin scheme or Iljin–Allen–Southwell (Il’in (1969); Allen and Southwell(1955)) scheme. In some applications it is called also Scharfetter–Gummel scheme(Scharfetter and Gummel (1969)). 2
Theorem 3.57 Uniform convergence of the Iljin–Allen–Southwell scheme.The Iljin–Allen–Southwell scheme converges in [0, 1] uniformly of first order in thediscrete maximum norm, i.e., it holds
maxi=1,...,N−1
|u(xi)− ui| ≤ Ch
with a constant C that is independent of ε and h.
Proof: The proof is rather long and involved, e.g., see Roos et al. (2008).
Example 3.58 Iljin–Allen–Southwell scheme. Consider
−εu′′ + u′ = 1 in (0, 1), u(0) = u(1) = 0,
with the solution
u(x) = x−exp
(− 1−x
ε
)− exp
(− 1
ε
)1− exp
(− 1
ε
) .
For ε = 10−3, one obtains the errors in the discrete maximum norm given in Ta-ble 3.1. It can be seen that the Iljin–Allen–Southwell scheme gives always the mostaccurate results. If the nodes are sufficiently away from the layer, then the computedsolution is even exact in the nodes. 2
Table 3.1: Example 3.58, errors in the discrete maximum norm.# intervals central diff. simple upwind IAS scheme
2 124.5 0.00199 04 31.004 0.00398 08 7.715 0.00793 0
16 2.0235 0.01574 032 0.91132 0.03100 2.2204e-1664 0.77305 0.06015 1.5543e-15
128 0.59276 0.11307 8.3598e-15256 0.34287 0.18371 1.2388e-14512 0.12997 0.19679 1.0976e-14
1024 0.03277 0.12933 5.8457e-142048 0.00750 0.07486 1.5675e-134096 0.00183 0.04076 2.8882e-13
3.4.2 Layer-Adapted Grids
Remark 3.59 Motivation. As already mentioned, the solution of a singularly per-turbed problem consists of two parts:
38
• the solution of the reduced problem, which is generally smooth and easily toapproximate numerically,• the correction term, which enforces the boundary condition at the outflow
boundary. This term is responsible for the appearance of the layer, i.e., forthe dramatic change of the solution in a very small interval.
Consider as typical example the two-point boundary value problem from Exam-ple 3.58. In the interval [0, 1 − ε], the solution has practically the form u(x) = x.This solution can be easily approximated on a coarse grid. The interesting part ofthe solution is in the interval [1− ε, 1]. If one chooses an equidistant grid with themesh width h, then it is generally h ε. Hence, the interval [1− ε, 1] is containedin [xN−1, xN ] = [1 − h, 1]. Consequently, one cannot expect with this choice toresolve the solution in [1− ε, 1] in a good way.
The idea of layer-adapted grids is to choose in the layer region a considerablyfiner grid than outside the layer region. This approach offers the possibility toapproximate the solution in the layer well. 2
Remark 3.60 Shishkin1 mesh. Consider, for simplicity of notation, a problemwhere the layer is situated at x = 0. In addition, let b = −β, β ∈ R+, a constant.The grid points are distributed in the form
xi = φ(i/N),
where one has to choose the function φ(ξ) in such a way that one obtains in aneighborhood of x = 0 a sufficiently fine mesh. The number N of intervals is given.
A mesh of Shishkin type is defined by
φ(ξ) =
σε
βφ(ξ) for ξ ∈ [0, 1/2],
1− 2
(1− σε
βln(N)
)(1− ξ) for ξ ∈ [1/2, 1],
with φ(1/2) = ln(N) and the parameter σ > 0. The Shishkin mesh (1988) isobtained with
φ(ξ) = 2 ln(N)ξ.
With this choice, one has for the nodes x0, . . . , xN/2, i ≥ 1,
xi − xi−1 = φ
(i
N
)− φ
(i− 1
N
)=σε
β2 ln(N)
(i
N− i− 1
N
)= 2
σε
β
ln(N)
N,
independent of i. For the nodes xN/2+1, . . . , xN , it is for i ≥ N/2 + 1,
xi − xi−1 = φ
(i
N
)− φ
(i− 1
N
)= 1− 2
(1− σε
βln(N)
)(1− i
N
)− 1 + 2
(1− σε
βln(N)
)(1− i− 1
N
)=
2
N− 2
σε
β
ln(N)
N,
independent of i. Hence, a piecewise equidistant mesh is defined. The transitionpoint from the very fine to the coarse grid is located at
τ = xN/2 =σε
βln(N).
The use of the Shishkin mesh is supported by results from the numerical analysis.2
1Grigory I. Shishkin
39
Theorem 3.61 Convergence of the simple upwind scheme on a Shishkinmesh. Consider the simple upwind scheme on a Shishkin mesh with the transitionpoint
τ = min
1
2,ε
βln(N)
,
i.e., with σ = 1. Then it holds the following error estimate
maxi=1,...,N−1
|u(xi)− ui| ≤ CN−1 ln(N),
where the constant C is independent of ε and N .
Proof: The proof is based on the decomposition of the solution in the part coming
from the reduced problem (smooth part) and the correction part. It is rather long and
involved, see Roos et al. (2008).
Remark 3.62 Layer-adapted meshes.
• The convergence is slightly suboptimal because of the factor ln(N). However,one can see in numerical examples that the given error estimate is sharp, i.e.,this factor cannot be omitted.• The idea to used layer-adapted mesh goes back to Bahvalov2 (Bahvalov (1969)).
In Bahvalov meshes, there is a smooth transition from the fine to the coarsemesh. However, the numerical analysis for schemes on Bahvalov meshes is ingeneral more complicated than on Shishkin meshes.• The a priori (before the numerical simulation) construction of appropriate layer-
adapted meshes needs more or less already the knowledge of the solution. Thisaspect is in applications not given, in particular for problems in two or threedimensions. Then, one needs an a posteriori (during the numerical simulation)construction of adapted grids. There are ways to perform this approach.• An essential finding of the analysis of numerical methods on a priori layer-
adapted grids is that one can use on an appropriate grids a simple numericalmethod and one obtains reasonable error estimates.• Using a Shishkin mesh, one has to define the finite differences in the node xN/2,
where the distances to the neighbor nodes are of different lengths.Let xi be a node and let the intervals [xi−1, xi] and [xi, xi+1] have the lengthhi and hi+1, respectively. There are no changes for the backward and forwardfinite difference compared with Definition 3.4, since for them one needs only oneof the neighbor intervals. Define
hi :=hi + hi+1
2,
then the central difference is the weighted average
D0v(xi) =1
2hi
(hiD
+v(xi) + hi+1D−v(xi)
).
The second derivative is approximated by
v′′(xi) ≈1
hi
(D+v(xi)−D−v(xi)
)=
1
hi
(vi+1 − vihi+1
− vi − vi−1
hi
).
• The matrices, which are obtained when using layer-adapted meshes, have gen-erally a very bad condition number.
2Nikolai Sergejewitsch Bahvalov (1934 – 2005)
40
• Layer adapted meshes can be applied in the finite element context in the samefashion as for finite difference methods.• A comprehensive monograph about numerical methods on layer-adapted meshes
is Linß (2010).
2
Example 3.63 Shishkin mesh. Consider again
−εu′′ + u′ = 1 in (0, 1), u(0) = u(1) = 0,
with the solution
u(x) = x−exp
(− 1−x
ε
)− exp
(− 1
ε
)1− exp
(− 1
ε
) ,
like in Example 3.58. In this example, the layer is at x = 1. Thus, the transitionpoint is chosen at
τ = xN/2 = 1− σε
βln(N) = 1− σε ln(N).
The errors in the discrete maximum norm for ε = 10−6 and σ = 2 are given inTable 3.2.
Table 3.2: Errors in the discrete maximum norm for simulations on a Shishkin mesh.
# intervals ‖u− uh‖∞,d
4 0.255848 0.16455
16 0.1083332 0.06912564 0.043656
128 0.026335256 0.015402512 0.0087902
1024 0.00492572048 0.00272254096 0.0014891
The results depend strongly on the choice of σ, exercise. 2
Remark 3.64 Summary. There are two ways to construct uniformly convergentmethods:
• by using an appropriately modified scheme on a simple grid,• by using a simple scheme on an appropriately chosen grid.
