chapter 5 finite element methods (fem)68 5 finite element methods (fem) for ﬁxed h. consequently,...

Chapter 5

Finite Element Methods (FEM)

5.1 Generalities

Remark 5.1. Finite element methods. Finite element methods were one ofthe main topics of Numerical Mathematics 3. The knowledge of the lecturenotes of Numerical Mathematics 3 is assumed. Only a few issues, which areimportant for this course on finite element methods for convection-dominatedproblems, will be reminded here.

Let {T h} be a family of regular triangulations consisting of mesh cells{K}. The triangulations are assumed to be quasi-uniform. The diameter ofa mesh cell K is denoted by hK and it is h = maxK{hK}. Parametric finiteelement spaces will be considered with affine maps between a reference cellK̂ and all physical cells K. ✷

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) is a polynomial space defined on K. Let p ∈ [1,∞) and (m+1−s)p > 1. Then there is a constant C, which is independent of v ∈Wm+1,p(K),such that

∥∥Dk(v − IKv)

∥∥Lp(K)

≤ Chm+1−kK∥∥Dm+1v

∥∥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 andlet p, q ∈ [1,∞]. Then there is a constant Cinv, which depends only onk, l, p, q, K̂, P̂ (K̂) 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. �

63

64 5 Finite Element Methods (FEM)

Fig. 5.1 Boris Grigorievich Galerkin (1871 – 1945).

5.2 The Galerkin Method

Remark 5.4. The Galerkin method. The standard finite element method,which just replaces in the variational formulation (4.2) the space V by afinite-dimensional space V h, is called Galerkin method: Find uh ∈ V h, suchthat for all vh ∈ V h

ε(∇uh,∇vh

)+(b · ∇uh + cuh, vh

)=(f, vh

). (5.3)

If V h ⊂ V , the the finite element method is called conforming. ✷

Theorem 5.5. Lemma of Cea. Let V h ⊂ V and assume the conditions ofthe Theorem of Lax–Milgram, Theorem 4.10. Then there is a unique solution

of the problem 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≤ Mm

infvh∈V h

∥∥u− vh

∥∥V, (5.5)

where u is the unique solution of the continuous problem (4.8) and the con-stants are defined in Definition 4.8, see also (4.6) and (4.7).

Proof. The existence of a unique solution of the discrete problem follows directly fromthe Theorem of Lax–Milgram, since the subspace of a Hilbert space is also a Hilbert space

and the properties of the bilinear form carry over from V to V h.Computing the difference of the continuous equation (4.8) and the discrete equation

(5.4) yields the error equation

a(u− uh, vh

)= 0 ∀ vh ∈ V h.

5.2 The Galerkin Method 65

Withm ‖v‖2V ≤ a(v, v) and |a(u, v)| ≤ M ‖u‖V ‖v‖V

it follows for all vh ∈ V h that∥∥u− uh

∥∥2

V≤ 1

ma(u− uh, u− uh

)=

1

ma(u− uh, u− vh

)

≤ Mm

∥∥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 prop-erties of the bilinear form from problems (4.2) and (5.3) were studied inExample 4.9. It was shown that with appropriate regularity assumptions andunder condition (4.5), the bilinear form is bounded with a constant M oforder max{‖b‖

∞, ‖c‖

∞} and it is coercive with m = ε. In this case, one can

apply the Lemma of Cea and one obtains the error estimate

∥∥u− uh

∥∥V≤ Cmax{‖b‖∞ , ‖c‖∞}

εinf

vh∈V h

∥∥u− vh

∥∥V, C ∈ R.

In the singularly perturbed case ε ≪ ‖b‖∞, the first factor of this estimate

becomes very large.Thus, from this error estimate one cannot expect that the Galerkin fi-

nite element solution is accurate unless the second factor, which is the bestapproximation error, is very small. On uniformly refined grids, the best ap-proximation error becomes very small only if the dimension of V h becomesvery large. ✷

Example 5.7. Galerkin method for P1 finite elements in one dimension. LetΩ = (0, 1) and consider the case that the coefficients of the problem areconstant, i.e., b(x) = b, c(x) = c and f(x) = f . For the P1 finite element onthe reference cell K̂ = [−1, 1], the basis functions and their derivatives aregiven by

φ̂1(x̂) =1

2(−x̂+ 1), φ̂′1(x̂) = −

1

2, φ̂2(x̂) =

1

2(x̂+ 1), φ̂′2(x̂) =

1

2.

Consider the matrix entry ai,i+1, which is computed with the test function

φi(x), which is transformed to φ̂1(x̂), and the ansatz function φi+1(x), which

is transformed to φ̂2(x̂). The common support is the mesh cell [xi, xi+1]. Leth denote the length of this cell, then one obtains with a transform of theintegration variable,


ai,i+1 =

∫ xi+1

xi

εφ′i(x)φ′i+1(x) + bφ

′i(x)φi+1(x) + cφi(x)φi+1(x) dx

=

∫ 1

−1

(

ε2

hφ̂′i(x̂)

2

hφ̂′i+1(x̂) + b

2

hφ′i(x̂)φi+1(x̂) + cφi(x̂)φi+1(x̂)

)h

2dx̂

=2ε

h

∫ 1

−1

1

2·(

−12

)

dx̂+ b

∫ 1

−1

1

2· 12(−x̂+ 1) dx̂

+ch

2

∫ 1

−1

1

2(x̂+ 1)

1

2(−x̂+ 1) dx̂

= − εh+b

2+ch

6.

For the i-th component of the right-hand side, one gets

fi = f

∫ xi+1

xi−1

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 obtains

ch

2

∫ 1

−1

1

2(−x̂+ 1)1

2(x̂+ 1) dx̂ =

ch

220 + 0

2= 0.

Then, it follows that

ai,i+1 = −ε

h+b

2.

