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106 3. Finite Element Methods for Linear Elliptic Problems

Indeed, using Theorem 3.13, we have

2 2 2 2 2
a(u, u) ≥ k 0 |u| + α 0 u dσ ≥ min{k 0 ,α 0 } |u| + u dσ ≥ C 5 u
1 1 1
˜ Γ ˜ Γ
with C 5 := C −2 min{k 0 ,α 0 }. Therefore, we obtain the existence and
F
uniqueness of a solution analogously to Theorem 3.12.
(III) General Case
First, we consider the case of a homogeneous Dirichlet boundary
condition on Γ 3 with |Γ 3 | d−1 > 0. For this, we define
1
V := v ∈ H (Ω) : γ 0 (v)=0 on Γ 3 . (3.30)
1
Here V is a closed subspace of H (Ω), since the trace mapping γ 0 :
2
1
2
2
H (Ω) → L (∂Ω) and the restriction of a function from L (∂Ω) to L (Γ 3 )
are continuous.
Suppose u is a solution of (3.12), (3.18)–(3.20); that is, in the sense
1 ¯
2
of classical solutions let u ∈ C (Ω) ∩ C (Ω) and the differential equation
(3.12) be satisfied pointwise in Ω and the boundary conditions (3.18)–
(3.20) pointwise on their respective parts of ∂Ω under the assumptions
2
(3.13), (3.21). However, the weaker case that u ∈ H (Ω) and the differential
2
equation is satisfied in the sense of L (Ω) and the boundary conditions
2
(3.18)–(3.20) are satisfied in the sense of L (Γ j ),j =1, 2, 3, under the
assumptions (3.14), (3.22) can also be considered here.
As in (I), according to (3.11),

a(u, v):= {K∇u ·∇v + c ·∇uv + ruv} dx + αuv dσ (3.31)
Ω Γ 2

= b(v):= fv dx + g 1 vdσ + g 2 vdσ for all v ∈ V.
Ω Γ 1 Γ 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 verification 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 2 1 2
c ·∇uudx = − ∇· cu dx + ν · cu dσ ,
Ω 2 Ω 2 Γ 1 ∪Γ 2
and therefore

1 2
a(u, u)= K∇u ·∇u + r − ∇· c u dx
Ω 2

1 2 1 2
+ ν · cu dσ + α + ν · c u dσ .
2 Γ 1 Γ 2 2