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))