For c = 0 (or with trapezoidal rule) these are, apart of a factor h, thesame matrix entries as for the central difference scheme, see Remark 3.10.One gets for c = 0

−hεD+D−ui + hbiD0ui = hfi ⇐⇒ −εD+D−ui + biD0ui = fi.

Hence, one obtains in this special case the same matrix and right-hand sidevector with the Galerkin finite element method and the central finite differ-ence scheme. For singularly perturbed problems, the corresponding resultsare very bad.

The equivalence of the central finite difference method and the Galerkinfinite element method generally does not hold if the coefficients are not con-stant. In higher dimensions, there are differences even for constant coeffi-cients. But still, these two methods give very similar results. At any rate, theGalerkin finite element method is not useful in the singularly perturbed casealso in more general situations. ✷

5.3 Stabilized Finite Element Methods 67

5.3 Stabilized Finite Element Methods

Remark 5.8. On the H1(Ω) norm for the numerical analysis of singularlyperturbed problems. Consider the problem: Find u ∈ V = H10 (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

−12∇ · b(x) + c(x) ≥ ω > 0 almost everywhere in Ω

be satisfied, which is stronger than condition (4.5). Then, an analogous calcu-lation as in Example 4.9 shows that a(·, ·) is uniformly coercive with respectto the following norm, which depends on ε,

‖v‖2ε := ε |v|21,2 + ω ‖v‖

20 = ε ‖∇v‖

2L2(Ω) + ω ‖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. (5.7)

Applying integration by parts, exercise, shows that there is a constant M ,which is also independent on ε, such that

|a(v, w)| ≤M ‖v‖ε ‖w‖H1(Ω) ∀ (v, w) ∈ V × V. (5.8)

However, there is no constant M̃ , which is independent of ε, with

|a(v, w)| ≤ M̃ ‖v‖ε ‖w‖ε ∀ (v, w) ∈ V × V.

Using the estimates (5.7) and (5.8) with constants that are independentof ε, one obtains in a 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 (piecewisepolynomial), then one can show that in layers it is for the best approximationerror

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 ‖·‖ε is not suited for the investigation of numerical methodsfor convection-dominated problems. It turns out that the use of appropriatenorms is important for the numerical analysis of finite element methods forconvection-dominated problems. ✷

5.3.1 Petrov–Galerkin Methods and Upwind Methods

Remark 5.9. Petrov–Galerkin method. A finite element method, whose ansatzand test space are different, is called Petrov–Galerkin method. Let Sh be theansatz space and Th be the test space with dim(Sh) = dim(Th), then thePetrov–Galerkin method reads as follows: Find uh ∈ Sh such that

a(uh, vh

)= f

(vh)

∀ vh ∈ Th.

✷

Example 5.10. Petrov–Galerkin method and upwind method. Consider

−εu′′(x) + bu′(x) = 0 in (0, 1),

with b ∈ R \ {0} and homogeneous Dirichlet boundary conditions. Use asfunctions in 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 calculationshows, 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 differencescheme, see Definition 3.33.


Fig. 5.2 Piecewise quadratic test function for κ = 1/2.

A test function ψi(x), defined in the nodes {0, 0.5, 1}, for κ = 1/2 ispresented in Figure 5.2. ✷

Remark 5.11. Fitted upwind schemes. It is also possible to define fitted up-wind methods with Petrov–Galerkin methods, even for non-constant coeffi-cients. However, one does not obtain better results as for the Iljin–Allen–Southwell method, Theorem 3.57, i.e., one obtains not more than linear con-vergence. ✷

5.3.2 The Streamline-Upwind Petrov–Galerkin(SUPG) Method

Remark 5.12. Goal. The goal consists in the construction of a method that ismore stable than the Galerkin finite element method and which can be usedwith finite elements of arbitrary order. The convergence of this method, inan appropriate norm, should be of higher order.

Consider problem (4.1) and assume at the moment that condition (4.5) issatisfied. Later, an even stronger assumption will be made. ✷

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. (5.9)

For residual-based stabilizations, a modification of Astr is needed which iswell-defined for finite element functions. This modification should be also alinear operator and it is denoted by Ahstr : V

h → L2(Ω). The (strong)residual is now defined by

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, instead of finding the solution of (5.9), the minimizer of the residual issearched, i.e, the following optimization problem is considered

argminuh∈V h

∥∥rh

(uh)∥∥2

L2(Ω)= argmin

uh∈V h

(rh(uh), rh

(uh)). (5.10)

The necessary condition for taking the minimum is the vanishing of theGâteaux derivative. This derivative is computed by using the linearity ofAhstr and the bilinearity of the inner product in L

2(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), Ahstrv

h)

∀ vh ∈ V h.

It follows that the necessary condition for the solution of (5.10) is

(rh(uh), Ahstrv

h)= 0 ∀ vh ∈ V h.

A generalization consists in considering the minimization problem

argminuh∈V h

∥∥∥δ1/2rh

(uh)∥∥∥

2

L2(Ω)= argmin

uh∈V h

(δrh

(uh), rh

(uh))

(5.11)

with the positive weighting function δ(x). Analogously to the derivation forthe special case, one obtains as necessary condition for the minimum

(δrh

(uh), Ahstrv

h)= 0 ∀ vh ∈ V h. (5.12)

The solutions of (5.10) or (5.11) will not be identical to the solution of theGalerkin discretization (5.9). It turns out that the reason for the Galerkindiscretization to fail is that the solution possesses structures (scales) that areimportant but which are not resolved by the used finite element space (grid).For convection-diffusion problems, such structures are layers, particularly atboundaries. The numerical methods should also compute sharp layers. How-


Fig. 5.3 Function with sharp layer (solid line) and optimal piecewise linear approximation

in a mesh cell K (dashed line). The equation that is fulfilled by the function in K is far from

being satisfied by the piecewise linear approximation. Hence, despite the approximation isof the type considered to be optimal, the residual will be large.

