ˆ

Therefore, let some reference element K with the nodes a1 , . . . , aL be

ˆ ˆ

chosen as ¬xed. By assumption, there exists some bijective, a¬ne-linear

134 3. Finite Element Methods for Linear Elliptic Problems

mapping

F = FK : K ’ K ,

ˆ

(3.74)

F (ˆ) = B x + d ,

x ˆ

(cf. (2.30) and (3.57)). By this transformation, functions v : K ’ R are

mapped to functions v : K ’ R by

ˆˆ

v (ˆ) := v(F (ˆ)) .

ˆx x (3.75)

This transformation is also compatible with the local interpolation operator

in the following sense:

IK (v) = IK (ˆ) for v ∈ C(K) .

ˆv (3.76)

This follows from the fact that the nodes of the elements as well as the

shape functions are mapped onto each other by F .

For a classically di¬erentiable function the chain rule (see (2.49)) implies

∇x v(F (ˆ)) = B ’T ∇x v (ˆ) ,

x ˆˆ x (3.77)

and corresponding formulas for higher-order derivatives, for instance,

Dx v(F (ˆ)) = B ’T Dx v (ˆ)B ’1 ,

2 2

x ˆˆ x

2

where Dx v(x) denotes the matrix of the second-order derivatives. These

chain rules hold also for corresponding v ∈ H l (K) (Exercise 3.22).

The situation becomes particularly simple in one space dimension (d =

1). The considered elements reduce to a polynomial ansatz on simplices,

which here are intervals. Thus

F : K = [0, 1] ’ K = [ai1 , ai2 ] ,

ˆ

x ’ hK x + ai1 ,

ˆ ˆ

where hK := ai2 ’ ai1 denotes the length of the element. Hence, for l ∈ N,

‚x v(F (ˆ)) = h’l ‚x v (ˆ) .

l l

x K ˆˆ x

By the substitution rule for integrals (cf. (2.50)) an additional factor

| det(B)| = hK arises such that, for v ∈ H l (K), we have

2l’1

1

|v|2 |ˆ|2 K .

= v l, ˆ

l,K

hK

Hence, for 0 ¤ m ¤ k + 1 it follows by (3.76) that

2m’1

1 2

IK (v)|2

|v ’ v ’ IK (ˆ)

= ˆ ˆv .

ˆ

m,K m,K

hK

Thus, what is missing, is an estimate of the type

v ’ IK (ˆ) ¤ C|ˆ|k+1,K

ˆ ˆv v (3.78)

ˆ

ˆ

m,K

3.4. Convergence Rate Estimates 135

for v ∈ H k+1 (K). In speci¬c cases this can partly be proven directly but

ˆ

ˆ

in the following a general proof, which is also independent of d = 1, will be

sketched. For this, the mapping

G : H k+1 (K) ’ H m (K) ,

ˆ ˆ

(3.79)

v ’ v ’ IK (ˆ) ,

ˆ ˆ ˆv

is considered. The mapping is linear but also continuous, since

L

¤

IK (ˆ)

ˆv v (ˆi )•i

ˆa ˆ

ˆ

m,K

ˆ

i=1 k+1,K

(3.80)

L

¤ ¤C v

•i

ˆ v

ˆ ˆ ,

ˆ ˆ ˆ

∞,K

k+1,K k+1,K

i=1

ˆ ˆ

where the continuity of the embedding of H k+1 (K) in H m (K) (see

ˆ ˆ

(3.8)) and of H k+1 (K) in C(K) (Theorem 3.10) is used, and the norm

contribution from the ¬xed basis functions •i is included in the constant.

ˆ

ˆ is chosen in such a way that Pk ‚ P , then G has

ˆ

If the ansatz space P

the additional property

G(p) = 0 for p ∈ Pk ,

since these polynomials are interpolated then exactly. Such mappings sat-

isfy the Bramble“Hilbert lemma, which will directly be formulated, for

further use, in a more general way.

Theorem 3.24 (Bramble“Hilbert lemma)

Suppose K ‚ Rd is open, k ∈ N0 , 1 ¤ p ¤ ∞, and G : Wp (K) ’ R is a

k+1

continuous linear functional that satis¬es

for all q ∈ Pk .

G(q) = 0 (3.81)

Then there exists some constant C > 0 independent of G such that for all

v ∈ Wp (K)

k+1

|G(v)| ¤ C G |v|k+1,p,K .

2

Proof: See [9, Theorem 28.1].

Here G denotes the operator norm of G (see (A4.25)). The estimate

with the full norm · k+1,p,K on the right-hand side (and C = 1) would

hence only be the operator norm™s de¬nition. The condition (3.81) allows

the reduction to the highest seminorm.

