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104 3. Finite Element Methods for Linear Elliptic Problems

Taking into account (3.16) and using the simple decomposition k 0 =
k 0 C P 2
+ k 0 we can further conclude that
1+ C 2 1+ C 2
P P

2
a(u, u) ≥ k 0 |∇u| dx (3.26)
Ω
2
k 0 2 C P 1 2 2
≥ 2 |∇u| dx + 2 k 0 2 |u| dx = C 4 u ,
1
1+ C 1+ C C
P Ω P P Ω
k 0
where C 4 := > 0 .
1+ C 2
P
For this estimate it is essential that u satisfies the homogeneous Dirichlet
boundary condition.

(B) |c| > 0:
∞ 1
First of all, we consider a smooth function u ∈ C (Ω). From u∇u = ∇u 2
∞
0 2
we get by integrating by parts

1 2 1 2
c ·∇uudx = c ·∇u dx = − ∇· cu dx .
Ω 2 Ω 2 Ω
Since according to Theorem 3.7 the space C (Ω) is dense in V ,the above
∞
0
relation also holds for u ∈ V . Consequently, by virtue of (3.16) and (3.17)
we obtain

1 2
a(u, u)= K∇u ·∇u + r − ∇· c u dx
Ω 2
(3.27)
2
2
≥ {k 0 |∇u| + r 0 |u| } dx for all u ∈ V.
Ω
Hence, a distinction concerning r 0 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):
d
Theorem 3.12 Suppose Ω ⊂ R is a bounded Lipschitz domain. Under
the assumptions (3.15)–(3.17) the homogeneous Dirichlet problem has one
1
and only one weak solution u ∈ H (Ω).
0
(II) Mixed Boundary Conditions
1
∂Ω= Γ 2 ,V = H (Ω)
Suppose u is a solution of (3.12), (3.19); that is, in the classical sense
1 ¯
2
let u ∈ C (Ω) ∩ C (Ω) and the differential equation (3.12) be satisfied
pointwise in Ω and (3.19) pointwise on ∂Ω under the assumptions (3.13),
(3.21). However, the weaker case can again be considered, now under the
2
assumptions (3.14), (3.22), that u ∈ H (Ω) and the differential equation is
2
satisfied in the sense of L (Ω) as well as the boundary condition (3.19) in
2
the sense of L (∂Ω).