ever the sharpness of layers in numerical solutions is restricted by the resolu-tion, which is generally much coarser than the layer width. Hence, even for anumerical solution with sharp layers, the residual in the layer regions are verylarge. In particular, a numerical solution with sharp layers (with respect tothe resolution of the finite element space) will not be the minimizer of (5.10)or (5.11), see Figure 5.3. The minimizers of (5.10) or (5.11) tend to possessstrongly smeared layers and these solutions are useless in applications. Forthis reason, one considers in residual-based stabilizations a combination ofthe Galerkin discretization (5.9), which possesses not sufficient diffusion, andthe minimization of the residual, which is over-diffusive,

ah(uh, vh

)+(δrh

(uh), Ahstrv

h)=(f, vh

)∀ vh ∈ V h. (5.13)

The goal of numerical analysis consists in determining the weighting functionδ optimally in an asymptotic sense. ✷

Definition 5.14. Streamline-Upwind Petrov–Galerkin FEM, SUPG

method, Streamline-Diffusion FEM, SDFEM. The Streamline-UpwindPetrov–Galerkin (SUPG) FEM or Streamline-Diffusion FEM (SDFEM) hasthe form: Find uh ∈ V h, such that

ah(uh, vh

)= fh

(vh)

∀ vh ∈ V h (5.14)

with


ah(v, w) := a(v, w) (5.15)

+∑

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 parametersor SUPG parameters. ✷

Remark 5.15. Concerning the SUPG method.

• The method was developed in Hughes & Brooks (1979); Brooks & Hughes(1982).

• The name “SUPG” comes from the fact that the method can be consideredas 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 the so-calledstreamline direction b(x) ·∇w(x). From this property, the name “Stream-line Diffusion FEM” originates.

• The operator Ahstr is given in the second part of the bilinear form (5.15).The second derivative for finite element functions is defined only piecewise.

• In the stabilization term of the SUPG method, not the strong operatorAhstr applied to the test function is used, as in (5.13), but only the firstorder term contained in this expression. However, for singularly perturbedproblems, the first order term is the dominating term of the strong oper-ator applied to the test function. The numerical analysis presented belowwill show that using only the first order term suffices.It is also possible to define a method with the strong operator appliedto the test function, the so-called Galerkin least squared (GLS) method.In general, one gets similar results with the SUPG and the GLS method,but the SUPG method is easier to implement.

• Generally, the SUPG parameter is a general function. However, in practiceit is often chosen as a piecewise constant function. The goal of the finiteelement error analysis consists in proposing a good asymptotic choice ofthis parameter.

✷

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 allcoefficients are constant, c = 0, and if one chooses the SUPG parameter alsoas a constant, then the 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 anequation with left-hand side

−(ε+ δb2

)u′′(x) + bu′(x).

It is known from Example 5.7 that the Galerkin finite element method isequivalent to a central finite difference scheme. The right-hand side of theSUPG 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

linear combination of the basis functions {φi(x)} of P1 and the integral ofthe derivative of each basis function vanishes. Alternatively, one can applyintegration by parts to check this fact.

Altogether, the SUPG method with the conditions stated above is equiv-alent to the fitted finite difference scheme (3.11)

−ε(

1 + δb2

ε

)

D+D−ui + bD0ui,= fi,

i.e., σ(q) = 1 + δb2/ε = 1 + 2δbq/h with q = bh/(2ε). Choosing the SUPGparameter by

δ(q) =h

2b

(

coth(q)− 1q

)

,

then it is

σ(q) = 1 +hb2

2bε

(

coth(q)− 1q

)

= 1 + q

(

coth(q)− 1q

)

= q coth(q).

One obtains the Iljin–Allen–Southwell scheme. With δ = h/(2b), one gets thesimple upwind scheme.

These simple connections do not hold in higher dimensions. ✷

Definition 5.17. Consistent finite element method. Let u(x) be a suf-ficiently smooth 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 func-tional. A finite element method related to this problem: Find uh ∈ V h suchthat

ah(uh, vh

)= fh

(vh)

∀ vh ∈ V h

is called consistent, if

ah(u, vh

)= fh

(vh)

∀ vh ∈ V h. (5.16)

✷

Remark 5.18. Consistency. Note that consistency of a finite element methodis not the same as consistency of a finite difference method, see Definition 3.6.For finite element methods, consistency means that a sufficiently smoothsolution satisfies also the discrete equation. ✷

Lemma 5.19. Galerkin orthogonality. A consistent finite element method

has the property of the Galerkin orthogonality

ah(u− uh, vh

)= 0 ∀ vh ∈ V h. (5.17)

The error is “orthogonal” to the finite element space.

Proof. The statement of the lemma follows immediately by subtracting (5.14) and (5.16).�

Lemma 5.20. Consistency of the SUPG method. The SUPG method

(5.14) – (5.15) is consistent.

Proof. A sufficiently smooth solution u(x) of (4.2) satisfies the strong form of the equationeven pointwise. Hence, the residual is pointwise zero. Inserting this solution in the SUPG

formulation (5.14) – (5.15) results in a vanishing of the stabilization term. It remains

a(u, vh

)= f

(vh)

∀ vh ∈ V h,

which is satisfied by any weak solution since V h ⊂ V . That means, the smooth solutionsatisfies also the discrete equation. �

Definition 5.21. SUGP norm. Let for almost all x ∈ Ω hold

− 12∇ · b(x) + c(x) ≥ ω > 0. (5.18)

In V h, the SUPG norm is defined by

∣∣∣∣∣∣vh∣∣∣∣∣∣SUPG

:=

ε∣∣vh∣∣2

H1(Ω)+ ω

∥∥vh

∥∥2

L2(Ω)+∑

K∈T h

∥∥∥δ

1/2K

(b · ∇vh

)∥∥∥

2

L2(K)

1/2

.(5.19)


✷

Theorem 5.22. Coercivity of the SUPG bilinear form. Assume that

b ∈W 1,∞(Ω), c ∈ L∞(Ω), (5.18), and let

0 < δK ≤1

2min

{

h2KεC2inv

,ω

‖c‖2L∞(K)

}

, (5.20)

where Cinv is the constant in the inverse estimate (5.2) Then, the SUPGbilinear form is coercive with respect to the SUPG norm, i.e., it is

ah(vh, vh

)≥ 1

2

∣∣∣∣∣∣vh∣∣∣∣∣∣2

SUPG∀ vh ∈ V h.

