Inserting the estimates of Theorem 3.32 (for v ’ IK (v)), Theorem 3.31,

and Theorem 3.33 into each other and recalling (3.96), (3.97), the following

theorem follows from C´a™s lemma (Theorem 2.17):

e

Theorem 3.35 Consider the linear ansatz (3.53) on a family of triangu-

lations (Th )h of triangles that satis¬es the maximum angle condition. Then

there exists some constant C > 0 such that for v ∈ H 2 („¦),

v ’ Ih (v) ¤ C h |v|2 .

1

If the solution u of the boundary value problem (3.12), (3.18)“(3.20) belongs

to H 2 („¦), then for the ¬nite element approximation uh de¬ned in (3.39)

we have the estimate

u ’ uh ¤ Ch|u|2 . (3.98)

1

Exercise 3.26 shows the necessity of the maximum angle condition. Again,

a remark analogous to Remark 3.30 holds. For an analogous investigation

of tetrahedra we refer to [58].

With a modi¬cation of the above considerations and an additional

condition anisotropic error estimates of the form

d

|v ’ Ih (v)|1 ¤ C hi |‚i v|1

i=1

can be proven for v ∈ H 2 („¦), where the hi denote length parameter de-

pending on the element type. In the case of triangles, these are the longest

edge (h1 = hK ) and the height on it as shown in Figure 3.12 (see [41]).

3.4.3 L2 Error Estimates

The error estimate (3.89) also contains a result about the approximation

of the gradient (and hence of the ¬‚ux), but it is linear only for k = 1, in

3.4. Convergence Rate Estimates 145

contrast to the error estimate of Chapter 1 (Theorem 1.6). The question is

whether an improvement of the convergence rate is possible if we strive only

for an estimate of the function values. The duality argument of Aubin and

Nitsche shows that this is correct, if the adjoint boundary value problem

is regular, where we have the following de¬nition:

De¬nition 3.36 The adjoint boundary value problem for (3.12), (3.18)“

(3.20) is de¬ned by the bilinear form

(u, v) ’ a(v, u) for u, v ∈ V

with V from (3.30). It is called regular if for every f ∈ L2 („¦) there exists

a unique solution u = uf ∈ V of the adjoint boundary value problem

for all v ∈ V

a(v, u) = f, v 0

and even uf ∈ H 2 („¦) is satis¬ed, and for some constant C > 0 a stability

estimate of the form

|uf |2 ¤ C f for given f ∈ L2 („¦)

0

is satis¬ed.

The V -ellipticity and the continuity of the bilinear form (3.2), (3.3) di-

rectly carry over from (3.31) to the adjoint boundary value problem, so

that in this case the unique existence of uf ∈ V is ensured. More pre-

cisely, the adjoint boundary value problem is obtained by an exchange of

the arguments in the bilinear form, which does not e¬ect any change in its

symmetric parts. The nonsymmetric part of (3.31) is „¦ c · ∇u v dx, which

becomes „¦ c · ∇v u dx. By virtue of

c · ∇v u dx = ’ ∇ · (cu) v dx + c · ν uv dσ

„¦ „¦ ‚„¦

the transition to the adjoint boundary value problem therefore means the

exchange of the convective part c · ∇u by a convective part, now in diver-

gence form and in the opposite direction ’c, namely ∇ · (’cu), with the

correponding modi¬cation of the boundary condition. Hence, in general we

may expect a similar regularity behavior to that in the original boundary

value problem, which was discussed in Section 3.2.3. For a regular adjoint

problem we get an improvement of the convergence rate in · 0 :

Theorem 3.37 (Aubin and Nitsche)

Consider the situation of Theorem 3.29 or Theorem 3.35 and suppose the

adjoint boundary value problem is regular. Then there exists some constant

C > 0 such that for the solution u of the boundary value problem (3.12),

(3.18)“(3.20) and its ¬nite element approximation uh de¬ned by (3.39),

u ’ uh ¤ Ch u ’ uh

(1) ,

0 1

u ’ uh ¤ Ch u

(2) 1,

0

146 3. Finite Element Methods for Linear Elliptic Problems

u ’ uh ¤ Chk+1 |u|k+1 , if u ∈ H k+1 („¦).

(3) 0

Proof: The assertions (2) and (3) follow directly from (1). On the one

hand, by using u ’ uh 1 ¤ u 1 + uh 1 and the stability estimate (2.44),

on the other hand directly from (3.89) and (3.98), respectively.

