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98 3. Finite Element Methods for Linear Elliptic Problems

1
For v, w ∈ H (Ω) and i =1,... ,d,

∂ i vwdx = − v∂ i wdx + vw ν i dσ .
Ω Ω ∂Ω
Proof: See, forexample, [14] or[37].

2
2
If v ∈ H (Ω), then due to the above theorem, v| ∂Ω := γ 0 (v) ∈ L (∂Ω)
1
2
and ∂ i v| ∂Ω := γ 0 (∂ i v) ∈ L (∂Ω), since also ∂ i v ∈ H (Ω). Hence, the normal
derivative
d

∂ ν v| ∂Ω := ∂ i v| ∂Ω ν i
i=1
2
is well-defined and belongs to L (∂Ω).
Thus, the trace mapping
2
2
2
γ : H (Ω) → L (∂Ω) × L (∂Ω) ,
v → (v| ∂Ω ,∂ ν v| ∂Ω ) ,
is well-defined and continuous. The continuity of this mapping follows from
the fact that it is a composition of continuous mappings:

2
2
1
v ∈ H (Ω) continuous ∂ i v ∈ H (Ω) continuous ∂ i v| ∂Ω ∈ L (∂Ω)
→
→
continuous 2
→ ∂ i v| ∂Ω ν i ∈ L (∂Ω) .
d
Corollary 3.9 Suppose Ω ⊂ R is a bounded Lipschitz domain.
1
1
(1) Let w ∈ H (Ω), q i ∈ H (Ω), i =1,... ,d.Then

q ·∇wdx = − ∇· qw dx + q · νw dσ . (3.10)
Ω Ω ∂Ω
1
2
(2) Let v ∈ H (Ω), w ∈ H (Ω).Then

∇v ·∇wdx = − ∆vwdx + ∂ ν vw dσ .
Ω Ω ∂Ω
The integration by parts formulas also hold more generally if only it is
1
ensured that the function whose trace has to be formed belongs to H (Ω).
1 2
For example, if K =(k ij ) ,where k ij ∈ W (Ω) and v ∈ H (Ω), w ∈
ij ∞
1
H (Ω), it follows that

K∇v ·∇wdx = − ∇· (K∇v) wdx + K∇v · νw dσ (3.11)
Ω Ω ∂Ω
with conormal derivative (see (0.41))
d

T
v := K∇v · ν = ∇v · K ν = k ij ∂ j vν i .
∂ ν K
i,j=1