2
41
Chapter 4
Weak Solution Theory
Remark 4.1 Motivation. This chapter presents an extension of the notation of asolution of partial differential equations, the so-called weak or variational solution.This extension is necessary for the following reasons:
• In general, one cannot expect that a partial differential equation has a classicalsolution. For the existence of a classical solution, all parameters have to besufficiently smooth. In higher dimensions, also the domain has to satisfy certainregularity conditions. Such smoothness or regularity conditions are often notsatisfied in applications. Nevertheless, the processes which are modeled with thepartial differential equations occur and there is obviously a solution. However,this solution will not possess the (regularity) properties of the classical solutionand therefore one needs an extension of the notation of the solution.• It is already known from Numerical Mathematics 3 that finite element methods
are based on a weak or variational formulation of the partial differential equation.
2
Remark 4.2 Tools from functional analysis. The study of variational equationsand of finite element methods requires many tools from functional analysis, likeLebesgue spaces, Sobolev spaces, and a number of inequalities, like the Poincare–Friedrichs inequality. For a detailed introduction to the these tools, it will bereferred to the lecture notes on Numerical Mathematics 3. 2
4.1 Variational Formulation
Remark 4.3 Convection-diffusion-reaction equation. Let Ω ⊂ Rd, d ∈ 1, 2, 3,be a bounded domain with Lipschitz boundary ∂Ω. A linear convection-diffusion-reaction equation with homogeneous Dirichlet boundary conditions is given by
−ε∆u+ b · ∇u+ cu = f in Ω,u = 0 on ∂Ω.
(4.1)
In (4.1), b is the convection field. 2
Remark 4.4 Derivation of the variational or weak formulation. Consider problem(4.1). Multiplication of the differential equation with an appropriate function v(x),with v = 0 on ∂Ω, integration of the resulting equation on Ω, and integration by
42
Figure 4.1: Left: Henri Lebesgue (1875 – 1941), right: Serge Leewards Sobolev(1908 – 1989).
Figure 4.2: David Hilbert (1862 – 1943).
parts (Gaussian theorem) yields∫Ω
(−ε∆u+ b · ∇u+ cu) (x)v(x) dx
=
∫∂Ω
−ε(∇u · n)(s)v(s) ds +
∫Ω
(ε∇u · ∇v + (b · ∇u+ cu) v) (x) dx
=
∫Ω
(ε∇u · ∇v + (b · ∇u+ cu) v) (x) dx
=
∫Ω
f(x)v(x) dx.
Here, n is the outward pointing unit normal vector on ∂Ω. The integral on theboundary vanishes because of the boundary condition of the test function. Thehighest order derivative of u(x) has been transferred to v(x). Let (·, ·) denote theinner product of L2(Ω), then this equation can be written in a more compact form.
2
43
Figure 4.3: Left: Jules Henri Poincare (1854 – 1912), right: Kurt Otto Friedrichs(1901 – 1982).
Definition 4.5 Variational or weak formulation. Let b, c ∈ L∞(Ω) and f ∈H−1(Ω). The variational or weak formulation of the convection-diffusion-reactionequation (4.1) is: Find u ∈ H1
0 (Ω) such that for all v ∈ H10 (Ω)
ε (∇u,∇v) + (b · ∇u+ cu, v) = (f, v), (4.2)
where (with some abuse of notation, the dual pairing of H10 (Ω) and H−1(Ω) is also
denoted by (·, ·). A solution of (4.2) is called variational or weak solution. The spacein which the solution is searched is called solution or ansatz space. The functionsv(x) are called test functions and the space from which they come is the test space.
2
Remark 4.6 The variational formulation.
• The space H−1(Ω) is the dual space of H10 (Ω) and not of H1(Ω).
• With the given assumptions, all terms are well defined.• For the weak solution, only the first derivative, in the weak sense, is required.• Each classical solution is a weak solution. The other direction holds only for
sufficiently regular coefficients, right-hand side, and domain.
2
Remark 4.7 Other boundary conditions.
• Consider first inhomogeneous Dirichlet boundary conditions
u(x) = g(x) on ∂Ω.
These are so-called essential boundary conditions. Such boundary conditionsare included into the definition of the ansatz space
Va =v ∈ H1(Ω) : v|∂Ω = g
,
where the restriction to the boundary is understood in the sense of traces. Thetest space is still V = H1
0 (Ω). Then, the weak formulation reads as follows: Findu ∈ Va such that
ε (∇u,∇v) + (b · ∇u+ cu, v) = (f, v) ∀ v ∈ V.
44
A different way of writing the variational problem uses an extension ug ∈ H1(Ω)of g(x) to Ω. Then, one seeks u ∈ H1(Ω) such that u− ug ∈ V and
ε (∇u,∇v) + (b · ∇u+ cu, v) = (f, v) ∀ v ∈ V.
• Neumann boundary conditions appear in a straightforward way in the variationalformulation since the integral on Neumann boundaries appears in the integrationby parts. They are called natural boundary conditions. Let ∂Ω = ∂ΩD ∪ ∂ΩN
with ∂ΩD ∩ ∂ΩN = ∅, and assume for simplicity that u(x) = 0 for all x ∈ ∂ΩD.Let V0 = v ∈ H1(Ω) : v|∂ΩD
= 0, then the variational formulation has theform: Find u ∈ V0 such that
ε (∇u,∇v) + (b · ∇u+ cu, v) = (f, v) +
∫∂ΩN
ε (∇u · n) (s)v(s) ds ∀ v ∈ V0.
2
Definition 4.8 Properties of bilinear forms. Let (V, ‖·‖V ) be a Banach space.A map a : V × V → R is called
1. bilinear, if a(·, ·) is linear in both arguments,2. symmetric, if a(u, v) = a(v, u) for all u, v ∈ V ,3. positive, if a(v, v) ≥ 0 for all v ∈ V ,4. strictly positive or coercive or V -elliptic or positive definite if there is a m > 0
such that a(v, v) ≥ m ‖v‖2V for all v ∈ V ,5. bounded if there is a M > 0 such that
|a(u, v)| ≤M ‖u‖V ‖v‖V
for all u, v ∈ V .
2
Figure 4.4: Stefan Banach (1892 – 1945).
Example 4.9 Properties of the bilinear form of problem (4.2). Consider the bound-ary value problem (4.2).
45
• It is
a(u, v) :=
∫Ω
(ε∇u(x) · ∇v(x) + b(x) · ∇u(x)v(x) + c(x)u(x)v(x)
)dx (4.3)
a bilinear form in the space V = H10 (Ω). This property follows directly from the
linearity of integration and differentiation. It will be assumed that u, v ∈ H10 (Ω)
for the remainder of this example.• If b(x) = 0 for all x ∈ Ω, then a(u, v) is symmetric.• Let b ∈ C1(Ω) and c ∈ C(Ω). Applying integration by parts and the product
rule, one obtains
1
2
∫Ω
b(x) · ∇v(x)v(x) dx = −1
2
∫Ω
∇ · (b(x)v(x))v(x) dx
= −1
2
∫Ω
(∇ · b(x)) v(x)v(x) dx− 1
2
∫Ω
b(x) · ∇v(x)v(x) dx.
It follows that∫Ω
b(x) · ∇v(x)v(x) dx = −1
2
∫Ω
∇ · b(x)v(x)v(x) dx.
Inserting this relation into (4.3) with u(x) = v(x) yields
a(v, v) =
∫Ω
(ε (∇v(x))
2+
(−∇ · b
2(x) + c(x)
)(v(x))
2
)dx.
If
− 1
2∇ · b(x) + c(x) ≥ 0 (4.4)
for all x ∈ Ω, then it is for all v ∈ H10 (Ω)
a(v, v) ≥∫
Ω
ε (∇v(x))2dx = ε ‖∇v‖2L2(Ω) = ε ‖v‖2V .
Hence, a(·, ·) is coercive under condition (4.4) since ‖∇v‖L2(Ω) is a norm in
H10 (Ω).