For the application of the Bramble“Hilbert lemma (Theorem 3.24), which

was formulated only for functionals, to the operator G according to (3.79)

an additional argument is required (alternatively, Theorem 3.24 could be

generalized):

136 3. Finite Element Methods for Linear Elliptic Problems

Generally, for w ∈ H m (K) (as in every normed space) we have

ˆ

ˆ

w

ˆ = sup •(w) ,

ˆ (3.82)

ˆ

m,K

ˆ

•∈(H m (K))

• ¤1

where the norm applying to • is the operator norm de¬ned in (A4.25).

For any ¬xed • ∈ (H m (K)) the linear functional on H k+1 (K) is de¬ned

ˆ ˆ

by

for v ∈ H k+1 (K) .

˜v ˆ

G(ˆ) := •(G(ˆ))

v ˆ (3.83)

˜

According to (3.80), G is continuous and it follows that

G¤•

˜ G.

˜

Theorem 3.24 is applicable to G and yields

|G(ˆ)| ¤ C • G |ˆ|k+1,K .

˜v v ˆ

By means of (3.82) it follows that

¤ C G |ˆ|k+1,K .

G(ˆ)

v v

ˆ ˆ

m,K

The same proof can also be used in the proof of Theorem 3.31 (3.94).

Applied to G de¬ned in (3.79), the estimate (3.80) shows that the

operator norm Id ’ IK can be estimated independently from m (but

ˆ

dependent on k and the •i ) and can be incorporated in the constant that

ˆ

gives (3.78) in general, independent of the one-dimensional case.

Therefore, in the one-dimensional case we can continue with the

estimation and get

2m’1

1

IK (v)|2

|v ’ ¤ C|ˆ|2 K ¤ C(hK )1’2m+2(k+1)’1 |v|2

v k+1, ˆ k+1,K .

m,K

hK

Since due to Ih (v) ∈ H 1 („¦) we have for m = 0, 1

|v ’ IK (v)|2

m,K = |v ’ Ih (v)|m ,

2

K∈Th

we have proven the following Theorem:

Theorem 3.25 Consider in one space dimension „¦ = (a, b) the polyno-

mial Lagrange ansatz on elements with maximum length h and suppose that

for the respective local ansatz spaces P , the inclusion Pk ‚ P is satis¬ed

for some k ∈ N. Then there exists some constant C > 0 such that for all

v ∈ H k+1 („¦) and 0 ¤ m ¤ k + 1,

1/2

|v ’ IK (v)|2 ¤ Chk+1’m |v|k+1 .

m,K

K∈Th

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

to H k+1 („¦), then we have for the ¬nite element approximation uh according

3.4. Convergence Rate Estimates 137

to (3.39),

u ’ uh ¤ Chk |u|k+1 .

1

Note that for d = 1 a direct proof is also possible (see Exercise 3.21).

Now we address to the general d-dimensional situation: The seminorm

| · |1 is transformed, for instance, as follows (cf. (2.49)):

B ’T ∇x v · B ’T ∇x v | det(B)| dˆ .

|v|2 = |∇x v|2 dx = ˆˆ ˆˆ x (3.84)

1,K

ˆ

K K

From this, it follows for v ∈ H 1 (K) that

ˆ

ˆ

|v|1,K ¤ C B ’1 | det(B)|1/2 |ˆ|1,K .

vˆ

Since d is one of the mentioned “¬xed” quantities and all norms on Rd,d

are equivalent, the matrix norm · can be chosen arbitrarily, and it is

also possible to change between such norms. In the above considerations K

and K had equal rights; thus similarly for v ∈ H 1 (K), we have

ˆ

|ˆ|1,K ¤ C B | det(B)|’1/2 |v|1,K .

vˆ

In general, we have the following theorem:

Theorem 3.26 Suppose K and K are bounded domains in Rd that are

ˆ

mapped onto each other by an a¬ne bijective linear mapping F , de¬ned in

(3.74). If v ∈ Wp (K) for l ∈ N and p ∈ [1, ∞], then we have for v (de¬ned

l

ˆ

in (3.75)), v ∈ Wp (K), and for some constant C > 0 independent of v,

lˆ

ˆ

| det(B)|’1/p |v|l,p,K ,

|ˆ|l,p,K ¤ l

v CB (3.85)

ˆ

C B ’1

|v|l,p,K ¤ | det(B)|1/p |ˆ|l,p,K .

l

v (3.86)

ˆ

2

Proof: See [9, Theorem 15.1].

For further use, also this theorem has been formulated in a more general

way than would be necessary here. Here, only the case p = 2 is relevant.

Hence, if we use the estimate of Theorem 3.24, then the value B (for

some matrix norm) has to be related to the geometry of K. For this, let

for K ∈ Th ,

:= sup diam (S) S is a ball in Rd and S ‚ K .

K

Hence, in the case of a triangle, hK denotes the longest edge and K the

diameter of the inscribed circle. Similarly, the reference element has its

ˆ

(¬xed) parameters h and ˆ. For example, for the reference triangle with

ˆ

the vertices a1 = (0, 0), a2 = (1, 0), a3 = (0, 1) we have that h = 21/2 and

ˆ ˆ ˆ

ˆ = 2 ’ 21/2 .

