solutions let u ∈ C 2 („¦) © C(„¦) and the di¬erential equation (3.12) be

¯

satis¬ed pointwise in „¦ under the assumptions (3.13) as well as u = 0

pointwise on ‚„¦. However, the weaker case in which u ∈ H 2 („¦) © V and

the di¬erential equation is satis¬ed in the sense of L2 („¦), now under the

assumptions (3.14), can also be considered.

∞

Multiplying (3.12) by v ∈ C0 („¦) (in the classical case) or by v ∈ V ,

respectively, then integrating by parts according to (3.11) and taking into

∞ 1

account that v = 0 on ‚„¦ by virtue of the de¬nition of C0 („¦) and H0 („¦),

respectively, we obtain

{K∇u · ∇v + c · ∇u v + r uv} dx

a(u, v) := (3.23)

„¦

∞

for all v ∈ C0 („¦) or v ∈ V .

= b(v) := f v dx

„¦

The bilinear form a is symmetric if c vanishes (almost everywhere).

For f ∈ L2 („¦),

b is continuous on (V, · 1) . (3.24)

This follows directly from the Cauchy“Schwarz inequality, since

|b(v)| ¤ |f | |v| dx ¤ f ¤f for v ∈ V .

v v

0 0 0 1

„¦

Further, by (3.15),

a is continuous (V, · 1) . (3.25)

Proof: First, we obtain

|a(u, v)| ¤ {|K∇u| |∇v| + |c| |∇u||v| + |r| |u| |v|} dx .

„¦

Here | · | denotes the absolute value of a real number or the Euclidean

norm of a vector. Using also · 2 for the (associated) spectral norm, and

· ∞ for the L∞ („¦) norm of a function, we further introduce the following

notations:

< ∞, C2 := |c| < ∞.

C1 := max K 2 ∞, r ∞ ∞

By virtue of

|K(x)∇u(x)| ¤ K(x) |∇u(x)|,

2

3.2. Elliptic Boundary Value Problems 103

we continue to estimate as follows:

|a(u, v)| ¤ C1 {|∇u| |∇v| + |u| |v|} dx + C2 |∇u| |v| dx .

„¦ „¦

=:A1 =:A2

The integrand of the ¬rst addend is estimated by the Cauchy“Schwarz

inequality for R2 , and then the Cauchy“Schwarz inequality for L2 („¦) is

applied:

1/2 1/2

A1 ¤ C1 |∇u|2 + |u|2 |∇v|2 + |v|2 dx

„¦

1/2 1/2

¤ C1 |u| + |∇u| dx |v| + |∇v| dx

2 2 2 2

= C1 u v 1.

1

„¦ „¦

Dealing with A2 , we can employ the Cauchy“Schwarz inequality for L2 („¦)

directly:

1/2 1/2

¤ |∇u| dx |v| dx

2 2

A2 C2

„¦ „¦

¤ ¤ C2 u for all u, v ∈ V .

C2 u v v

1 0 1 1

2

Thus, the assertion follows.

Remark 3.11 In the proof of the propositions (3.24) and (3.25) it has not

been used that the functions u, v satisfy homogeneous Dirichlet boundary

conditions. Therefore, under the assumptions (3.15) these properties hold

for every subspace V ‚ H 1 („¦).

Conditions for the V -Ellipticity of a

(A) a is symmetric; that is c = 0 (a.e.): Condition (3.17) then has the

simple form r(x) ≥ r0 for almost all x ∈ „¦.

(A1) c = 0, r0 > 0:

Because of (3.16) we directly get

a(u, u) ≥ {k0 |∇u|2 + r0 |u|2 } dx ≥ C3 u for all u ∈ V ,

2

1

„¦

where C3 := min{k0 , r0 }. This also holds for every subspace V ‚ H 1 („¦).

(A2) c = 0, r0 ≥ 0:

According to the Poincar´ inequality (Theorem 2.18), there exists some

e

constant CP > 0, independent of u, such that for u ∈ H0 („¦)

1

1/2

¤ CP |∇u| dx 2

u .