Considerations of this form can be performed with the weaker assumptionsb,∇b, c ∈ L∞(Ω), since all integrals are still well defined under these assump-tions. Then, condition (4.4) for coercivity has to hold almost everywhere inΩ.• Let b, c ∈ L∞(Ω). One gets, using the Cauchy–Schwarz inequality, Holder’s
inequality, and the Poincare–Friedrichs inequality
|a(u, v)| ≤ ε ‖∇u‖L2(Ω) ‖∇v‖L2(Ω) + ‖b‖L∞(Ω) ‖∇u‖L2(Ω) ‖v‖L2(Ω)
+ ‖c‖L∞(Ω) ‖u‖L2(Ω) ‖v‖L2(Ω)
≤ ε ‖∇u‖L2(Ω) ‖∇v‖L2(Ω) + CPF ‖b‖L∞(Ω) ‖∇u‖L2(Ω) ‖∇v‖L2(Ω)
+C2PF ‖c‖L∞(Ω) ‖∇u‖L2(Ω) ‖∇v‖L2(Ω)
= C ‖∇u‖L2(Ω) ‖∇v‖L2(Ω) .
Hence, the bilinear form is bounded.
2
Theorem 4.10 Theorem of Lax–Milgram. Let a(·, ·) : V × V → R be abounded and coercive bilinear form on the Hilbert space V . Then, for each boundedlinear functional f ∈ V ′ there is exactly one u ∈ V with
a(u, v) = f(v) ∀ v ∈ V. (4.5)
46
Figure 4.5: Left: Augustin Louis Cauchy (1789 – 1857), right: Hermann AmandusSchwarz (1843 – 1921).
Proof: The proof can be found in the lecture notes of Numerical Mathematics 3.
Corollary 4.11 Existence and uniqueness of a solution of the weak prob-lem (4.3). Let V = H1
0 (Ω) and assume f ∈ V ′, b,∇b, c ∈ L∞(Ω) and (4.4) almosteverywhere in Ω. Then, (4.3) has a unique solution.
Proof: The statement follows directly from the Theorem of Lax–Milgram and the
properties of the bilinear form which were proved in Example 4.9.
47
Chapter 5
Finite Element Methods(FEM)
5.1 Generalities
Remark 5.1 Finite element methods. Finite element methods were one of themain topics of Numerical Mathematics 3. The knowledge of the lecture notes ofNumerical Mathematics 3 is assumed. Only a few issues, which are important forthis course on finite element methods for convection-dominated problems, will bereminded here.
Let T h be a family of regular triangulations consisting of mesh cells K. Thetriangulations are assumed to be quasi-uniform. The diameter of a mesh cell K isdenoted by hK and it is h = maxKhK. Parametric finite element spaces will beconsidered with affine maps between a reference cell K and all physical cells K. 2
Theorem 5.2 Local interpolation error estimate. Let IK : Cs(K)→ P (K)be an interpolation operator as defined in Numerical Mathematics 3, where P (K) isa polynomial space defined on K. Let p ∈ [1,∞] and (m+ 1− s)p > 1. Then thereis a constant c, which is independent of v ∈Wm+1,p(K), such that∥∥∥(v − IKv)(k)
∥∥∥Lp(K)
≤ chm+1−kK
∥∥∥v(m+1)∥∥∥Lp(K)
, 0 ≤ k ≤ m+ 1, (5.1)
for all v ∈Wm+1,p(K).
Proof: See lecture notes of Numerical Mathematics 3.
Theorem 5.3 Inverse estimate. Let 0 ≤ k ≤ l be natural numbers and letp, q ∈ [1,∞]. Then there is a constant Cinv, which depends only on k, l, p, q, K, P (K)such that∥∥Dlvh
∥∥Lq(K)
≤ Cinvh(k−l)−d(p−1−q−1)K
∥∥Dkvh∥∥Lp(K)
∀ vh ∈ P (K). (5.2)
Proof: See lecture notes of Numerical Mathematics 3.
5.2 The Galerkin Method
Remark 5.4 The Galerkin method. The standard finite element method, whichjust replaces in the variational formulation (4.2) the space V by V h ⊂ V , is calledGalerkin method: Find uh ∈ V h, such that for all vh ∈ V h
ε(∇uh,∇vh
)+(b · ∇uh + cuh, vh
)=(f, vh
). (5.3)
48
2
Figure 5.1: Boris Grigorievich Galerkin (1871 – 1945).
Theorem 5.5 Lemma of Cea. Let V h ⊂ V and assume the conditions of theTheorem of Lax–Milgram, Theorem 4.10. Then there is a unique solution of theproblem to find uh ∈ V h such that
a(uh, vh
)= f
(vh)∀ vh ∈ V h (5.4)
and it holds the error estimate∥∥u− uh∥∥V≤ M
minf
vh∈V h
∥∥u− vh∥∥V, (5.5)
where u is the unique solution of the continuous problem (4.5) and the constantsare defined in Definition 4.8.
Proof: The existence of a unique solution of the discrete problem follows directlyfrom the Theorem of Lax–Milgram, since the subspace of a Hilbert space is also a Hilbertspace and the properties of the bilinear form carry over from V to V h.
Computing the difference of the continuous equation (4.5) and the discrete equation(5.4) yields
a(u− uh, vh
)= 0 ∀ vh ∈ V h.
Withm ‖v‖2V ≤ a(v, v) and |a(u, v)| ≤M ‖u‖V ‖v‖V
if follows for all vh ∈ V h that∥∥∥u− uh∥∥∥2
V≤ 1
ma(u− uh, u− uh
)=
1
ma(u− uh, u− vh
)≤ M
m
∥∥∥u− uh∥∥∥V
∥∥∥u− vh∥∥∥V.
This inequality is equivalent to the statement of the theorem.
Remark 5.6 Application to the Galerkin finite element method. The properties ofthe bilinear form from problems (4.2) and (5.3) were studied in Example 4.9. Itwas shown that with appropriate regularity assumptions and under condition (4.4),the bilinear form is bounded with a constant M of order max‖b‖∞ , ‖c‖∞ and
49
it is coercive with m = ε. In this case, one can apply the Lemma of Cea and oneobtains the error estimate∥∥u− uh∥∥
V≤ C
max‖b‖∞ , ‖c‖∞ε
infvh∈V h
∥∥u− vh∥∥V, C ∈ R.
In the singularly perturbed case ε ‖b‖∞, the first factor of this estimate becomesvery large.
Thus, from this error estimate one cannot expect that the Galerkin finite elementsolution is accurate unless the second factor, which is the best approximation error,is very small. On uniformly refined grids, the best approximation error becomesvery small only if the dimension of V h becomes very large. 2
Example 5.7 Galerkin method for P1 finite elements in one dimension. Let Ω =(0, 1) and consider the case that the coefficients of the problem are constant, i.e.,b(x) = b, c(x) = c and f(x) = f . For the P1 finite element on the reference cellK = [−1, 1], the basis functions and their derivatives are given by
φ1(x) =1
2(−x+ 1), φ′1(x) = −1
2, φ2(x) =
1
2(x+ 1), φ′2(x) =
1
2.
Consider the matrix entry ai,i+1, which is computed with the test function φi(x),
which is transformed to φ1(x), and the ansatz function φi+1(x), which is transformed
to φ2(x). The common support is the mesh cell [xi, xi+1]. Let h denote the lengthof this cell, then one obtains
ai,i+1 =2ε
h
∫ 1
−1
1
2·(−1
2
)dx+ b
∫ 1
−1
1
2· 1
2(−x+ 1) dx
+ch
2
∫ 1
−1
1
2(x+ 1)
1
2(−x+ 1) dx
= − εh
+b
2+ch
6.
For the i-th component of the right-hand side, one gets
fi = f
∫ 1
0
φi(x) dx = hf.
The other matrix entries can be calculated in a similar way, exercise.If one applies the trapezoidal rule for the approximation of the integrals, one
obtainsch
2
∫ 1
−1
1
2(−x+ 1)
1
2(x+ 1) dx =
ch
22
0 + 0
2= 0.
Then, it follows that
ai,i+1 = − εh
+b
2.
For c = 0 (or with trapezoidal rule) these are, apart of a factor h, the samematrix entries as for the central difference scheme, see Remark 3.10. One gets forc = 0
−hεD+D−ui + hbiD0ui = hfi ⇐⇒ −εD+D−ui + biD
0ui = fi.