138 3. Finite Element Methods for Linear Elliptic Problems

Theorem 3.27 For F = FK according to (3.74), in the spectral norm · 2 ,

we have

ˆ

hK h

B ’1

¤ ¤

B and .

2 2

ˆ K

ˆ

Proof: Since K and K have equal rights in the assertion, it su¬ces to

prove one of the statements: We have (cf. (A4.25))

1 1

sup |Bξ|2 .

B = sup B ξ =

2

ˆ ˆ |ξ|2 = ˆ

|ξ|2 = ˆ 2

For every ξ ∈ Rd with |ξ|2 = ˆ there exist some points y , z ∈ K such that

ˆˆ ˆ

y ’ˆ = ξ. Since Bξ = F (ˆ)’F (ˆ) and F (ˆ), F (ˆ) ∈ K, we have |Bξ|2 ¤ hK .

ˆz y z y z

2

Consequently, by the above identity we get the ¬rst inequality.

If we combine the local estimates of (3.78), Theorem 3.26, and

Theorem 3.27, we obtain for v ∈ H k+1 (K) and 0 ¤ m ¤ k + 1,

m

hK

|v ’ IK (v)|m,K ¤ C hk+1’m |v|k+1,K , (3.87)

K

K

where ˆ and ˆ are included in the constant C. In order to obtain some

h

convergence rate result, we have to control the term hK / K . If this term is

bounded (uniformly for all triangulations), we get the same estimate as in

the one-dimensional case (where even hK / K = 1). Conditions of the form

≥ σh1+±

K K

for some σ > 0 and 0 ¤ ± < k+1 ’ 1 for m ≥ 1 would also lead to

m

convergence rate results. Here we pursue only the case ± = 0.

De¬nition 3.28 A family of triangulations (Th )h is called regular if there

exists some σ > 0 such that for all h > 0 and all K ∈ Th ,

≥ σhK .

K

From estimate (3.87) we conclude directly the following theorem:

Theorem 3.29 Consider a family of Lagrange ¬nite element discretiza-

tions in Rd for d ¤ 3 on a regular family of triangulations (Th )h in the

generality described at the very beginning. For the respective local ansatz

spaces P suppose Pk ‚ P for some k ∈ N.

Then there exists some constant C > 0 such that for all v ∈ H k+1 („¦)

and 0 ¤ m ¤ k + 1,

1/2

|v ’ IK (v)|2 ¤ Chk+1’m |v|k+1 . (3.88)

m,K

K∈Th

3.4. Convergence Rate Estimates 139

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

to H k+1 („¦), then for the ¬nite element approximation uh de¬ned in (3.39),

it follows that

u ’ uh ¤ Chk |u|k+1 . (3.89)

1

Remark 3.30 Indeed, here and also in Theorem 3.25 a sharper estimate

has been shown, which, for instance for (3.89), has the following form:

1/2

u ’ uh ¤C h2k |u|2 . (3.90)

1 K k+1,K

K∈Th

In the following we will discuss what the regularity assumption means in

the two simplest cases:

For a rectangle and the cuboid K, whose edge lengths can be assumed,

without any loss of generality, to be of order h1 ¤ h2 [¤ h3 ], we have

1/2

2 2

hK h2 h3

= 1+ + .

h1 h1

K

This term is uniformly bounded if and only if there exists some constant

±(≥ 1) such that

¤ ¤

h1 h2 ±h1 ,

(3.91)

¤ ¤

h1 h3 ±h1 .

In order to satisfy this condition, a re¬nement in one space direction has

to imply a corresponding one in the other directions, although in certain

anisotropic situations only the re¬nement in one space direction is recom-

mendable. If, for instance, the boundary value problem (3.12), (3.18)“(3.20)

with c = r = 0, but space-dependent conductivity K, is interpreted as the

simplest ground water model (see (0.18)), then it is typical that K varies

discontinuously due to some layering or more complex geological structures

(see Figure 3.11).

K1

K2

K1

Figure 3.11. Layering and anisotropic triangulation.

If thin layers arise in such a case, on the one hand they have to be resolved;

that is, the triangulation has to be compatible with the layering and there

140 3. Finite Element Methods for Linear Elliptic Problems

have to be su¬ciently many elements in this layer. On the other hand, the

solution often changes less strongly in the direction of the layering than over

the boundaries of the layer, which suggests an anisotropic triangulation,

that is, a strongly varying dimensioning of the elements. The restriction

(3.91) is not compatible with this, but in the case of rectangles this is

due only to the techniques of proof. In this simple situation, the local

interpolation error estimate can be performed directly, at least for P =

Q1 (K), without any transformation such that the estimate (3.89) (for k =

1) is obtained without any restrictions like (3.91).

