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The powers of hK are due to the transformation steps.
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 .


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 .

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,

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

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