Hence, one obtains in this special case the same results with the Galerkin finiteelement method and the central finite difference scheme. For singularly perturbedproblems, these results are very bad.
The equivalence of the central finite difference method and the Galerkin finiteelement method generally does not hold if the coefficients are not constant. In higherdimensions, there are differences even for constant coefficients. But still, these twomethods give very similar results. At any rate, the Galerkin finite element methodis not useful in the singularly perturbed case also in more general situations. 2
50
5.3 Stabilized Finite Element Methods
Remark 5.8 On the H1(Ω) norm for the numerical analysis of singularly perturbedproblems. Consider the problem: Find u ∈ V = H1
0 (Ω) such that
a(u, v) = f(v) ∀ v ∈ V (5.6)
with
a(u, v) :=
∫Ω
(ε∇u(x) · ∇v(x) + b(x) · ∇u(x)v(x) + c(x)u(x)v(x)
)dx,
f(v) :=
∫Ω
f(x)v(x) dx.
Let the condition
−1
2∇ · b(x) + c(x) ≥ µ0 > 0 almost everywhere in Ω
be satisfied, which is stronger than condition (4.4). Then, an analogous calcula-tion as in Example 4.9 shows that a(·, ·) is uniformly coercive with respect to thefollowing norm, which depends on ε,
‖v‖2ε := ε |v|21,2 + µ0 ‖v‖20 = ε ‖∇v‖2L2(Ω) + µ0 ‖v‖2L2(Ω) ,
i.e., there is a constant m which does not depend on ε such that
a(v, v) ≥ m ‖v‖2ε ∀ v ∈ V.
Applying integration by parts, exercise, shows that there is a constant M , which isalso independent on ε, such that
|a(v, w)| ≤M ‖v‖ε ‖w‖H1(Ω) ∀ (v, w) ∈ V × V.
However, there is no constant M , which is independent of ε, with
|a(v, w)| ≤ M ‖v‖ε ‖w‖ε ∀ (v, w) ∈ V × V.
Applying the estimates with constants that are independent of ε, one obtains ina similar way as in the proof of the Lemma of Cea that∥∥u− uh∥∥
ε≤ C inf
vh∈V h
∥∥u− vh∥∥H1(Ω)
,
with C independent of ε. If V h is a standard finite element space (piecewise poly-nomial), then one can show that in layers it is
infvh∈V h
∥∥u− vh∥∥H1(Ω)
→∞ for ε→ 0,
for fixed h. Consequently, there is no uniform convergence∥∥u− uh∥∥
ε→ 0 for
h→ 0. The norm ‖·‖H1(Ω) is not suited for the investigation of numerical methodsfor convection-dominated problems. It turns out that the use of appropriate normsis important for the numerical analysis of finite element methods for convection-dominated problems. 2
51
5.3.1 Petrov–Galerkin Methods and Upwind Methods
Remark 5.9 Petrov–Galerkin method. A finite element method, whose ansatz andtest space are different, is called Petrov–Galerkin method. Let Sh be the ansatzspace and Th be the test space with dim(Sh) = dim(Th), then the Petrov–Galerkinmethod reads as follows: Find uh ∈ Sh such that
a(uh, vh
)= f
(vh)∀ vh ∈ Th.
2
Example 5.10 Petrov–Galerkin method and upwind method. Consider
−εu′′(x) + bu′(x) = 0
with b ∈ R \ 0 and homogeneous Dirichlet boundary conditions. Use as functionsin the ansatz space continuous piecewise linear functions
φi(x) =
(x− xi−1)/h for x ∈ [xi−1, xi],(xi+1 − x)/h for x ∈ [xi, xi+1],0 else,
, i = 1, . . . , N − 1.
Define the bubble functions
σi−1/2(x) =
4(x− xi−1)(xi − x)/h2 for x ∈ [xi−1, xi],0 else.
The test functions will be defined as piecewise quadratic functions
ψi(x) = φi(x) +3
2κ(σi−1/2(x)− σi+1/2(x)
), i = 1, . . . , N − 1,
where κ is an upwind parameter which has to be chosen. A direct calculation shows,exercise, that one obtains the following method
−εD+D−ui + b
[(1
2− κ)D+ui +
(1
2+ κ
)D−ui
]= 0.
For the choice κ = sgn(b)/2, one obtains the simple upwind finite difference scheme,see Definition 3.33.
A test function ψi(x), defined in the nodes 0, 0.5, 1, for κ = 1/2 is presentedin Figure 5.2.
2
Remark 5.11 Fitted upwind schemes. It is also possible to define fitted upwindmethods with Petrov–Galerkin methods, even for non-constant coefficients. How-ever, one does not obtain better results as for the Iljin–Allen–Southwell method,Theorem 3.57, i.e., one obtains not more than linear convergence. 2
5.3.2 The Streamline-Upwind Petrov–Galerkin (SUPG)Method
Remark 5.12 Goal. The goal consists in the construction of a method that is morestable than the Galerkin finite element method and which can be used with finiteelements of arbitrary order. The convergence of this method, in an appropriatenorm, should be of higher order.
Consider the problem (4.1) and assume at the moment that condition (4.4) issatisfied. Later, an even stronger assumption will be made. 2
52
Figure 5.2: Piecewise quadratic test function for κ = 1/2.
Remark 5.13 The basic idea. The basic idea consists in a penalization of largevalues of the so-called strong residual. Such methods are called residual-basedstabilizations.
Given a linear partial differential equation in strong form
Astrustr = f, f ∈ L2(Ω),
and its Galerkin finite element discretization: Find uh ∈ V h such that
ah(uh, vh
)=(f, vh
)∀ vh ∈ V h. (5.7)
For residual-based stabilizations, a modification of Astr is needed which is well-defined for finite element functions. This modification should be also a linear oper-ator and it is denoted by Ah
str : V h → L2(Ω). The (strong) residual is now definedby
rh(uh)
= Ahstru
h − f ∈ L2(Ω).
In general, it holds rh(uh)6= 0, but a good numerical approximation of the solution
of the continuous problem should have in some sense a small residual. Now, insteadof finding the solution of (5.7), the minimizer of the residual is searched, i.e, thefollowing optimization problem is considered
arg minuh∈V h
∥∥rh (uh)∥∥2
L2(Ω)= arg min
uh∈V h
(rh(uh), rh
(uh)). (5.8)
The necessary condition for taking the minimum is the vanishing of the Gateauxderivative. This derivative is computed by using the linearity of Ah
str and the bilin-earity of the inner product in L2(Ω)
0 = limε→0
(rh(uh + εvh
), rh
(uh + εvh
))−(rh(uh), rh
(uh))
ε
= limε→0
(rh(uh)
+ εAhstrv
h, rh(uh)
+ εAhstrv
h)−(rh(uh), rh
(uh))
ε
= 2(rh(uh), Ah
strvh)∀ vh ∈ V h.
It follows that the necessary condition for the solution of (5.8) is(rh(uh), Ah
strvh)
= 0.
53
A generalization consists in considering the minimization problem
arg minuh∈V h
∥∥∥δ1/2rh(uh)∥∥∥2
L2(Ω)= arg min
uh∈V h
(δrh
(uh), rh
(uh)). (5.9)
with the positive weighting function δ(x). Analogously to the derivation for thespecial case, one obtains as necessary condition for the minimum(
δrh(uh), Ah
strvh)
= 0. (5.10)
The solutions of (5.8) or (5.9) will not be identical to the solution of the Galerkindiscretization (5.7). It turns out that the reason for the Galerkin discretization tofail is that the solution possesses structures (scales) that are important but whichare not resolved by the used finite element space (grid). For convection-diffusionproblems, such structures are layers, particularly at boundaries. The numericalmethods should also compute sharp layers. However the sharpness of layers innumerical solutions is restricted by the resolution, which is generally much coarserthan the layer width. Hence, even for a numerical solution with sharp layers, theresidual in the layer regions are very large. In particular, a numerical solution withsharp layers (with respect to the resolution of the finite element space) will not bethe minimizer of (5.8) or (5.9), see Figure 5.3. The minimizers of (5.8) or (5.9) tendto possess strongly smeared layers and these solutions are useless in applications.For this reason, one considers in residual-based stabilizations a combination of theGalerkin discretization (5.7) and the minimization of the residual
ah(uh, vh
)+(δrh
(uh), Ah
strvh)
=(f, vh
)∀ vh ∈ V h. (5.11)
The goal of numerical analysis consists in determining the weighting function δoptimally in an asymptotic sense. 2
Figure 5.3: Function with sharp layer (solid line) and optimal piecewise linear ap-proximation in a mesh cell K (dashed line). The equation which is fulfilled bythe function in K is far from being satisfied by the piecewise linear approxima-tion. Hence, despite the approximation is of the type considered to be optimal, theresidual will be large.