The next simple example is a triangle K: The smallest angle ±min =

±min (K) includes the longest edge hK , and without loss of generality, the

situation is as illustrated in Figure 3.12.

a3

± min

a1

h2

hK

a2

Figure 3.12. Triangle with the longest edge and the height as parameters.

For the 2 — 2 matrix B = (a2 ’ a1 , a3 ’ a1 ), in the Frobenius norm · F

(see (A3.5)) we have

1

B ’1 = B ,

F F

| det(B)|

and further, with the height h2 over hK ,

det(B) = hK h2 , (3.92)

since det(B)/2 is the area of the triangle, as well as

= |a2 ’ a1 |2 + |a3 ’ a1 |2 ≥ h2 ,

2

B F 2 2 K

such that

B ’1 ≥ hK /h2 ,

B F F

and thus by virtue of cot ±min < hK /h2 ,

B ’1

B > cot ±min .

F F

Since we get by analogous estimates

B ’1 ¤ 4 cot ±min ,

B F F

it follows that cot ±min describes the asymptotic behavior of B B ’1 for

a ¬xed chosen arbitrary matrix norm. Therefore, from Theorem 3.27 we

3.4. Convergence Rate Estimates 141

get the existence of some constant C > 0 independent of h such that for

all K ∈ Th ,

hK

≥ C cot ±min (K) . (3.93)

K

Consequently, a family of triangulations (Th )h of triangles can only be reg-

ular if all angles of the triangles are uniformly bounded from below by

some positive constant. This condition sometimes is called the minimum

angle condition. In the situation of Figure 3.11 it would thus not be al-

lowed to decompose the ¬‚at rectangles in the thin layer by means of a

Friedrichs“Keller triangulation. Obviously, using directly the estimates of

Theorem 3.26 we see that the minimum angle condition is su¬cient for the

estimates of Theorem 3.29. This still leaves the possibility open that less

severe conditions are also su¬cient.

3.4.2 The Maximum Angle Condition on Triangles

In what follows we show that the condition (3.93) is due only to the tech-

niques of proof, and at least in the case of the linear ansatz, it has indeed

only to be enssured that the largest angle is uniformly bounded away from

π. Therefore, this allows the application of the described approach in the

layer example of Figure 3.11.

The estimate (3.87) shows that for m = 0 the crucial part does not arise;

hence only for m = k = 1 do the estimates have to be investigated. It turns

out to be useful to prove the following sharper form of the estimate (3.78):

ˆ

Theorem 3.31 For the reference triangle K with linear ansatz functions

there exists some constant C > 0 such that for all v ∈ H 2 (K) and j = 1, 2,

ˆ

ˆ

‚ ‚

v ’ IK (ˆ) ¤C

ˆ ˆv v

ˆ .

‚ xj

ˆ ‚ xj

ˆ

ˆ ˆ

0,K 1,K

Proof: In order to simplify the notation, we drop the hat ˆ in the notation

of the reference situation in the proof. Hence, we have K = conv {a1 , a2 , a3 }

with a1 = (0, 0)T , a2 = (1, 0)T , and a3 = (0, 1)T . We consider the following

linear mappings: F1 : H 1 (K) ’ L2 (K) is de¬ned by

1

F1 (w) := w(s, 0) ds ,

0

and, analogously, F2 as the integral over the boundary part conv {a1 , a3 }.

The image is taken as constant function on K. By virtue of the Trace The-

orem (Theorem 3.5), and the continuous embedding of L2 (0, 1) in L1 (0, 1),

the Fi are well-de¬ned and continuous. Since we have for w ∈ P0 (K),

Fi (w) = w ,

142 3. Finite Element Methods for Linear Elliptic Problems

the Bramble“Hilbert lemma (Theorem 3.24) implies the existence of some

constant C > 0 such that for w ∈ H 1 (K),

Fi (w) ’ w ¤ C|w|1,K . (3.94)

0,K

This can be seen in the following way: Let v ∈ H 1 (K) be arbitrary but

¬xed, and for this, consider on H 1 (K) the functional

G(w) := Fi (w) ’ w, Fi (v) ’ v for w ∈ H 1 (K) .

We have G(w) = 0 for w ∈ P0 (K) and

|G(w)| ¤ Fi (w) ’ w Fi (v) ’ v ¤ C Fi (v) ’ v w

0,K 0,K 0,K 1,K

by the above consideration. Thus by Theorem 3.24,

|G(w)| ¤ C Fi (v) ’ v |w|1,K .

0,K

For v = w this implies (3.94). On the other hand, for w := ‚1 v it follows

that

v(1, 0) ’ v(0, 0) = (IK (v))(1, 0) ’ (IK (v))(0, 0) =

F1 (‚1 v) =

= ‚1 (IK (v))(x1 , x2 )

for (x1 , x2 ) ∈ K and, analogously, F2 (‚2 v) = ‚2 (IK (v))(x1 , x2 ). This,

2

substituted into (3.94), gives the assertion.