For the proof of (1), we consider the solution uf of the adjoint problem

with the right-hand side f = u ’ uh ∈ V ‚ L2 („¦). Choosing the test

function u ’ uh and using the error equation (2.39) gives

u ’ uh = u ’ uh , u ’ uh = a(u ’ uh , uf ) = a(u ’ uh , uf ’ vh )

2

0 0

for all vh ∈ Vh . If we choose speci¬cally vh = Ih (uf ), then from the con-

tinuity of the bilinear form, Theorem 3.29, and Theorem 3.35, and the

regularity assumption it follows that

u ’ uh ¤ C u ’ uh 1 uf ’ Ih (uf ) 1

2

0

¤ C u ’ uh 1 h|uf |2 ¤ C u ’ uh 1 h u ’ uh 0.

Division by u ’ uh gives the assertion, which is trivial in the case u ’

0

2

uh 0 = 0.

Thus, if a rough right-hand side in (3.12) prevents convergence from

being ensured by Theorem 3.29 or Theorem 3.35, then the estimate (2) can

still be used to get a convergence estimate (of lower order).

In the light of the considerations from Section 1.2, the result of Theo-

rem 3.37 is surprising, since we have only (pointwise) consistency of ¬rst

order. On the other hand, Theorem 1.6 also raises the question of conver-

gence rate results in · ∞ which then would give a result stronger, in

many respects, than Theorem 1.6. Although the considerations described

here (as in Section 3.9) can be the starting point of such L∞ estimates, we

get the most far-reaching results with the weighted norm technique (see [9,

pp. 155 ¬.]), whose description is not presented here.

The above theorems contain convergence rate results under regularity

assumptions that may often, even though only locally, be violated. In fact,

there also exist (weaker) results with less regularity assumptions. However,

the following observation seems to be meaningful: Estimate (3.90) indicates

that on subdomains, where the solution has less regularity, on which the

(semi) norms of the solutions thus become large, local re¬nement is advan-

tageous (without improving the convergence rate by this). Adaptive mesh

re¬nement strategies on the basis of a posteriori error estimates described

in Chapter 4 provide a systematical approach in this direction.

Exercises

3.21 Prove for the linear ¬nite element ansatz (3.53) in one space di-

mension that for K ∈ Th and v ∈ H 2 (K), the following estimate

Exercises 147

holds:

|v ’ IK (v)|1,K ¤ hK |v|2,K .

Hint: Rolle™s theorem and Exercise 2.5 (b) (Poincar´ inequality).

e

Generalize the considerations to an arbitrary polynomial ansatz P = Pk

in one space dimension by proving

|v ’ IK (v)|1,K ¤ hk |v|k+1,K for v ∈ H k+1 (K) .

K

3.22 Prove the chain rule (3.77) for v ∈ H 1 (K).

3.23 Derive analogously to Theorem 3.29 a convergence rate result for

the Hermite elements (3.64) and (3.65) (Bogner“Fox“Schmit element) and

the boundary value problem (3.12) with Dirichlet boundary conditions.

3.24 Derive analogously to Theorem 3.29 a convergence rate result for

the Bogner“Fox“Schmit element (3.65) and the boundary value problem

(3.36).

3.25 Let a triangle K with the vertices a1 , a2 , a3 and a function u ∈

C 2 (K) be given. Show that if u is interpolated by a linear polynomial

IK (u) with (IK (u))(ai ) = u(ai ), i = 1, 2, 3, then, for the error the estimate

h2

sup |u(x) ’ (IK (u))(x)| + h sup |∇(u ’ IK (u))(x)| ¤ 2M

cos(±/2)

x∈K x∈K

holds, where h denotes the diameter, ± the size of the largest interior angle

of K and M an upper bound for the maximum of the norm of the Hessian

matrix of u on K.

3.26 Consider a triangle K with the vertices a1 := (’h, 0), a2 := (h, 0),

a3 := (0, µ), and h, µ > 0. Suppose that the function u(x) := x2 is linearly

1

interpolated on K such that (Ih (u))(ai ) = u(ai ) for i = 1, 2, 3.

Determine ‚2 (Ih (u) ’ u) 2,K as well as ‚2 (Ih (u) ’ u) ∞,K and discuss

the consequences for of di¬erent orders of magnitude of h and µ.