0

„¦

104 3. Finite Element Methods for Linear Elliptic Problems

Taking into account (3.16) and using the simple decomposition k0 =

2

k0 CP

2 + 1 + C 2 k0 we can further conclude that

1 + CP P

a(u, u) ≥ k0 |∇u|2 dx (3.26)

„¦

2

k0 CP 1

≥ |∇u| dx + |u|2 dx = C4 u

2 2

k0 2 1,

2 2

1 + CP 1 + CP CP

„¦ „¦

k0

where C4 := 2 > 0.

1 + CP

For this estimate it is essential that u satis¬es the homogeneous Dirichlet

boundary condition.

(B) |c| ∞ > 0 :

∞

First of all, we consider a smooth function u ∈ C0 („¦). From u∇u = 1 ∇u2

2

we get by integrating by parts

1 1

c · ∇u u dx = c · ∇u2 dx = ’ ∇ · c u2 dx .

2 2

„¦ „¦ „¦

∞

Since according to Theorem 3.7 the space C0 („¦) is dense in V , the above

relation also holds for u ∈ V . Consequently, by virtue of (3.16) and (3.17)

we obtain

1

K∇u · ∇u + r ’ ∇ · c u2 dx

a(u, u) =

2

„¦

(3.27)

≥ {k0 |∇u| + r0 |u| } dx for all u ∈ V .

2 2

„¦

Hence, a distinction concerning r0 as in (A) with the same results

(constants) is possible.

Summarizing, we have therefore proven the following application of the

Lax“Milgram Theorem (Theorem 3.1):

Theorem 3.12 Suppose „¦ ‚ Rd is a bounded Lipschitz domain. Under

the assumptions (3.15)“(3.17) the homogeneous Dirichlet problem has one

and only one weak solution u ∈ H0 („¦).

1

(II) Mixed Boundary Conditions

‚„¦ = “2 , V = H 1 („¦)

Suppose u is a solution of (3.12), (3.19); that is, in the classical sense

let u ∈ C 2 („¦) © C 1 („¦) and the di¬erential equation (3.12) be satis¬ed

¯

pointwise in „¦ and (3.19) pointwise on ‚„¦ under the assumptions (3.13),

(3.21). However, the weaker case can again be considered, now under the

assumptions (3.14), (3.22), that u ∈ H 2 („¦) and the di¬erential equation is

satis¬ed in the sense of L2 („¦) as well as the boundary condition (3.19) in

the sense of L2 (‚„¦).

3.2. Elliptic Boundary Value Problems 105

As in (I), according to (3.11),

{K∇u · ∇v + c · ∇u v + r uv} dx +

a(u, v) := ± uv dσ (3.28)

„¦ ‚„¦

for all v ∈ V .

= b(v) := f v dx + g2 v dσ

„¦ ‚„¦

Under the assumptions (3.15), (3.22) the continuity of b and a, respec-

tively, ((3.24) and (3.25)) can easily be shown. The additional new terms

can be estimated, for instance under the assumptions (3.15), (3.22), by

the Cauchy“Schwarz inequality and the Trace Theorem (Theorem 3.4) as

follows:

g2 v dσ ¤ g2 ¤ C g2 for all v ∈ V

v|‚„¦ v

0,‚„¦ 0,‚„¦ 0,‚„¦ 1

‚„¦

and

±uv dσ ¤ ± ¤ C2 ±

u|‚„¦ v|‚„¦ u v ,

∞,‚„¦ ∞,‚„¦

0,‚„¦ 0,‚„¦ 1 1

‚„¦

respectively, for all u, v ∈ V, where C > 0 denotes the constant appearing

in the Trace Theorem.