Compared with estimate (3.78), for example in the case j = 1 the term

2

‚

v does not arise on the right-hand side: The derivatives and thus the

ˆ

‚ x2

ˆ2

space directions are therefore treated “more separately.”

Next, the e¬ect of the transformation will be estimated more precisely.

For this, let ±max = ±max (K) be the largest angle arising in K ∈ Th ,

supposed to include the vertex a1 , and let h1 = h1K := |a2 ’ a1 |2 , h2 =

h2K := |a3 ’ a1 | (see Figure 3.13).

a1

±max

h2

h1

a3

a2

Figure 3.13. A general triangle.

As a variant of (3.86) (for l = 1) we have the following:

3.4. Convergence Rate Estimates 143

Theorem 3.32 Suppose K is a general triangle. With the above notation

for v ∈ H 1 (K) and the transformed v ∈ H 1 (K),

ˆ

ˆ

1/2

√ 2 2

‚ ‚

¤ 2 | det(B)|’1/2

|v|1,K h2 + h2

v

ˆ v

ˆ .

2 1

‚ x1

ˆ ‚ x2

ˆ

ˆ ˆ

0,K 0,K

Proof: We have

b11 b12

B = (a2 ’ a1 , a3 ’ a1 ) =:

b21 b22

and hence

b11 b12

= h1 , = h2 . (3.95)

b21 b22

From

’b21

b22

1

B ’T =

’b12

det(B) b11

and (3.84) it thus follows that

2

’b21

1 b22 ‚ ‚

|v|2 = v+

ˆ v

ˆ dˆ

x

| det(B)| ’b12

1,K

‚ x1

ˆ b11 ‚ x2

ˆ

ˆ

K

2

and from this the assertion.

In modi¬cation of the estimate (3.85) (for l = 2) we prove the following

result:

Theorem 3.33 Suppose K is a general triangle with diameter hK =

diam (K). With the above notation for v ∈ H 2 (K) and the transformed

ˆ

ˆ

v ∈ H 2 (K),

‚

¤ 4| det(B)|’1/2 hi hK |v|2,K

v

ˆ for i = 1, 2 .

‚ xi

ˆ ˆ

1,K

ˆ

Proof: According to (3.84) we get by exchanging K and K,

B T ∇x w · B T ∇x w dx | det(B)|’1

|w|2 K =

ˆ 1, ˆ

K

thus by (3.77) for w = (B T ∇x v)i ,

‚

and, consequently, for w =

ˆ ‚ xi v ,

ˆˆ

2

‚ 2

dx | det(B)|’1 .

B T ∇x B T ∇x v

v

ˆ = i

‚ xi

ˆ ˆ K

1,K

According to (3.95), the norm of the ith row vector of B T is equal to hi ,

2

which implies the assertion.

144 3. Finite Element Methods for Linear Elliptic Problems

Instead of the regularity of the family of triangulations and hence

the uniform bound for cot ±min (K) (see (3.93)) we require the following

de¬nition:

De¬nition 3.34 A family of triangulations (Th )h of triangles satis¬es the

maximum angle condition if there exists some constant ± < π such that for

all h > 0 and K ∈ Th the maximum angle ±max (K) of K satis¬es

±max (K) ¤ ± .

Since ±max (K) ≥ π/3 is always satis¬ed, the maximum angle condition

is equivalent to the existence of some constant s > 0, such that

˜

sin(±max (K)) ≥ s for all K ∈ Th and h > 0 .

˜ (3.96)

The relation of this condition to the above estimates is given by (cf. (3.92))

det(B) = h1 h2 sin ±max . (3.97)

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

plying (3.112) assumes the evaluability of v in ˆi , which is in the following

ˆb

ensured by the continuity of v . This implies the same assumption for the

ˆ

coe¬cients, since the shape functions Ni and their derivatives are continu-

ous. In order to ensure the numerical stability of a quadrature formula, it

is usually required that

ωi > 0 for all i = 1, . . . , R ,

ˆ (3.113)

which we will also do. Since all the considered ¬nite elements are such

that their faces with the enclosed degrees of freedom represent again a ¬-

nite element (in Rd’1 ) (see (3.42)), the boundary integrals are included

in a general discussion. In principle, di¬erent quadrature formulas can be

applied for each of the above integrals, but here we will disregard this pos-

sibility (with the exception of distinguishing between volume and boundary

integrals because of their di¬erent dimensions).