Definition 5.14 Streamline-Upwind Petrov–Galerkin FEM, SUPG meth-od, Streamline-Diffusion FEM, SDFEM. The Streamline-Upwind Petrov–Galerkin (SUPG) FEM or Streamline-Diffusion FEM (SDFEM) has the form: Finduh ∈ V h, such that
ah(uh, vh
)= fh
(vh)∀ vh ∈ V h (5.12)
54
with
ah(v, w) := a(v, w) (5.13)
+∑
K∈T h
∫K
δK
(− ε∆v(x) + b(x) · ∇v(x) + c(x)v(x)
)(b(x) · ∇w(x)
)dx,
fh(w) := (f, w) +∑
K∈T h
∫K
δKf(x)(b(x) · ∇w(x)
)dx.
Here, δK are user-chosen weights, which are called stabilization parameters orSUPG parameters. 2
Remark 5.15 Concerning the SUPG method.
• The method was developed in Hughes and Brooks (1979); Brooks and Hughes(1982).• The name “SUPG” comes from the fact that the method can be considered as
a Petrov–Galerkin method with the test space
span
w(x) +∑
K∈T h
δKb(x) · ∇w(x)
.
• The SUPG method introduces artificial diffusion only in streamline directionb(x) · ∇w(x). From this property, the name “Streamline Diffusion FEM” origi-nates.• The operator Ah
str is given in the second part of the bilinear form (5.13). Thesecond derivative for finite element functions is defined only piecewise.• In the stabilization term of the SUPG method, not the strong operator Ah
str
applied to the test function is used, as in (5.11), but only the first order termcontained in this expression. However, for singularly perturbed problems, thefirst order term is the dominating term of the strong operator applied to thetest function.It is also possible to define a method with the strong operator applied to thetest function, the so-called Galerkin least squared (GLS) method. In general,one gets similar results with the SUPG and the GLS method, but the SUPGmethod is easier to implement.
• Generally, the SUPG parameter is a general function. However, in practice itis often chosen as a piecewise constant function. The goal of the finite elementerror analysis consists in proposing a good choice of this parameter.
2
Example 5.16 SUPG in one dimension for P1 finite elements. Consider Ω = (0, 1)and V h = P1 on an equidistant grid with hi = h, i = 1, . . . , N . If all coefficients areconstant, c = 0, and if one chooses the SUPG parameter also as a constant, thenthe left-hand side of the SUPG method reduces to
ε((uh)′,(vh)′
) + (b(uh)′, vh) +
N∑i=1
δ
∫ xi
xi−1
(− ε · 0 + b
(uh)′
(x))(b(vh)′
(x))dx
= ε((uh)′,(vh)′
) + b((uh)′, vh) + δb2(
(uh)′,(vh)′
).
This expression is of the form of a Galerkin finite element method for an equationwith left-hand side
−(ε+ δb2
)u′′(x) + bu′(x).
55
It is known from Example 5.7 that the Galerkin finite element method is equivalentto a central finite difference scheme. The right-hand side of the SUPG method is
(f, vh) +
N∑i=1
δ
∫ xi
xi−1
fb(vh)′
(x) dx = (f, vh) + δfb
N∑i=1
∫ xi
xi−1
(vh)′
(x) dx︸ ︷︷ ︸=0
= (f, vh) = hfi.
The sum vanishes, since each test function(vh)′
(x) can be written as a linearcombination of the basis functions φi(x) of P1 and the integral of the derivativeof each basis function vanishes. Alternatively, one can apply integration by partsto check this fact.
Altogether, the SUPG method with the conditions stated above is equivalent tothe fitted finite difference scheme (3.10)
−ε(
1 + δb2
ε
)D+D−ui + bD0ui,= fi,
i.e., σ(q) = 1 + δb2/ε, q = bh/(2ε). Choosing the SUPG parameter by
δ(q) =h
2b
(coth(q)− 1
q
),
then it is
σ(q) = 1 +hb2
2bε
(coth(q)− 1
q
)= 1 + q
(coth(q)− 1
q
)= q coth(q).
One obtains the Iljin–Allen–Southwell scheme. With δ = h/(2b), one gets the simpleupwind scheme.
These simple connections do not hold in higher dimensions. 2
Definition 5.17 Consistent finite element method. Let u(x) be a sufficientlysmooth solution of: Find u ∈ V such that
a(u, v) = f(v) ∀ v ∈ V,
where a(·, ·) is an appropriate bilinear form and f(·) an appropriate functional. Afinite element method related to this problem: Find uh ∈ V h such that
ah(uh, vh
)= fh
(vh)∀ vh ∈ V h
is called consistent, if
ah(u, vh
)= fh
(vh)∀ vh ∈ V h. (5.14)
2
Remark 5.18 Consistency. Note that consistency of a finite element method is notthe same as consistency of a finite difference method, see Definition 3.6. For finiteelement methods, consistency means that a sufficiently smooth solution satisfiesalso the discrete equation. 2
56
Lemma 5.19 Galerkin orthogonality. A consistent finite element method hasthe property of the Galerkin orthogonality
ah(u− uh, vh
)= 0 ∀ vh ∈ V h. (5.15)
The error is “orthogonal” to the finite element space.
Proof: The statement of the lemma follows immediately by subtracting (5.12) and
(5.14).
Lemma 5.20 Consistency of the SUPG method. The SUPG method (5.12)– (5.13) is consistent.
Proof: A sufficiently smooth solution u(x) of (4.2) satisfies the strong form of theequation even pointwise. Hence, the residual is pointwise zero. Inserting this solution intothe SUPG formulation (5.12) – (5.13) results in a vanishing of the stabilization term. Itremains
a(u, vh
)= f
(vh)∀ vh ∈ V h,
which is satisfied by any weak solution since V h ⊂ V . That means, the smooth solution
satisfies also the discrete equation.
Definition 5.21 SUGP norm. Let
− 1
2∇ · b(x) + c(x) ≥ ω > 0. (5.16)
In V h, the SUPG norm is defined by
∣∣∣∣∣∣vh∣∣∣∣∣∣SUPG
:=
ε ∣∣vh∣∣21
+ ω∥∥vh∥∥2
0+∑
K∈T h
∥∥∥δ1/2K
(b · ∇vh
)∥∥∥2
0,K
1/2
,
where ‖·‖0,K denotes the norm in L2(K). 2
Theorem 5.22 Coercivity of the SUPG bilinear form. Assume b ∈W 1,∞(Ω),c ∈ L∞(Ω), (5.16), and let
0 < δK ≤1
2min
h2K
εC2inv
,ω
‖c‖2L∞(K)
, (5.17)
where Cinv is the constant in the inverse estimate (5.2) Then, the SUPG bilinearform is coercive with respect to the SUPG norm, i.e., it is
ah(vh, vh
)≥ 1
2
∣∣∣∣∣∣vh∣∣∣∣∣∣2SUPG
∀ vh ∈ V h.
Proof: Integration by parts gives, see Example 4.9,(b · ∇vh + cvh, vh
)=
((−∇ · b
2+ c
)vh, vh
)∀ vh ∈ V h.