3.27 Suppose that no further regularity properties are known for the

solution u ∈ V of the boundary value problem (3.12). Show under the

assumptions of Section 3.4 that for the ¬nite element approximation

uh ∈ Vh

u ’ uh ’ 0 for h ’ 0 .

1

148 3. Finite Element Methods for Linear Elliptic Problems

3.5 The Implementation of the Finite Element

Method: Part 2

3.5.1 Incorporation of Dirichlet Boundary Conditions: Part 2

In the theoretical analysis of boundary value problems with inhomogeneous

Dirichlet boundary conditions u = g3 on “3 , the existence of a function

w ∈ H 1 („¦) with w = g3 on “3 has been assumed so far. The solution

u ∈ V (with homogeneous Dirichlet boundary conditions) is then de¬ned

according to (3.31) such that u = u + w satis¬es the variational equation

˜

with test functions in V :

a(u + w, v) = b(v) for all v ∈ V . (3.99)

For the Galerkin approximation uh , which has been analyzed in Section 3.4,

this means that the parts ’a(w, •i ) with nodal basis functions •i , i =

1, . . . , M1 , go into the right-hand side of the system of equations (2.34), and

then uh := uh +w has to be considered as the solution of the inhomogeneous

˜

problem

for all v ∈ Vh .

a(uh + w, v) = b(v) (3.100)

If we complete the basis of Vh by the basis functions •M1 +1 , . . . , •M for the

Dirichlet boundary nodes aM1 +1 , . . . , aM and denote the generated space

by Xh ,

Xh = span {•1 , . . . , •M1 , •M1 +1 , . . . , •M } , (3.101)

that is the ansatz space without taking into account boundary conditions,

then in particular, uh ∈ Xh does not hold in general. This approach does

˜

not correspond to the practice described in Section 2.4.3. That practice,

applied to a general variational equation, reads as follows:

For all degrees of freedom 1, . . . , M1 , M1 + 1, . . . , M the system of

equations is built with the components

a(•j , •i ) , i, j = 1, . . . , M , (3.102)

for the sti¬ness matrix and

b(•i ) , i = 1, . . . , M , (3.103)

for the load vector. The vector of unknowns is therefore

ξ

˜ ˆ

with ξ ∈ RM1 , ξ ∈ RM2 .

ξ= ˆ

ξ

For Dirichlet boundary conditions the equations M1 +1, . . . , M are replaced

by

˜

ξi = g3 (ai ) , i = M1 + 1, . . . , M ,

3.5. The Implementation of the Finite Element Method: Part 2 149

and the concerned variables are eliminated in equations 1, . . . , M1 . Of

course, it is assumed here that g3 ∈ C(“3 ). This procedure can also be

interpreted in the following way: If we set

ˆ

Ah := (a(•j , •i ))i,j=1,...,M1 , Ah := (a(•j , •i ))i=1,...,M1 , j=M1 +1,...,M ,

then the ¬rst M1 equations of the generated system of equations are

ˆˆ

Ah ξ + Ah ξ = q h ,

where q h ∈ RM1 consists of the ¬rst M1 components according to (3.103).

Hence the elimination leads to

ˆˆ

Ah ξ = q h ’ Ah ξ (3.104)

ˆ

with ξ = (g3 (ai ))i=M1 +1,...,M2 . Suppose

M

g3 (ai ) •i ∈ Xh

wh := (3.105)

i=M1 +1

is the ansatz function that satis¬es the boundary conditions in the Dirichlet

nodes and assumes the value 0 in all other nodes. The system of equations

(3.104) is then equivalent to

for all v ∈ Vh

a(ˇh + wh , v) = b(v)

u (3.106)

for uh = M1 ξi •i ∈ Vh (that is, the “real” solution), in contrast to the

ˇ i=1

variational equation (3.100) was used in the analysis. This consideration

also holds if another h-dependent bilinear form ah and analogously a lin-

ear form bh instead of the linear form b is used for assembling. In the

following we assume that there exists some function w ∈ C(„¦) that sat-

¯

is¬es the boundary condition on “3 . Instead of (3.106), we consider the

¬nite-dimensional auxiliary problem of ¬nding some uh ∈ Vh , such that

ˇ

for all v ∈ Vh .