Proof. Integration by parts gives, see Example 4.9,

(b · ∇vh + cvh, vh

)=

((

−∇ · b2

+ c

)

vh, vh)

∀ vh ∈ V h.

With the definition of ω, one obtains

ah(vh, vh

)

= ε∣∣vh∣∣2

1+

∫

Ω

(

c(x)− ∇ · b(x)2

)

︸︷︷︸

≥ω>0

(vh)2

(x) dx+∑

K∈T h

∥∥∥δ

1/2K

(b · ∇vh

)∥∥∥

2

L2(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


) (b(x) · ∇vh(x)

)dx

∣∣∣∣∣∣

.

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 (5.20) on the SUPG parameter. It is for each K ∈ T h


∣∣∣∣

∫

K


) (b · ∇vh(x)

)dx

∣∣∣∣

≤∫

K

(

δ1/2K ε

∣∣∆vh(x)

∣∣)(

δ1/2K

∣∣b · ∇vh(x)

∣∣)

dx

+

∫

K

(

δ1/2K |c(x)|

∣∣vh(x)

∣∣)(

δ1/2K

∣∣b · ∇vh(x)

∣∣)

dx

CS≤

(

δ1/2K ε

∥∥∆vh

∥∥L2(K)

+ δ1/2K ‖c‖L∞(K)

∥∥vh

∥∥L2(K)

)∥∥∥δ

1/2K

(b · ∇vh

)∥∥∥L2(K)

(5.2)

≤(

δ1/2K

εCinv

hK

∥∥∇vh

∥∥L2(K)

+ δ1/2K ‖c‖L∞(K)

∥∥vh

∥∥L2(K)

)∥∥∥δ

1/2K

(b · ∇vh

)∥∥∥L2(K)

(5.20)

≤(

hK√2εCinv

εCinv

hK

∥∥∇vh

∥∥L2(K)

+

√ω√

2 ‖c‖L∞(K)‖c‖L∞(K)

∥∥vh

∥∥L2(K)

)

×∥∥∥δ

1/2K

(b · ∇vh

)∥∥∥L2(K)

=

(√ε

2

∥∥∇vh

∥∥L2(K)

+

√ω

2

∥∥vh

∥∥L2(K)

)∥∥∥δ

1/2K

(b · ∇vh

)∥∥∥L2(K)

Young≤ ε

2

∥∥∇vh

∥∥2

L2(K)+

1

4

∥∥∥δ

1/2K

(b · ∇vh

)∥∥∥

2

L2(K)+

ω

2

∥∥vh

∥∥2

L2(K)

+1

4

∥∥∥δ

1/2K

(b · ∇vh

)∥∥∥

2

L2(K)

=1

2

∣∣∣∣∣∣vh∣∣∣∣∣∣SUPG,K

.

Now, the proof is finished by summing over all mesh cells and inserting this estimate inthe first estimate of the proof. �

Corollary 5.23. Coercivity of the SUPG bilinear form for P1 finiteelements. Let the assumptions of Theorem 5.22 with respect to the coeffi-

cients of the problem be valid. For piecewise linear finite elements, the SUPG

bilinear form (5.15) is coercive with respect to the SUPG norm if

0 < δK ≤ω

‖c‖2L∞(K). (5.21)

Proof. The proof is the same as for Theorem 5.22, where one uses that for piecewiselinear finite elements ∆vh(x)|K = 0 for all K ∈ T h. Thus, the corresponding terms do notappear 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, respec-

tively, be valid. Then, the SUPG finite element method (5.14) – (5.15) has aunique 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 stabi-lized finite element methods. One tries to get rid of the troubling termsby estimating them with the used norm. This approach works only if oneuses an appropriate norm. In particular, the stabilization has to appearin the norm.

• Theorem 5.22 provides an upper bound for the SUPG parameter. Thisbound is generally not critical in applications.

• From Theorem 5.22, one obtains the stability of the SUPG method withrespect to the SUPG norm. Stability means that an appropriate norm ofthe solution can 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‖L2(Ω)

√ω∥∥uh

∥∥L2(Ω)

+2∑

K∈T h

∥∥∥δ

1/2K f

∥∥∥L2(K)

∥∥∥δ

1/2K

(b · ∇uh

)∥∥∥L2(K)

Young

≤ C ‖f‖2L2(Ω) +1

2

ω∥∥uh

∥∥2

L2(Ω)+∑

K∈T h

∥∥∥δ

1/2K

(b · ∇uh

)∥∥∥

2

L2(K)

≤ C ‖f‖2L2(Ω) +1

2

∣∣∣∣∣∣uh∣∣∣∣∣∣2

SUPG.

In the last estimate, it was used that the terms are part of the SUPG normand thus they can be estimated by the whole norm. Now, the second termon the right-hand side can be absorbed in the left-hand side and stabilityis proved. The stability constant depends on ω and on the upper boundof δK .

• All vh ∈ V h satisfy∣∣∣∣∣∣vh∣∣∣∣∣∣SUPG

≥ min{1, ω}∥∥vh

∥∥ε.

Hence, the SUPG method is also stable with respect to the norm ‖·‖ε.With respect to this norm, also the Galerkin finite element method is sta-ble, however this method is not stable with respect to the SUPG norm.That means, the stability of the SUPG method is stronger than the sta-bility of the Galerkin finite element method.

✷

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), let theassumptions of Theorem 5.22 be satisfied, and consider the SUPG method for


Fig. 5.4 Joseph–Louis Lagrange (1736 – 1813).