With the definition of ω, one obtains
ah(vh, vh
)= ε
∣∣∣vh∣∣∣21
+
∫Ω
(c(x)− ∇ · b(x)
2
)︸ ︷︷ ︸
≥ω>0
(vh)2
(x) dx +∑K∈T h
∥∥∥δ1/2K
(b · ∇vh
)∥∥∥2
0,K
+∑K∈T h
∫K
δK(−ε∆vh(x) + c(x)vh(x)
)(b(x) · ∇vh(x)
)dx
≥∣∣∣∣∣∣∣∣∣vh∣∣∣∣∣∣∣∣∣2
SUPG−
∣∣∣∣∣∣∑K∈T h
∫K
δK(−ε∆vh(x) + c(x)vh(x)
)(b(x) · ∇vh(x)
)dx
∣∣∣∣∣∣ .57
Now, the last term will be estimated from above. Then, one obtains altogether an estimatefrom below if the estimate of the last term is subtracted from the first term. In the followingestimate, one uses the conditions (5.17) on the SUPG parameter. It is for each K ∈ T h∣∣∣∣∫K
δK(−ε∆vh(x) + c(x)vh(x)
)(b · ∇vh(x)
)dx
∣∣∣∣≤
∫K
(δ
1/2K ε
∣∣∣∆vh(x)∣∣∣ )(δ1/2
K
∣∣∣b · ∇vh(x)∣∣∣ ) dx
+
∫K
(δ
1/2K |c(x)|
∣∣∣vh(x)∣∣∣ )(δ1/2
K
∣∣∣b · ∇vh(x)∣∣∣ ) dx
CS
≤(δ
1/2K ε
∥∥∥∆vh∥∥∥
0,K+ δ
1/2K ‖c‖L∞(K)
∥∥∥vh∥∥∥0,K
)∥∥∥δ1/2K
(b · ∇vh
)∥∥∥0,K
(5.2)
≤(δ
1/2K
εCinv
hK
∥∥∥∇vh∥∥∥0,K
+ δ1/2K ‖c‖L∞(K)
∥∥∥vh∥∥∥0,K
)∥∥∥δ1/2K
(b · ∇vh
)∥∥∥0,K
(5.17)
≤
(hK√
2εCinv
εCinv
hK
∥∥∥∇vh∥∥∥0,K
+
√ω√
2 ‖c‖L∞(K)
‖c‖L∞(K)
∥∥∥vh∥∥∥0,K
)∥∥∥δ1/2K
(b · ∇vh
)∥∥∥0,K
=
(√ε
2
∥∥∥∇vh∥∥∥0,K
+
√ω
2
∥∥∥vh∥∥∥0,K
)∥∥∥δ1/2K
(b · ∇vh
)∥∥∥0,K
Young
≤ ε
2
∥∥∥∇vh∥∥∥2
0,K+
1
4
∥∥∥δ1/2K
(b · ∇vh
)∥∥∥2
0,K+ω
2
∥∥∥vh∥∥∥2
0,K+
1
4
∥∥∥δ1/2K
(b · ∇vh
)∥∥∥2
0,K
=1
2
∣∣∣∣∣∣∣∣∣vh∣∣∣∣∣∣∣∣∣SUPG,K
.
Now, the proof is finished by summing over all mesh cells and inserting this estimate in
the first estimate of the proof.
Corollary 5.23 Coercivity of the SUPG bilinear form for P1 finite ele-ments. Let the assumptions of Theorem 5.22 with respect to the coefficients ofthe problem be valid. For piecewise linear finite elements, the SUPG bilinear form(5.13) is coercive with respect to the SUPG norm if
0 < δK ≤ω
‖c‖2L∞(K)
. (5.18)
Proof: The proof is the same as for Theorem 5.22, where one uses that for piecewise
linear finite elements ∆vh(x)|K = 0 for all K ∈ T h. Thus, the corresponding terms do
not appear in the proof.
Corollary 5.24 Existenz and uniqueness of a solution of the SUPG method.Let the assumptions of Theorem 5.22 and Corollary 5.23, respectively, be valid.Then, the SUPG finite element method (5.12) – (5.13) has a unique solution.
Proof: The statement is obtained by the application of the Theorem of Lax–Milgram,
Theorem 4.10. The coercivity of the bilinear form was proved in Theorem 5.22 and Corol-
lary 5.23, respectively. For the boundedness, one uses similar estimates as in the proof of
Theorem 5.22 and in Example 4.9, exercise.
Remark 5.25 On the coercivity of the SUPG bilinear form.
• The proof of Theorem 5.22 is typical for the numerical analysis of stabilized finiteelement methods. One tries to get rid of the troubling terms by estimating themwith the used norm. This approach works only if one uses an appropriate norm.In particular, the stabilization has to appear in the norm.• Theorem 5.22 provides an upper bound for the SUPG parameter. This bound
is generally not critical in applications.
58
• From Theorem 5.22 one obtains the stability of the SUPG method with respectto the SUPG norm. Stability means that an appropriate norm of the solutioncan be estimated with the data of the problem. It is∣∣∣∣∣∣uh∣∣∣∣∣∣2
SUPG
≤ 2ah(uh, uh
)= 2fh
(uh)
= 2(f, uh
)+ 2
∑K∈T h
∫K
δKf(x)(b(x) · ∇uh(x)
)dx
CS≤ 2√
ω‖f‖0
√ω∥∥uh∥∥
0+ 2
∑K∈T h
∥∥∥δ1/2K f
∥∥∥0,K
∥∥∥δ1/2K
(b · ∇uh
)∥∥∥0,K
Young
≤ C ‖f‖20 +1
2
ω ∥∥uh∥∥2
0+∑
K∈T h
∥∥∥δ1/2K
(b · ∇uh
)∥∥∥2
0,K
.
Now, the last terms on the right-hand side can be absorbed in the left-hand sideand one has stability. The stability constant depends on ω and on the upperbound of δK .• All vh ∈ V h satisfy ∣∣∣∣∣∣vh∣∣∣∣∣∣
SUPG≥ min1, ω
∥∥vh∥∥ε.
Hence, the SUPG method is also stable with respect to the norm ‖·‖ε. Withrespect to this norm, also the Galerkin finite element method is stable, howeverthis method is not stable with respect to the SUPG norm. That means, thestability of the SUPG method is stronger than the stability of the Galerkinfinite element method.
2
Theorem 5.26 Convergence of the SUPG method. Let the solution of (4.2)satisfy u ∈ Hk+1(Ω), k ≥ 1, let b ∈W 1,∞(Ω), c ∈ L∞(Ω), and consider the SUPGmethod for Pk finite elements. Let the SUPG parameter be given as follows
δK =
C0h2K
εfor hK < ε,
C0hK for ε ≤ hK ,(5.19)
where the constant C0 > 0 is sufficiently small such that (5.17) is satisfied for k ≥ 2or (5.18) for k = 1, respectively. Then, the solution uh ∈ Pk of the SUPG method(5.12) satisfies the following error estimate∣∣∣∣∣∣u− uh∣∣∣∣∣∣
SUPG≤ C
(ε1/2hk + hk+1/2
)|u|k+1 ,
where the constant C does not depend on ε and h.
Proof: Let uhI ∈ V h be the Lagrange interpolant of u(x). One obtains with thetriangle inequality∣∣∣∣∣∣∣∣∣u− uh∣∣∣∣∣∣∣∣∣
SUPG≤∣∣∣∣∣∣∣∣∣u− uhI ∣∣∣∣∣∣∣∣∣
SUPG+∣∣∣∣∣∣∣∣∣uhI − uh∣∣∣∣∣∣∣∣∣
SUPG.
The first term on the right-hand side is the interpolation error. Using the interpolationerror estimate (5.1), which is applied for each term of the SUPG norm individually, one
59
Figure 5.4: Joseph–Louis Lagrange (1736 – 1813).
gets ∣∣∣∣∣∣∣∣∣u− uhI ∣∣∣∣∣∣∣∣∣SUPG
≤
Cεh2k |u|2k+1 + Cωh2(k+1) |u|2k+1 + C∑K∈T h
δK ‖b‖2∞,K h2kK |u|2k+1,K
1/2
≤ C(εh2k + h2(k+1) + h2k+1
)1/2
|u|k+1
≤ C(ε1/2hk + hk+1/2
)|u|k+1 .
Here, it was used that for both regimes it is δK ≤ C0hK ≤ Ch.Consider now the second term on the right-hand side. The coercivity, Theorem 5.22,
and the Galerkin orthogonality yield
1
2
∣∣∣∣∣∣∣∣∣uhI − uh∣∣∣∣∣∣∣∣∣2SUPG
≤ ah(uhI − uh, uhI − uh
)= ah
(uhI − u, uhI − uh
).