ˆ

A quadrature formula on K generates a quadrature formula on a general

element K, recalling

v (ˆ) dˆ | det(B)|

v(x) dx = ˆx x

ˆ

K K

by

R

ωi,K v(bi,K ) ,

i=1

152 3. Finite Element Methods for Linear Elliptic Problems

:= F (ˆi ) are dependent on

where ωi = ωi,K = ωi | det(B)| and bi = bi,K

ˆ b

K. The positivity of the weights is preserved. Here, again F (ˆ) = B x + d

x ˆ

ˆ to K. The errors of the

denotes the a¬ne-linear transformation from K

quadrature formulas

R

ωi v (ˆi ) ,

v (ˆ) dˆ ’

ˆv

E(ˆ) := ˆx x ˆ ˆb

ˆ

K i=1

(3.114)

R

v(x) dx ’

EK (v) := ωi v(bi )

K i=1

are related to each other by

EK (v) = | det(B)|E(ˆ) .

ˆv (3.115)

The accuracy of a quadrature formula will be de¬ned by the requirement

that for l as large as possible,

E(ˆ) = 0 for p ∈ Pl (K)

ˆp ˆ

ˆ

is satis¬ed, which transfers directly to the integration over K. A quadrature

formula should further provide the desired accuracy by using quadrature

nodes as less as possible, since the evaluation of the coe¬cient functions is

often expensive. In contrast, for the shape functions and their derivatives

a single evaluation is su¬cient. In the following we discuss some exam-

ples of quadrature formulas for the elements that have been introduced in

Section 3.3.

The most obvious approach consists in using nodal quadrature formu-

ˆ ˆˆ

las, which have the nodes a1 , . . . , aL of the reference element (K, P , Σ) as

ˆ ˆ

ˆ

quadrature nodes. The requirement of exactness in P is then equivalent to

ωi =

ˆ Ni (ˆ) dˆ ,

xx (3.116)

ˆ

K

so that the question of the validity of (3.113) remains.

ˆ

We start with the unit simplex K de¬ned in (3.47). Here, the weights

of the quadrature formulas can be given directly on a general simplex K: If

the shape functions are expressed by their barycentric coordinates »i , the

integrals can be computed by

±1 !±2 ! · · · ±d+1 ! vol (K)

±

»±1 »±2 · · · »d+1 (x) dx =

d+1

(3.117)

(±1 + ±2 + · · · + ±d+1 + d)! vol (K)

1 2 ˆ

K

(see Exercise 3.28).

If P = P1 (K) and thus the quadrature nodes are the vertices, it follows

that

1

ωi = »i (x) dx = vol (K) for all i = 1, . . . , d + 1 . (3.118)

d+1

K

3.5. The Implementation of the Finite Element Method: Part 2 153

For P = P2 (K) and d = 2 we get, by the shape functions »i (2»i ’ 1), the

weights 0 for the nodes ai and, by the shape functions 4»i »j , the weights

1

ωi = vol (K) for bi = aij , i, j = 1, . . . , 3 , i > j ,

3

so that we have obtained here a quadrature formula that is superior to

(3.118) (for d = 2). However, for d ≥ 3 this ansatz leads to negative weights

and is thus useless. We can also get the exactness in P1 (K) by a single

quadrature node, by the barycentre (see (3.52)):

d+1

1

ω1 = vol (K) and b1 = aS = ai ,

d+1 i=1

which is obvious due to (3.117).

As a formula that is exact for P2 (K) and d = 3 (see [53]) we present

R = 4, ωi = 1 vol (K), and the bi are obtained by cyclic exchange of the

4

barycentric coordinates:

√ √ √ √

5’ 5 5’ 5 5’ 5 5+3 5

, , , .

20 20 20 20

ˆ

On the unit cuboid K we obtain nodal quadrature formulas, which are

ˆ

exact for Qk (K), from the Newton“Cˆtes formulas in the one-dimensional

o

situation by

i1 id

for ˆi1 ...id =

ωi1 · · · ωid

ωi1 ...id

ˆ = ˆ ˆ b ,..., (3.119)

k k

for ij ∈ {0, . . . , k} and j = 1, . . . , d .

1

Here the ωij are the weights of the Newton“Cˆtes formula for 0 f (x)dx

ˆ o

(see [30, p. 128]). As in (3.118), for k = 1 we have here a generalization

of the trapezoidal rule (cf. (2.38), (8.31)) with the weights 2’d in the 2d

vertices. From k = 8 on, negative weights arise. This can be avoided and

the accuracy for a given number of points increased if the Newton“Cˆtes o

integration is replaced by the Gauss“(Legendre) integration: In (3.119), ij /k

has to be replaced by the jth node of the kth Gauss“Legendre formula

(see [30, p. 156] there on [’1, 1]) and analogously ωij . In this way, by

ˆ

ˆ

ˆ

(k + 1)d quadrature nodes the exactness in Q2k+1 (K), not only in Qk (K),

is obtained.

Now the question as to which quadrature formula should be chosen arises.