Conditions for the V -Ellipticity of a

For the proof of the V -ellipticity we proceed similarly to (I)(B), but now

taking into account the mixed boundary conditions. For the convective

term we have

1 1 1

c · ∇u u dx = c · ∇u2 dx = ’ ∇ · c u2 dx + ν · c u2 dσ ,

2„¦ 2„¦ 2 ‚„¦

„¦

and thus

1 1

K∇u · ∇u + r ’ ∇ · c u2 dx+ ± + ν · c u2 dσ.

a(u, u) =

2 2

„¦ ‚„¦

This shows that ± + 1 ν · c ≥ 0 on ‚„¦ should additionally be assumed. If

2

r0 > 0 in (3.17), then the V -ellipticity of a follows directly. However, if only

r0 ≥ 0 is valid, then the so-called Friedrichs™ inequality, a re¬ned version

of the Poincar´ inequality, helps (see [25, Theorem 1.9]).

e

Theorem 3.13 Suppose „¦ ‚ Rd is a bounded Lipschitz domain and let

the set “ ‚ ‚„¦ have a positive (d ’ 1)-dimensional measure. Then there

˜

exists some constant CF > 0 such that for all v ∈ H 1 („¦),

1/2

¤ CF |∇v| dx

2 2

v v dσ + . (3.29)

1

˜ „¦

“

If ± + 1 ν · c ≥ ±0 > 0 for x ∈ “ ‚ “2 and “ has a positive (d ’ 1)-

˜ ˜

2

dimensional measure, then r0 ≥ 0 is already su¬cient for the V -ellipticity.

106 3. Finite Element Methods for Linear Elliptic Problems

Indeed, using Theorem 3.13, we have

a(u, u) ≥ k0 |u|2 + ±0 u2 dσ ≥ min{k0 , ±0 } |u|2 + ≥ C5 u

u2 dσ 2

1 1 1

˜ ˜

“ “

’2

with C5 := CF min{k0 , ±0 }. Therefore, we obtain the existence and

uniqueness of a solution analogously to Theorem 3.12.

(III) General Case

First, we consider the case of a homogeneous Dirichlet boundary

condition on “3 with |“3 |d’1 > 0. For this, we de¬ne

V := v ∈ H 1 („¦) : γ0 (v) = 0 on “3 . (3.30)

Here V is a closed subspace of H 1 („¦), since the trace mapping γ0 :

H 1 („¦) ’ L2 (‚„¦) and the restriction of a function from L2 (‚„¦) to L2 (“3 )

are continuous.

Suppose u is a solution of (3.12), (3.18)“(3.20); that is, in the sense

of classical solutions let u ∈ C 2 („¦) © C 1 („¦) and the di¬erential equation

¯

(3.12) be satis¬ed pointwise in „¦ and the boundary conditions (3.18)“

(3.20) pointwise on their respective parts of ‚„¦ under the assumptions

(3.13), (3.21). However, the weaker case that u ∈ H 2 („¦) and the di¬erential

equation is satis¬ed in the sense of L2 („¦) and the boundary conditions

(3.18)“(3.20) are satis¬ed in the sense of L2 (“j ), j = 1, 2, 3, under the

assumptions (3.14), (3.22) can also be considered here.

As in (I), according to (3.11),

{K∇u · ∇v + c · ∇u v + r uv} dx +

a(u, v) := ± uv dσ (3.31)

„¦ “2

for all v ∈ V .

= b(v) := f v dx + g1 v dσ + g2 v dσ

„¦ “1 “2

Under the assumptions (3.15), (3.22) the continuity of a and b, (3.25)) and

((3.24) can be proven analogously to (II).

Conditions for V -Ellipticity of a

For the veri¬cation of the V -ellipticity we again proceed similarly to (II),

but now the boundary conditions are more complicated. Here we have for

the convective term

1 1

c · ∇u u dx = ’ ∇ · c u2 dx + ν · cu2 dσ ,

2 2 “1 ∪“2

„¦ „¦

and therefore

1

K∇u · ∇u + r ’ ∇ · c u2 dx

a(u, u) =

2

„¦

1 1

ν · c u2 dσ + ± + ν · c u2 dσ .