Now, the triangle inequality is applied to ah(uhI − u, uhI − uh
)and every term is estimated
individually. In these estimates, the interpolation estimate (5.1) plays an important role.Let wh = uhI − uh. One obtains for the diffusion term∣∣∣ε(∇(uhI − u) ,∇wh)∣∣∣
CS
≤ ε∥∥∥∇(uhI − u)∥∥∥
0
∥∥∥∇wh∥∥∥0
= ε1/2∥∥∥∇(uhI − u)∥∥∥
0ε1/2
∥∥∥∇wh∥∥∥0
(5.1)
≤ Cε1/2hk |u|k+1 ε1/2∥∥∥∇wh∥∥∥
0≤ Cε1/2hk |u|k+1
∣∣∣∣∣∣∣∣∣wh∣∣∣∣∣∣∣∣∣SUPG
.
For the reactive term, one obtains in a similar way∣∣∣(c(uhI − u), wh)∣∣∣ CS
≤ ‖c‖∞∥∥∥uhI − u∥∥∥
0
∥∥∥wh∥∥∥0
= ω−1/2 ‖c‖∞∥∥∥uhI − u∥∥∥
0ω1/2
∥∥∥wh∥∥∥0
(5.1)
≤ Chk+1 |u|k+1
∣∣∣∣∣∣∣∣∣wh∣∣∣∣∣∣∣∣∣SUPG
.
Next, the terms are considered which come from the SUPG stabilization. Since for both
60
regimes it is εδK ≤ C0h2K , one gets∣∣∣∣∣∣
∑K∈T h
(−ε∆
(uhI − u
), δKb · ∇wh
)K
∣∣∣∣∣∣CS
≤∑K∈T h
ε1/2∥∥∥∆(uhI − u
)∥∥∥0,K
ε1/2δ1/2K
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥0,K
≤ C1/20
∑K∈T h
hKε1/2∥∥∥∆(uhI − u
)∥∥∥0,K
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥0,K
CS
≤ C1/20 ε1/2h
∑K∈T h
∥∥∥∆(uhI − u
)∥∥∥2
0,K
1/2 ∑K∈T h
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥2
0,K
1/2
(5.1)
≤ Cε1/2h
∑K∈T h
h2(k−1)K |u|2k+1,K
1/2 ∑K∈T h
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥2
0,K
1/2
≤ Cε1/2hk |u|k+1
∣∣∣∣∣∣∣∣∣wh∣∣∣∣∣∣∣∣∣SUPG
.
For the other terms, one obtains with the relation δK ≤ C0hK , which holds for bothregimes,∣∣∣∣∣∣
∑K∈T h
(b · ∇
(uhI − u
)+ c
(uhI − u
), δK
(b · ∇wh
))∣∣∣∣∣∣CS
≤∑K∈T h
‖b‖∞∥∥∥∇(uhI − u)∥∥∥
0,Kδ
1/2K
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥0,K
+∑K∈T h
‖c‖∞∥∥∥uhI − u∥∥∥
0,Kδ
1/2K
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥0,K
≤ C
∑K∈T h
h1/2K
∥∥∥∇(uhI − u)∥∥∥0,K
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥0,K
+∑K∈T h
h1/2K
∥∥∥uhI − u∥∥∥0,K
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥0,K
CS
≤ Ch1/2
∑K∈T h
∥∥∥∇(uhI − u)∥∥∥2
0,K
1/2
+
∑K∈T h
∥∥∥uhI − u∥∥∥2
0,K
1/2
×
∑K∈T h
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥2
0,K
1/2
(5.1)
≤ C(hk+1/2 + hk+3/2
)|u|k+1
∣∣∣∣∣∣∣∣∣wh∣∣∣∣∣∣∣∣∣SUPG
.
To obtain an optimal estimate for the convective term, one has to apply first integrationby parts(
b · ∇(uhI − u
), wh
)=
(∇(uhI − u
),bwh
)= −
(uhI − u,∇ ·
(bwh
))= −
(uhI − u, (∇ · b)wh
)−(uhI − u,b · ∇wh
).
Both terms on the right-hand side are estimated separately. Using the same tools as for
61
the other estimates, ones obtains
∣∣∣(uhI − u, (∇ · b)wh)∣∣∣ ≤ ω−1/2 ‖∇ · b‖∞
∑K∈T h
∥∥∥uhI − u∥∥∥2
0,K
1/2
ω1/2∥∥∥wh∥∥∥
0
≤ Chk+1 |u|k+1
∣∣∣∣∣∣∣∣∣wh∣∣∣∣∣∣∣∣∣SUPG
.
In the estimate of the other term, one has to distinguish if in the mesh cell K it is ε ≤ hKor ε > hK . One gets∣∣∣(uhI − u,b · ∇wh)∣∣∣
CS
≤∑ε≤hK
δ−1/2K
∥∥∥uhI − u∥∥∥0,K
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥0,K
+∑ε>hK
‖b‖∞∥∥∥uhI − u∥∥∥
0,K
∥∥∥∇wh∥∥∥0,K
(5.1)
≤ C
∑ε≤hK
δ−1/2K hk+1
K |u|k+1,K
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥0,K
+∑ε>hK
hk+1K |u|k+1,K
∥∥∥∇wh∥∥∥0,K
C0hK≤δK ,ε>hK
≤ C
∑ε≤hK
C−1/20 h
−1/2K hk+1
K |u|k+1,K
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥0,K
+∑ε>hK
hk+1/2K |u|k+1,K ε
1/2∥∥∥∇wh∥∥∥
0,K
CS
≤ Chk+1/2 |u|k+1
∑K∈T h
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥2
0,K
1/2
+ ε∣∣∣wh∣∣∣
1
≤ Chk+1/2 |u|k+1
∣∣∣∣∣∣∣∣∣wh∣∣∣∣∣∣∣∣∣SUPG
.
Summarizing all estimates, the statement of the theorem is proved.
Remark 5.27 Concerning the error estimate.
• In the convection-dominated regime ε h, the order of convergence in theSUPG norm is k+ 1/2 and in the diffusion-dominated case it is k. In the lattercase, the SUPG norm is essentially the H1(Ω) semi norm such that order k isoptimal.
• It is essential for obtaining an estimate with a constant C which is independentof ε that the term ∑
K∈T h
∥∥∥δ1/2K
(b · ∇wh
)∥∥∥2
0,K
1/2
is part of the norm, which is used for estimating the error. Such an estimatedoes not hold for the norm ‖·‖ε.
• For the interpretation of the results one has to take into account that differentstabilization parameters by choosing different values of C0 lead also to differentnorms on the left-hand side of the estimate.
• On the other hand, the value of a constant which is independent of ε is ques-tionable since in general |u|k+1 depends on ε.
62
• In numerical simulations, often one can observe even convergence of order hk+1
for the error in L2(Ω), in particular on structured grids. However, in Zhou(1997) examples were constructed which show that the estimate of Theorem 5.26is sharp also for the error in L2(Ω).
2
Example 5.28 SUPG in one dimension, the user-chosen constant. The standardexample
−εu′′ + u′ = 1 on (0, 1), u(0) = u(1) = 0,
does not fit into the theory of the SUPG method since c(x)− b′(x)2 = 0. Nevertheless,
one can apply the SUPG method also for this example. An error estimate in thenorm (
ε∣∣vh∣∣2
1+
N∑i=1
∥∥∥δ1/2K b
(vh)′∥∥∥2
0,K
)1/2
can be proved. One looses the control on the error in the L2(0, 1) norm.A fundamental problem in the application of the SUPG method is the free
constant C0 in the definition of the parameter (5.19). At the present example,for ε = 10−6, one can observe very well that one obtains for different constantsrather different numerical results, see Figure 5.5. If C0 is too large, then the layeris smeared, for an appropriate value of C0 one obtains a solution which is almostexact in the nodes, and if C0 is too small, then one can observe spurious (unphysical)oscillations in the layer.
Figure 5.5: Results obtained with the SUPG method for the standard one-dimensional example, C0 = 1, C0 = 0.5, C0 = 0.25 from left to right, h = 1/32, P1
finite elements.