For this, di¬erent criteria can be considered (see also (8.29)). Here, we re-

quire that the convergence rate result that was proved in Theorem 3.29

should not be deteriorated. In order to investigate this question we have

to clarify which problem is solved by the approximation uh ∈ Vh based on

¯

quadrature. To simplify the notation, from now on we do not consider

boundary integrals, that is, only Dirichlet and homogeneous Neumann

154 3. Finite Element Methods for Linear Elliptic Problems

boundary conditions are allowed. However, the generalization should be

clear. Replacing the integrals in (3.111) and (3.111) by quadrature formu-

las R ωi v (ˆi ) leads to some approximation Ah of the sti¬ness matrix

¯

i=1 ˆ ˆ b

¯

and q h of the load vector in the form

¯ ¯

Ah = (ah (•j , •i ))i,j , q h = (bh (•i ))i ,

for i, j = 1, . . . , M . Here the •i are the basis functions of Xh (see (3.101))

without taking into account the Dirichlet boundary condition and

R

ωl,K (K∇v · ∇w)(bl,K )

ah (v, w) :=

K∈Th l=1

R

ωl,K (c · ∇vw)(bl,K ) +

+ ωl,K (rvw)(bl,K )

K∈Th l=1 K∈Th l=1

for v, w ∈ Xh , (3.120)

for v ∈ Xh .

bh (v) := ωl,K (f v)(bl,K )

K∈Th l=1

The above-given mappings ah and bh are well-de¬ned on Xh — Xh and Xh ,

respectively, if the coe¬cient functions can be evaluated in the quadrature

nodes. Here we take into account that for some element K, ∇v for v ∈

Xh can have jump discontinuities on ‚K. Thus, for the quadrature nodes

bl,K ∈ ‚K in ∇v(bl,K ) we have to choose the value “belonging to bl,K ” that

corresponds to the limit of sequences in the interior of K. We recall that

in general ah and bh are not de¬ned for functions of V . Obviously, ah is

bilinear and bh is linear. If we take into account the analysis of incorporating

the Dirichlet boundary conditions in (3.99)“(3.106), we get a system of

¯

equations for the degrees of freedom ξ = (ξ1 , . . . , ξM1 )T , which is equivalent

M1 ¯

to the variational equation on Vh for uh = i=1 ξi •i ∈ Vh :

¯

ah (¯h , v) = bh (v) ’ ah (wh , v) for all v ∈ Vh

u (3.121)

with wh according to (3.105). As has been shown in (3.109), (3.121) is

equivalent, in the sense of the total approximation uh + wh of u + w, to the

¯

¯h ∈ Vh ,

variational equation for u

ah (uh , v) = ¯h (v) := bh (v) ’ ah (Ih (w), v) for all v ∈ Vh ,

¯

¯ b (3.122)

if this system of equations is uniquely solvable.

Exercises

ˆ

3.28 Prove equation (3.117) by ¬rst proving the equation for K = K

and then deducing from this the assertion for the general simplex by

Exercise 3.18.

3.6. Convergence Rate Results in Case of Quadrature and Interpolation 155

3.29 Let K be a triangle with vertices a1 , a2 , a3 . Further, let a12 , a13 , a23

denote the corresponding edge midpoints, a123 the barycenter and |K| the

area of K. Check that the quadrature formula

3

|K|

Qh (u) := 3 u(ai ) + 8 u(aij ) + 27u(a123 )

60 i=1 i<j

computes the integral Q(u) := u dx exactly for polynomials of third

K

degree.

3.6 Convergence Rate Results in the Case of

Quadrature and Interpolation

The purpose of this section is to analyze the approximation quality of a

¯

¯

solution uh + Ih (w) according to (3.122) and thus of uh + wh according to

¯

(3.121) of the boundary value problem (3.12), (3.18)“(3.20).

Hence, we have left the ¬eld of Galerkin methods, and we have to

investigate the in¬‚uence of the errors

a ’ ah , b ’ a(w, ·) ’ bh + ah (Ih (w), ·).

¯

To this end, we consider in general the variational equation in a normed

space (V, · )

u ∈ V satis¬es for all v ∈ V ,

a(u, v) = l(v) (3.123)

and the approximation in subspaces Vh ‚ V for h > 0,

uh ∈ Vh satis¬es for all v ∈ Vh .

ah (uh , v) = lh (v) (3.124)

Here a and ah are bilinear forms on V — V and Vh — Vh , respectively, and

l, lh are linear forms on V and Vh , respectively. Then we have the following

theorem

Theorem 3.38 (First Lemma of Strang)

Suppose there exists some ± > 0 such that for all h > 0 and v ∈ Vh ,

¤ ah (v, v) ,

2

±v (3.125)

and let a be continuous in V — V .