+

2 “1 2

“2

3.2. Elliptic Boundary Value Problems 107

In order to ensure the V -ellipticity of a we need, besides the obvious

conditions

1

ν · c ≥ 0 on “1 and ± + ν · c ≥ 0 on “2 , (3.32)

2

the following corollary from Theorem 3.13.

Corollary 3.14 Suppose „¦ ‚ Rd is a bounded Lipschitz domain and “ ‚

˜

‚„¦ has a positive (d ’ 1)-dimensional measure. Then there exists some

constant CF > 0 such that for all v ∈ H 1 („¦) with v|“ = 0,

˜

1/2

¤ CF |∇v| dx = CF |v|1 .

2

v 0

„¦

This corollary yields the same results as in the case of homogeneous

Dirichlet boundary conditions on the whole of ‚„¦.

If |“3 |d’1 = 0, then by tightening conditions (3.32) for c and ±, the

application of Theorem 3.13 as done in (II) may be successful.

Summary

We will now present a summary of our considerations for the case of

homogeneous Dirichlet boundary conditions.

Theorem 3.15 Suppose „¦ ‚ Rd is a bounded Lipschitz domain. Under the

assumptions (3.15), (3.16), (3.22) with g3 = 0, the boundary value problem

(3.12), (3.18)“(3.20) has one and only one weak solution u ∈ V , if

(1) r ’ 1 ∇ · c ≥ 0 in „¦ .

2

(2) ν · c ≥ 0 on “1 .

(3) ± + 1 ν · c ≥ 0 on “2 .

2

(4) Additionally, one of the following conditions is satis¬ed:

(a) |“3 |d’1 > 0 .

(b) There exists some „¦ ‚ „¦ with |„¦|d > 0 and r0 > 0 such that

˜ ˜

r ’ 1 ∇ · c ≥ r0 on „¦.

˜

2

(c) There exists some “1 ‚ “1 with |“1 |d’1 > 0 and c0 > 0 such

˜ ˜

that ν · c ≥ c0 on “1 .

˜

(d) There exists some “2 ‚ “2 with |“2 |d’1 > 0 and ±0 > 0 such

˜ ˜

that ± + 1 ν · c ≥ ±0 on “2 .

˜

2

Remark 3.16 We point out that by using di¬erent techniques in the proof,

it is possible to weaken conditions (4)(b)“(d) in such a way that only the

following has to be assumed:

x ∈ „¦ : r ’ 1∇ · c > 0

(b) > 0,

2 d

{x ∈ “1 : ν · c > 0}

(c) > 0,

d’1

x ∈ “2 : ± + 1 ν · c > 0

(d ) > 0.

2 d’1

108 3. Finite Element Methods for Linear Elliptic Problems

However, we stress that the conditions of Theorem 3.15 are only su¬-

cient, since concerning the V -ellipticity, it might also be possible to balance

an inde¬nite addend by some “particular de¬nite” addend. But this would

require conditions in which the constants CP and CF are involved.

Note that the pure Neumann problem for the Poisson equation

’∆u =f in „¦ ,

(3.33)

‚ν u =g on ‚„¦

is excluded by the conditions of Theorem 3.15. This is consistent with the

fact that not always a solution of (3.33) exists, and if a solution exists, it

obviously is not unique (see Exercise 3.8).

Before we investigate inhomogeneous Dirichlet boundary conditions, the

application of the theorem will be illustrated by an example of a natural

situation described in Chapter 0.

For the linear stationary case of the di¬erential equation (0.33) in the

form

∇ · (c u ’ K∇u) + r u = f

˜

we obtain, by di¬erentiating and rearranging the convective term,

’∇ · (K∇u) + c · ∇u + (∇ · c + r ) u = f ,

˜

which gives the form (3.12) with r := ∇ · c + r . The boundary ‚„¦ consists

˜

only of two parts “1 and “2 . Therein, “1 an out¬‚ow boundary and “2 an