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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))

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