Then, there exists some constant C independent of Vh such that

|a(v, w) ’ ah (v, w)|

u ’ uh ¤ u ’ v + sup

C inf

w

v∈Vh w∈Vh

|l(w) ’ lh (w)|

+ sup .

w

w∈Vh

(3.126)

156 3. Finite Element Methods for Linear Elliptic Problems

Proof: Let v ∈ Vh be arbitrary. Then it follows from (3.123)“(3.125) that

± uh ’ v ¤ ah (uh ’ v, uh ’ v)

2

= a(u ’ v, uh ’ v) + a(v, uh ’ v) ’ ah (v, uh ’ v)

+ lh (uh ’ v) ’ l(uh ’ v)

and moreover, by the continuity of a (cf. (3.2)),

|a(v, w) ’ ah (v, w)|

± uh ’ v ¤ M u ’ v + sup

w

w∈Vh

|lh (w) ’ l(w)|

for v ∈ Vh .

+ sup

w

w∈Vh

By means of u ’ uh ¤ u ’ v + uh ’ v and taking the in¬mum over

all v ∈ Vh , the assertion follows. 2

For ah = a and lh = l the assertion reduces to C´a™s lemma (Theo-

e

rem 2.17), which was the initial point for the analysis of the convergence

rate in Section 3.4. Here we can proceed analogously. For that purpose, the

following conditions must be ful¬lled additionally:

• The uniform Vh -ellipticity of ah according to (3.125) must be ensured.

• For the consistency errors

|a(v, w) ’ ah (v, w)|

Ah (v) := sup (3.127)

w

w∈Vh

for an arbitrarily chosen comparison function v ∈ Vh and for

|l(w) ’ lh (w)|

sup

w

w∈Vh

the behavior in h must be analyzed.

The ¬rst requirement is not crucial if only a itself is V -elliptic and Ah

tends suitably to 0 for h ’ 0 :

Lemma 3.39 Suppose the bilinear form a is V -elliptic and there exists

some function C(h) with C(h) ’ 0 for h ’ 0 such that

Ah (v) ¤ C(h) v for v ∈ Vh .

¯

Then there exists some h > 0 such that ah is uniformly Vh -elliptic for

¯

h ¤ h.

Proof: By assumption, there exists some ± > 0 such that for v ∈ Vh ,

¤ ah (v, v) + a(v, v) ’ ah (v, v)

2

±v

and

|a(v, v) ’ ah (v, v)| ¤ Ah (v) v ¤ C(h) v 2

.

3.6. Convergence Rate Results in Case of Quadrature and Interpolation 157

¯ ¯

Therefore, for instance, choose h such that C(h) ¤ ±/2 for h ¤ h. 2

We concretely address the analysis of the in¬‚uence of numerical quadra-

ture, that is, ah is de¬ned as in (3.120) and lh corresponds to ¯h in (3.122)

b

with the approximate linear form bh according to (3.120). Since this is an

extension of the convergence results (in · 1 ) given in Section 3.4, the as-

sumptions about the ¬nite element discretization are as summarized there

at the beginning. In particular, the triangulations Th consist of elements

that are a¬ne equivalent to each other. Furthermore, for a simpli¬cation of

the notation, let again d ¤ 3 and only Lagrange elements are considered. In

particular, let the general assumptions about the boundary value problems

which are speci¬ed at the end of Section 3.2.1 be satis¬ed.

According to Theorem 3.38, the uniform Vh -ellipticity of ah must be

ensured and the consistency errors (for an appropriate comparison element

v ∈ Vh ) must have the correct convergence behavior. If the step size h is

small enough, the ¬rst proposition is implied by the second proposition

by virtue of Lemma 3.39. Now, simple criteria that are independent of this

restriction will be presented. The quadrature formulas satisfy the properties

(3.112), (3.113) introduced in Section 3.5; in particular, the weights are

positive.

Lemma 3.40 Suppose the coe¬cient function K satis¬es (3.16) and let

c = 0 in „¦, let |“3 |d’1 > 0, and let r ≥ 0 in „¦. If P ‚ Pk (K) for the

ansatz space and if the quadrature formula is exact for P2k’2 (K), then ah

is uniformly Vh -elliptic.

Proof: Let ± > 0 be the constant of the uniform positive de¬niteness of

K(x). Then we have for v ∈ Vh :

R

ah (v, v) ≥ ± ωl,K |∇v|2 (bl,K ) = ± |∇v|2 (x) dx = ±|v|2 ,

1

„¦

K∈Th l=1

since |∇v|2 ∈ P2k’2 (K). The assertion follows from Corollary 3.14. 2

K

Further results of this type can be found in [9, pp. 194]. To investigate

the consistency error we can proceed similarly to the estimation of the

interpolation error in Section 3.4: The error is split into the sum of the errors

over the elements K ∈ Th and there transformed by means of (3.115) into

ˆ

the error over the reference element K. The derivatives (in x) arising in the

ˆ

ˆ

error estimation over K are backtransformed by using Theorem 3.26 and

Theorem 3.27, which leads to the desired hK -factors. But note that powers

of B ’1 or similar terms do not arise. If the powers of det(B) arising in