Pk finite elements. Let the SUPG parameter be given as follows

δK =

C0h2Kε

for hK < ε,

C0hK for ε ≤ hK ,(5.22)

where the constant C0 > 0 is sufficiently small such that (5.20) is satisfiedfor k ≥ 2 or (5.21) for k = 1, respectively. Then, the solution uh ∈ Pk of theSUPG method (5.14) satisfies the following error estimate

∣∣∣∣∣∣u− uh

∣∣∣∣∣∣SUPG

≤ C(

ε1/2hk + hk+1/2)

|u|Hk+1(Ω) ,

where the constant C does not depend on ε and h.

Proof. Let uhI ∈ V h be the Lagrangian interpolant of u(x). One obtains with the triangleinequality ∣

∣∣∣∣∣u− uh


≤∣∣∣∣∣∣u− uhI


+∣∣∣∣∣∣uhI − uh


.

The first term on the right-hand side is the interpolation error. Note that for both regimes

it isδK ≤ C0hK ≤ Ch. (5.23)

Using this property after having applied the interpolation error estimate (5.1) to each term

of the SUPG norm individually gives


∣∣∣∣∣∣u− uhI


≤(

Cεh2k |u|2Hk+1(Ω)

+ Cωh2(k+1) |u|2Hk+1(Ω)

+C∑

K∈T h

δK ‖b‖2L∞(K) h2kK |u|2Hk+1(K)

1/2

≤ C(

εh2k + h2(k+1) + h2k+1)1/2

|u|Hk+1(Ω)

≤ C(

ε1/2hk + hk+1/2)

|u|Hk+1(Ω) .

Consider now the second term on the right-hand side. The coercivity, Theorem 5.22,and the Galerkin orthogonality yield

1

2

∣∣∣∣∣∣uhI − uh

∣∣∣∣∣∣2

SUPG≤ 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 diffusive term∣∣ε(∇(uhI − u

),∇wh

)∣∣

CS≤ ε

∥∥∇(uhI − u

)∥∥L2(Ω)

∥∥∇wh

∥∥L2(Ω)

= ε1/2∥∥∇(uhI − u

)∥∥L2(Ω)

ε1/2∥∥∇wh

∥∥L2(Ω)

(5.1)

≤ Cε1/2hk |u|Hk+1(Ω) ε1/2∥∥∇wh

∥∥L2(Ω)

≤ Cε1/2hk |u|Hk+1(Ω)∣∣∣∣∣∣wh


.

For the reactive term, one computes in a similar way

∣∣(c(uhI − u), wh

)∣∣

CS≤ ‖c‖L∞(Ω)

∥∥uhI − u

∥∥L2(Ω)

∥∥wh

∥∥L2(Ω)

= ω−1/2 ‖c‖L∞(Ω)∥∥uhI − u

∥∥L2(Ω)

ω1/2∥∥wh

∥∥L2(Ω)

(5.1)

≤ Chk+1 |u|Hk+1(Ω)∣∣∣∣∣∣wh


.

Next, the terms are considered which come from the SUPG stabilization. Since for both

regimes it is

εδK ≤ C0h2K ,one gets


∣∣∣∣∣∣

∑

K∈T h

(−ε∆

(uhI − u

), δKb · ∇wh

)

K

∣∣∣∣∣∣

CS≤

∑

K∈T h

ε1/2∥∥∆(uhI − u

)∥∥L2(K)

ε1/2δ1/2K

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥L2(K)

≤ C1/20∑

K∈T h

hKε1/2∥∥∆(uhI − u

)∥∥L2(K)

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥L2(K)

CS≤ C1/20 ε1/2h

∑

K∈T h

∥∥∆(uhI − u

)∥∥2

L2(K)

1/2

×

∑

K∈T h

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥

2

L2(K)

1/2

(5.1)

≤ Cε1/2h

∑

K∈T h

h2(k−1)K |u|

2Hk+1(K)

1/2

∑

K∈T h

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥

2

L2(K)

1/2

≤ Cε1/2hk |u|Hk+1(Ω)∣∣∣∣∣∣wh


.

For the other terms, one obtains with (5.23), which holds for both regimes,

∣∣∣∣∣∣

∑

K∈T h

(b · ∇

(uhI − u

)+ c(uhI − u

), δK

(b · ∇wh

))

K

∣∣∣∣∣∣

CS≤

∑

K∈T h

‖b‖L∞(Ω)∥∥∇(uhI − u

)∥∥L2(K)

δ1/2K

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥L2(K)

+∑

K∈T h

‖c‖L∞(Ω)∥∥uhI − u

∥∥L2(K)

δ1/2K

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥L2(K)

≤ C

∑

K∈T h

h1/2K

∥∥∇(uhI − u

)∥∥L2(K)

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥L2(K)

+∑

K∈T h

h1/2K

∥∥uhI − u

∥∥L2(K)

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥L2(K)

CS≤ Ch1/2

∑

K∈T h

∥∥∇(uhI − u

)∥∥2

L2(K)

1/2

+

∑

K∈T h

∥∥uhI − u

∥∥2

L2(K)

1/2

×

∑

K∈T h

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥

2

L2(K)

1/2

(5.1)

≤ C(

hk+1/2 + hk+3/2)

|u|Hk+1(Ω)∣∣∣∣∣∣wh


.

To obtain an optimal estimate for the convective term, one has to apply first integration

by 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 forthe other estimates yields

∣∣(uhI − u, (∇ · b)wh

)∣∣

≤ ω−1/2 ‖∇ · b‖L∞(Ω)

∑

K∈T h

∥∥uhI − u

∥∥2

L2(K)

1/2

ω1/2∥∥wh

∥∥L2(Ω)

≤ Chk+1 |u|Hk+1(Ω)∣∣∣∣∣∣wh


.