¯

ˇ

a(uh + Ih (w), v) = b(v)

ˇ (3.107)

Here Ih : C(„¦) ’ Xh is the interpolation operator with respect to all

¯ ¯

degrees of freedom,

M1 +M2

¯

Ih (v) := v(ai )•i ,

i=1

whereas in Section 3.4 we considered the interpolation operator Ih for func-

tions that vanish on “3 . In the following, when analyzing the e¬ect of

quadrature, we will show that ” also for some approximation of a and b

”

uh := uh + Ih (w) ∈ Xh

¯

ˇ

˜ ˇ (3.108)

is an approximation of u + w of the quality established in Theorem 3.29

(see Theorem 3.42). We have wh ’ Ih (w) ∈ Vh and hence also uh + wh ’

¯ ˇ

150 3. Finite Element Methods for Linear Elliptic Problems

Ih (w) ∈ Vh . If (3.107) is uniquely solvable, which follows from the general

¯

assumption of the V -ellipticity of a (3.3), we have

uh + wh ’ Ih (w) = uh

¯ ˇ

ˇ ˇ

and hence for uh , according to (3.108),

˜

uh = uh + wh .

˜ ˇ (3.109)

In this way the described implementation practice for Dirichlet boundary

conditions is justi¬ed.

3.5.2 Numerical Quadrature

We consider again a boundary value problem in the variational formulation

(3.31) and a ¬nite element discretization in the general form described

in Sections 3.3 and 3.4. If we step through Section 2.4.2 describing the

assembling within a ¬nite element code, we notice that the general element-

to-element approach with transformation to the reference element is here

also possible, with the exception that due to the general coe¬cient functions

K, c, r and f , the arising integrals can not be evaluated exactly in general.

If Km is a general element with degrees of freedom in ar1 , . . . , arL , then

the components of the element sti¬ness matrix for i, j = 1, . . . , L are

(m)

K∇•rj · ∇•ri + c · ∇•rj •ri + r•rj •ri dx

Aij =

Km

+ ±•rj •ri dσ (3.110)

Km ©“2

=: vij (x) dx + wij (σ) dσ

Km ©“2

Km

vij (ˆ) dˆ | det(B)| + wij (ˆ ) dˆ | det(B)| .

˜

= ˆxx ˆσσ

ˆ ˆ

K K

ˆ

Here, Km is a¬ne equivalent to the reference element K by the mapping

F (ˆ) = B x + d. By virtue of the conformity of the triangulation (T6), the

x ˆ

boundary part Km © “2 consists of none, one, or more complete faces of

¯

Km . For simplicity, we restrict ourselves to the case of one face that is a¬ne

˜

ˆ ˜σ ˜ˆ

equivalent to the reference element K by some mapping F (ˆ ) = B σ + d

(cf. (3.42)). The generalization to the other cases is obvious. The functions

vij and analogously wij are the transformed functions de¬ned in (3.75).

ˆ ˆ

Correspondingly, we get as components for the right-hand side of the

system of equations, that is, for the load vector,

ˆx

f (ˆ)Ni (ˆ) dˆ | det(B)|

q (m) = xx (3.111)

i ˆ

K

g1 (ˆ )Ni (ˆ ) dˆ | det(B1 )| + g2 (ˆ )Ni (ˆ ) dˆ | det(B2 )| .

˜ ˜

+ ˆσ σσ ˆσ σσ

ˆ ˆ

K1 K2

3.5. The Implementation of the Finite Element Method: Part 2 151

i = 1, . . . , L. Here, the Ni , i = 1, . . . , L, are the shape functions; that is,

ˆ

the local nodal basis functions on K.

If the transformed integrands contain derivatives with respect to x, they

can be transformed into derivatives with respect to x. For instance, for the

ˆ

(m)

¬rst addend in Aij we get, as an extension of (2.50),

K(F (ˆ))B ’T ∇x Nj (ˆ) · B ’T ∇x Ni (ˆ) dˆ | det(B)| .

x x xx

ˆ ˆ

ˆ

K

ˆ

The shape functions, their derivatives, and their integrals over K are known

which has been used in (2.52) for the exact integration. Since general coef-

¬cient functions arise, this is in general, but also in the remaining special

cases no longer possible, for example for polynomial K(x) it is also not

recommendable due to the corresponding e¬ort. Instead, one should ap-

proximate these integrals (and, analogously, also the boundary integrals)

by using some quadrature formula.

ˆ

A quadrature formula on K for the approximation of K v (ˆ) dˆ has the

ˆˆx x

form

R

ωi v (ˆi )

ˆ ˆb (3.112)

i=1

with weights ωi and quadrature or integration points ˆi ∈ K. Hence, ap-

ˆ

ˆ b