For general problems, it is difficult to choose C0 appropriately. In higher di-mensions, it is in general not possible to find C0 such that the solution is (almost)exact in the nodes. Often, the solution computed with the SUPG method in higherdimensions exhibits spurious oscillations at the layers, e.g., see Example 5.30. 2
Remark 5.29 Different choices of the SUPG parameter. In practice, one takesinstead of (5.19) also the parameter
δK =hK
2 ‖b‖L∞(K)
(coth(PeK)− 1
PeK
), PeK =
‖b‖L∞(K) hK
2ε, (5.20)
where PeK is the local Peclet number, since in one dimensions one recovers undercertain conditions the Iljin–Allen–Southwell scheme, see Example 5.16. There isno user-chosen constant in this parameter. Asymptotically, both parameters (5.19)and (5.20) have the same behavior. 2
63
Example 5.30 SUPG in two dimensions. A standard test problem in two dimen-sions has the form
−ε∆u+ (1, 0)T · ∇u = 1 in Ω = (0, 1)2,u = 0 on ∂Ω.
Besides the layer at the outflow boundary x = 1, there are also two layers at theboundaries y = 0 and y = 1. The layer at the outflow boundary is often calledexponential layer and the layers parallel to the flow direction parabolic layers.
A numerical solution obtained for ε = 10−8 with the Q1 finite element methodon a rather coarse grid and the SUPG parameter (5.20) is shown in Figure 5.6. Onecan see very well large spurious oscillations, in particular at the parabolic layers.These oscillations are a typical feature of solutions obtained with the SUPG method.They might become smaller with higher order elements or on finer grids. But theywill generally vanish only if the layer is resolved.
Figure 5.6: Result obtained with the SUPG method, Q1 finite elements, and theSUPG parameter (5.20).
2
64
Chapter 6
Outlook
Remark 6.1 Further linear residual-based stabilizations. Besides the SUPG meth-od, there are other residual-based stabilizations for finite element methods wereproposed. As already mentioned, one can take the full residual in the stabilizationterm also for the test function, which gives the Galerkin least squares (GLS) method.Another proposal is the so-called unusual finite element method. The numericalanalysis of these methods is similar to the numerical analysis of the SUPG method.However, the numerical results with the other methods are generally not betterthan with the SUPG method. Since the SUPG method is easiest to implement, itis preferred and the other methods are only of little importance in practice.
The idea of residual-based stabilization techniques is also used for other prob-lems, e.g., for (turbulent) incompressible flow problems governed by the incompress-ible Navier–Stokes equations. 2
Remark 6.2 Spurious Oscillations at Layers Diminishing (SOLD) methods. Theapplicability of the SUPG method in practice is, however, restricted by the appear-ance of generally considerable spurious oscillations in the numerical solutions. Theseoscillations correspond to unphysical values, like negative concentrations, which donot appear in practical problems. In addition, if a numerical solution with unphys-ical values is used in coupled problems as a parameter in other equations, one canget easily instabilities in the numerical simulations.
There has been much effort to reduce the spurious oscillations in residual-basedstabilizations. A detailed investigation of the solutions computed with the SUPGmethod shows that the spurious oscillations appear orthogonal to the streamlinedirection. Remind that the SUPG method only introduces numerical diffusion instreamline direction. The basic idea consists now in extending the SUPG methodby introducing some numerical diffusion orthogonal to the streamline direction. Toachieve methods of higher order, this numerical diffusion has to depend on the finiteelement solution. Hence, one obtains a nonlinear term. There are many proposalsfor such terms and this class of methods is called Spurious Oscillations at LayersDiminishing (SOLD) methods or shock capturing methods. Altogether, one hasto solve a nonlinear discrete problem for a linear boundary value problem. Thequestions of existence and uniqueness of a solution of the nonlinear problem arise.There are only answers for very few SOLD methods. A competitive study Johnand Knobloch (2007) showed that most of the SOLD methods in fact reduce thesize of the spurious oscillations of the SUPG method. However, even the reducedoscillations are still considerable large. Altogether, none of the SOLD methodsproposed so far cures the drawback of the SUPG method. 2
Remark 6.3 Other stabilized finite element methods. In the last decade, severalapproaches for stabilized finite element methods have been proposed which do not
65
rely on the residual, like Local Projection Stabilization (LPS) methods and Contin-uous Interior Penalty (CIP) methods. A numerical analysis for these methods canbe performed, which is, however, more complicated than for the SUPG method. Insimulations, the results are generally not better than for the SUPG method, ofteneven worse. 2
Remark 6.4 Algebraic flux correction schemes. All mentioned stabilizations sofar modify the bilinear form of the Galerkin finite element method to introducesome numerical diffusion. A much different approach are algebraic flux correctionschemes, see Kuzmin (2007), which start with the matrix-vector equation obtainedwith the Galerkin finite element method. Then, the matrix is modified such thatone gets an M-matrix. This modification introduces numerical diffusion and theresulting scheme satisfies the discrete maximum principle, but the solutions aresmeared very much. Thus, the next step consists in modifying the right-hand sideto remove the numerical diffusion where it is not necessary. This step is a nonlinearstep. Almost nothing is known about the existence and uniqueness of the solutionof the nonlinear problem. In practice, one observes sometimes difficulties in theconvergence to compute the solution by a fixed point iteration. This class of methodis only well defined for linear and bilinear finite elements. 2
Remark 6.5 Summary. There are a lot of proposals for stabilized discretizationsfor linear convection-dominated convection-diffusion equations. But none of theproposals can be recommended for all practical purposes. For many methods, thereare open questions concerning their numerical analysis. Therefore, the design andanalysis of numerical methods for convection-dominated problems is still an activefield of research. 2
66
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68
Index
ansatz space, 44
backward difference, 20, 27Bahvalov mesh, 40bilinear form, 45boundary condition
Dirichlet, 4essential, 44natural, 45Neumann, 4Robin, 4
boundary value problemsingularly perturbed, 26
central difference, 20, 40coercive bilinear form, 45comparison principle, 16
discrete, 23consistency
of a difference scheme, 21consistent
finite element method, 56consistent finite difference operator,
20convective term, 4convergence
of a difference scheme, 22uniform, 35
difference schemecentral, 21
differential operator, 6diffusion
artificial, 31diffusive term, 4Dirichlet boundary condition, 4, 44discrete maximum norm, 20
essential boundary condition, 44
finite element methodconsistent, 56
formulationvariational, 44weak, 44
forward difference, 20function
Green’s, 13functions
linearly independent, 8
Galerkin orthogonality, 56Green’s function, 13grid function, 19
inverse monotonicity, 16
layer, 6exponential, 64parabolic, 64smearing of, 33
lemmaCea, 49
linearly independent functions, 8
M-matrix, 23M-matrix criterion, 23majorizing element, 23matrix
inverse-monotone, 22M-, 23
maximum normdiscrete, 20
maximum principle, 15strong, 17
meshBahvalov, 40Shishkin, 39
methodGalerkin, 48Galerkin least squared, 55Petrov–Galerkin, 52stabilized, 33
monotonicity, inverse, 16
natural boundary condition, 45Neumann boundary condition, 4, 45norm
SUPG, 57
69
operator, 6order
natural, 22oscillation
spurious, 63
Peclet number, 4, 63parameter
SUPG, 54Petrov–Galerkin method, 52positive definite bilinear form, 45
reactive term, 4reduced problem, 7reduced solution, 7Robin boundary condition, 4
schemefitted upwind, 32Iljin, 38Iljin–Allen–Southwell, 38, 56Samarskii upwind, 34Scharfetter–Gummel, 38upwind, 28, 40with artificial diffusion, 32
SDFEM, 54second order difference, 20Shishkin mesh, 39simple upwind scheme, 28solution
variational, 44weak, 44
stability, 16of a difference scheme, 22
stabilizationresidual-based, 53
Streamline-Diffusion FEM, 54Streamline-Upwind Petrov–Galerkin
FEM, 54strong maximum principle, 17super position principle, 9SUPG, 54SUPG norm, 57
test space, 44theorem
Lax–Milgram, 46transition point, 39two-point boundary value problem
linear, 3
upwind schemesimple, 28
variational formulation, 44
variational solution, 44
weak formulation, 44weak solution, 44Wronski determinant, 8
70
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