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

∥∥L2(K)

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥L2(K)

+∑

ε>hK

‖b‖L∞(Ω)∥∥uhI − u

∥∥L2(K)

∥∥∇wh

∥∥L2(K)

(5.1)

≤ C

∑

ε≤hK

δ−1/2K h

k+1K |u|Hk+1(K)

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥L2(K)

+∑

ε>hK

hk+1K |u|Hk+1(K)∥∥∇wh

∥∥L2(K)

C0hK=δK ,ε>hK≤ C

∑

ε≤hK

C−1/20 h

−1/2K h

k+1K |u|Hk+1(K)

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥L2(K)

+∑

ε>hK

hk+1/2K |u|Hk+1(K) ε

1/2∥∥∇wh

∥∥L2(K)

CS≤ Chk+1/2 |u|Hk+1(Ω)

∑

K∈T h

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥

2

L2(K)

1/2

+ ε1/2∣∣wh

∣∣1

≤ Chk+1/2 |u|Hk+1(Ω)∣∣∣∣∣∣wh


.

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 inthe SUPG norm is k+1/2 and in the diffusion-dominated case it is k. Inthe latter case, the SUPG norm is essentially the H1(Ω) semi norm suchthat order k is optimal.

• It is essential for obtaining an estimate with a constant C which is inde-pendent of ε that the term


∑

K∈T h

∥∥∥δ

1/2K

(b · ∇wh

)∥∥∥

2

L2(K)

1/2

is part of the norm, which is used for estimating the error. Such a uniformestimate does not hold for the norm ‖·‖ε.

• For the interpretation of the results one has to take into account thatdifferent stabilization parameters by choosing different values of C0 leadalso to different norms on the left-hand side of the estimate.

• On the other hand, the value of a constant which is independent of ε isquestionable since in general |u|Hk+1(Ω) depends on ε.

• In numerical simulations, often one can observe even convergence of orderhk+1 for the error in L2(Ω), in particular on structured grids. However,in Zhou (1997) examples were constructed that show that the estimate ofTheorem 5.26 is sharp also for the error in L2(Ω).

✷

Example 5.28. SUPG in one dimension, the user-chosen constant. The stan-dard example

−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. Anerror estimate in the norm

(

ε∣∣vh∣∣2

H1(Ω)+

N∑

i=1

∥∥∥δ

1/2K b

(vh)′∥∥∥

2

L2(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 (5.22). 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 thelayer is smeared, for an appropriate value of C0 one obtains a solution whichis almost exact in the nodes, and if C0 is too small, then one can observespurious (unphysical) oscillations in the layer.

For general problems, it is difficult to choose C0 appropriately. In higherdimensions, 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 SUPGmethod in higher dimensions exhibits spurious oscillations at the layers, e.g.,see Example 5.30. ✷

Remark 5.29. Different choices of the SUPG parameter.

• A refined analysis, taking the polynomial degree k of the finite elementinto account, proposes the stabilization parameter


Fig. 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.

δK =

C0hK

‖b‖L∞(K)for PeK ≥ 1,

C0h2Kε

else,

with PeK =‖b‖L∞(K) hK

2pε(5.24)

• In practice, one takes for linear and bilinear finite elements instead of(5.22) also the parameter

δK =hK

2 ‖b‖L∞(K)

(

coth(PeK)−1

PeK

)

, PeK =‖b‖L∞(K) hK

2ε, (5.25)

where PeK is the local Péclet number, since in one dimensions one recov-ers under certain conditions the Iljin–Allen–Southwell scheme, see Exam-ple 5.16. There is no user-chosen constant in this parameter. Asymptoti-cally, both parameters (5.22) and (5.25) have the same behavior.

✷

Example 5.30. SUPG in two dimensions. A standard test problem in twodimensions 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 atthe boundaries y = 0 and y = 1. The layer at the outflow boundary is oftencalled exponential layer and the layers parallel to the flow direction paraboliclayers.

A numerical solution obtained for ε = 10−8 with the Q1 finite elementmethod on a rather coarse grid and the SUPG parameter (5.25) is shownin Figure 5.6. One can see very well large spurious oscillations, in particularat the parabolic layers. These oscillations are a typical feature of solutionsobtained with the SUPG method. They might become smaller with higherorder elements or on finer grids. But they will generally vanish only if thelayer is resolved. ✷


Fig. 5.6 Result obtained with the SUPG method, Q1 finite elements, and the SUPGparameter (5.25).

5.3.3 Residual A Posteriori Error Estimation

Remark 5.31. Goals of a posteriori error estimators. So far, so called a priorierror estimates were proved, e.g., in Theorem 5.26. The right-hand side ofsuch estimates depends on the unknown solution of the continuous problemand on an unknown constant.

The goals of a posteriori error estimates are twofold. First, they shouldprovide computable estimates of errors between a computed solution uh andthe unknown solution u of the continuous problem (4.2). Usually, the errorsare measured in norms of Sobolev spaces defined on Ω. That means, estimatesof global errors from above have to be derived, which have the form

∥∥u− uh

∥∥Ω≤ Cη. (5.26)

In (5.26), η is a quantity that is computable with the information availablein the numerical solution process and C is a positive constant that should beindependent of the mesh width and the solution and for which one shouldhave an idea of its order of magnitude.

The second task of a posteriori error estimates consists in controlling anadaptive mesh refinement. The rationale behind this idea is that the errorof the computed solution to the solution of the continuous problem can bereduced best, or at least significantly, if the mesh in those subregions of thedomain is refined, where the local error is largest. Then, a local a posteriorierror estimate should identify these subregions. To this end, a local lowerestimate from below of the form

ηK ≤ C∥∥u− uh

∥∥ω(K)

(5.27)

is necessary, where ω(K) denotes a small neighborhood of a mesh cell Kand ηK is a computable quantity. For (5.27), it has to be proved that thepositive constant C can be bounded from below and above independently of


K. Estimate (5.27) tells that in subregions where the local error estimate ηKis large, also the local error is large. The derivation of estimates of type (5.27)is usually quite technical.

There are different ways for computing a posteriori error estimates, see,e.g., the monographs Ainsworth & Oden (2000); Verfürth (2013). A review ofresidual-based estimators can be found in Verfürth (2013). The presentationbelow follows John & Novo (2013). ✷

Remark 5.32. The model problem. This section considers stationary convection-diffusion-reaction equations of the form

−ε∆u+ b · ∇u+ cu = f in Ω,u = 0 on ΓD, (5.28)

ε∂u

∂n= ε∂nu = g on ΓN ,

where Ω is a polygonal domain in Rd, d ≥ 2, with Lipschitz boundary Γconsisting of two disjoint components ΓD and ΓN . The Dirichlet part ΓD has apositive (d−1)-dimensional Lebesgue measure and ∂Ω− ⊂ ΓD, ∂Ω− being theinflow boundary of Ω, i.e., the set of points x ∈ ∂Ω such that b(x) ·n(x) < 0.It will be assumed that 0 < ε, b ∈ W 1,∞(Ω), c ∈ L∞(Ω), f ∈ L2(Ω), and(5.28) is scaled such that ‖b‖L∞(Ω) = O(1), ‖c‖L∞(Ω) = O(1), and 0 < ε≪ 1.Hence, the convection-dominated regime is studied. Furthermore, it will beassumed that the (5.18) is fulfilled.

The model problem (5.28) possesses a unique weak solution u ∈ V =H1D(Ω) = {v ∈ H1(Ω) : v|ΓD = 0} that satisfies

ε(∇u,∇v) + (b · ∇u, v) + (cu, v) = (f, v) + (g, v)ΓN ∀ v ∈ V (5.29)

with

(g, v)ΓN =

∫

ΓN

g(s)v(s) ds.

The corresponding SUPG finite element method reads as follows: Finduh ∈ V h ⊂ V such that

aSUPG(uh, vh) = (f, vh)+(g, vh)ΓN+

∑

K∈T h

δK(f, b·∇vh)K ∀ vh ∈ V h. (5.30)

✷

Remark 5.33. Hypotheses for the analysis. It will be assumed that severalnorms of the interpolation error u − Ihu can be bounded by norms of theerror u− uh:


∑

K∈T h

δ−1K ‖u− Ihu‖2L2(K) ≤ 2|||u− uh|||2SUPG, (5.31)

∑

K∈T h

δK‖b · ∇(u− Ihu)‖2L2(K) ≤ 2|||u− uh|||2SUPG, (5.32)

∑

E∈Eh

‖b‖L∞(E)‖u− Ihu‖2L2(E) ≤ 2|||u− uh|||2SUPG, (5.33)

where Eh is the set of all faces of the triangulation.It can be argued that these hypotheses are satisfied in standard situations,

see John & Novo (2013) for details. ✷

Lemma 5.34. Coercivity of the SUPG bilinear form with the error

as argument. Let e = u−uh, u ∈ H2(Ω) being the solution of (5.29) and uhits SUPG approximation computed by solving (5.30). If the SUPG parametersare chosen such that

δK ≤h2K

8C2invε, δK ≤

ω

2‖c‖2L∞(K)C2inv, (5.34)

then

aSUPG(e, e) ≥1

2|||e|||2SUPG −

∑

K∈T h

4δKh−2K ε

2C2inv‖∇(u− Ihu)‖2L2(K)

−∑

K∈T h

2δKε2‖∆(u− Ihu)‖2L2(K). (5.35)

Proof. The proof is very similar to the proof of Theorem 5.22. exercise �

Remark 5.35. On Lemma 5.34. If ‖c‖L∞(K) = 0 in (5.34), then δK should bejust bounded on K.

Condition (5.34) is fulfilled in the convection-dominated case PeK ≫ 1since one gets with (5.24)

δK =C02kPe−1K

h2Kε.

✷

Remark 5.36. Jumps across faces. Consider a triangulation T h and two meshcells K1,K2 ∈ T h with a common face E = K1 ∩K2. Without loss of gen-erality, the normal nE at E should be the outward normal with respect toK1. Then, the jump of a function v across the face E in the point x ∈ E isdefined by

[|v|]E = limy→x,y∈K1

v(y)− limy→x,y∈K2

v(y), x ∈ E, (5.36)


if both limits are well defined. Changing the direction of nE changes the signof the jump.

Using the definition, it follows that

[|v + w|]E = [|v|]E + [|w|]E .

If w is continuous almost everywhere on E, then

[|vw|]E = [|v|]E w.

✷

Theorem 5.37. Residual a posteriori error estimate. Let the conditions

of Lemma 5.34 hold. Under the hypotheses from Remark 5.33, one gets the

following global upper bound

|||u− uh|||2SUPG ≤ η21 + η22 + η23 +∑

K∈T h

16δKh−2K ε


+∑

K∈T h

8δKε2‖∆(u− Ihu)‖2L2(K), (5.37)

where

η21 =∑

K∈T h

min

{C

ω,C

h2Kε, 24δK

}

‖RK(uh)‖2L2(K),

η22 =∑

K∈T h

24δK‖RK(uh)‖2L2(K),

η23 =∑

E∈Eh

min

{24

‖b‖L∞(E), C

hEε,

C

ε1/2ω1/2

}

‖RE(uh)‖2L2(E), (5.38)

where the mesh cell and the face residuals are defined by

RK(uh) := f + ε∆uh − b · ∇uh − cuh

∣∣K,

RE(uh) :=

−ε[∣∣∂nEu

h∣∣]

Eif E ∈ EhΩ ,

g − ε∂nEuh if E ∈ EhN ,0 if E ∈ EhD,

with

EhΩ − set of all interior faces,EhN − set of all faces on the Neumann boundary,EhD − set of all faces on the Dirichlet boundary.

Proof. A representation of the error will be derived. Using the Galerkin orthogonality(5.17), the weak form of the equation (5.29), and integration by parts gives for all v ∈


H1D(Ω)

aSUPG(u− uh, v)G.orth.

= aSUPG(u− uh, v − Ihv) = aSUPG(u, v − Ihv)− aSUPG(uh, v − Ihv)= (f, v − Ihv) + (g, v − Ihv)ΓN +

∑

K∈T h

δK(f, b · ∇(v − Ihv))K

−ε(∇uh,∇(v − Ihv))− (b · ∇uh, v − Ihv)− (cuh, v − Ihv)−∑

K∈T h

δK(−ε∆uh + b · ∇uh + cuh, b · ∇(v − Ihv))K

i.b.p.=

∑

K∈T h

(f + ε∆uh − b · ∇uh − cuh, v − Ihv)K

+∑

K∈T h

δK(f + ε∆uh − b · ∇uh − cuh, b · ∇(v − Ihv))K

+∑

E∈EhΩ

(−ε[∣∣∂nEu

h∣∣]

E, v − Ihv)E +

∑

E∈EhN

(g − ε∂nEuh, v − Ihv)E

=∑

K∈T h

(RK(uh), v − Ihv)K +

∑

K∈T h

δK(RK(uh), b · ∇(v − Ihv))K

+∑

E∈Eh

(RE(uh), v − Ihv)E . (5.39)

Setting v = u − uh and observing that v − Ihv = (u − uh) − Ih(u − uh) = u − Ihu,since Ihuh = uh, leads together with (5.35) to

1

2|||u− uh|||2SUPG

≤∑

K∈T h

(RK(uh), u− Ihu)K +

∑

K∈T h

δK(RK(uh), b · ∇(u− Ihu))K

+∑

E∈Eh

(RE(uh), u− Ihu)E +

∑

K∈T h

4δKh−2K ε


+∑

K∈T h

2δKε2‖∆(u− Ihu)‖2

L2(K). (5.40)

Now, the first three terms on the right-hand side of (5.40) have to be bounded such that

expressions with u − Ihu are absorbed by the left-hand side. The norm on the left-handside consists of three terms, respectively, see (5.19). There are different ways for absorbingu− Ihu.

For illustration, only the estimate for the first term of (5.40) is presented. In the firststep of the estimate, the Cauchy–Schwarz inequality gives

∑

K∈T h

(RK(uh), u− Ihu)K ≤

∑

K∈T h

‖RK(uh)‖L2(K)‖u− Ihu‖L2(K)

=∑

K∈T h

‖RK(uh)‖L2(K)‖v − Ihv‖L2(K).

Using (5.18), one can apply the stability of the Lagrangian interpolant in L2(Ω) and

Young’s inequality to get


∑

K∈T h


∑

K∈T h

C‖RK(uh)‖L2(K)‖u− uh‖L2(K) (5.41)

≤∑

K∈T h

C

ω‖RK(uh)‖2L2(K) +

1

12‖µ1/2(u− uh)‖2

L2(Ω).

Alternatively, one can use the interpolation estimate (5.1) with k = 0, m = 0 to obtain

∑

K∈T h


∑

K∈T h

C‖RK(uh)‖L2(K)hK‖∇(u− uh)‖0,K (5.42)

≤∑

K∈T h

Ch2Kε

‖RK(uh)‖2L2(K) +ε

12‖∇(u− uh)‖2

L2(Ω).

Finally, one can try to use the term with the streamline derivative for the bound. Young’s

inequality yields

∑

K∈T h

(RK(uh), u− Ihu)K

≤ 6∑

K∈T h

δK‖RK(uh)‖2L2(K) +1

24

∑

K∈T h

δ−1K ‖u− Ihu‖2

L2(K). (5.43)

Collecting the estimates (5.41), (5.42), and (5.43) together with hypothesis (5.31) leads

to

∑

K∈T h

(RK(uh), u− Ihu)K

≤∑

K∈T h

min

{C

ω,Ch2Kε

−1, 6δK

}

‖RK(uh)‖2L2(K) +1

12|||u− uh|||2SUPG. (5.44)

For the remainder of the proof, it is referred to John & Novo (2013). �

Remark 5.38. On the additional terms on the right hand side of (5.37). Be-sides the a posteriori terms, there are two extra terms on the right hand sideof (5.37). It can be shown that these terms are in the convection-dominatedregime negligible compared with the error of the SUPG approximation. Thispoint justifies the use of the quantity (η21 + η

22 + η

23)

1/2 as an upper errorestimate. ✷

Example 5.39. Smooth solution. This example studies the effectivity of thea posteriori error estimator (η21+η

22+η

23)

1/2 for the diffusion- and convection-dominated regime. To this end, equation (5.28) is considered in Ω = (0, 1)2

with b = (1,−4)T , c = 1, and with f chosen such that

u(x, y) = sin(πx) sin(πy)

is the solution of (5.28). Dirichlet boundary conditions were prescribed atx = 0, y = 0, and y = 1. At x = 1, Neumann boundary conditions wereapplied.

Uniformly refined grids were used in the simulations. The coarsest grids(level 0) are presented in Figure 5.7. The finest grid in all simulations was the


Fig. 5.7 Coarsest grids for the simulations on the unit square.

first grid with more than 106 degrees of freedom. Simulations were performedfor first, second, and third order finite elements.

Results with respect to the effectivity index

ηeff =η

‖u− uh‖ ,

for different finite elements and different diffusion parameters ε are presentedin Figure 5.8. In all cases, the effectivity index is in the interval [5, 12]. Consid-ering only the convection-dominated case, then the effectivity index is alwaysin the interval [6.5, 7]. In summary, one can observe that the effectivity indexbehaves robustly with respect to the Péclet number. ✷


Fig. 5.8 Example 5.39. Effectivity index for different values of the diffusion and for dif-

ferent finite elements.

chapter 5 finite element methods (fem)68 5 finite element methods (fem) for ﬁxed h. consequently,...

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