Partial Differential Equations - Modelling and ... - ResearchGate

176 B. Maury
Proposition 8. The problem (8) admits a unique weak solution u ∈ V =
H0 1 (Ω), which is characterized as the solution to the minimization problem
⎧
Find u ∈ K such that
⎪⎨
∫
J(u) = inf J(v), with J(v) =1
v∈K 2 Ω
⎪⎩
K = {v ∈ H0 1 (Ω) |∇v =0a.e. in O},
∫
|∇u| 2 − fv
Ω
where f has been extended by 0 inside O.
Furthermore, the restriction of u to Ω \ O is in H 2 (Ω \ O).
Proof. Existence and uniqueness are direct consequences of the Lax–Milgram
theorem applied in K = {v ∈ V |∇v =0a.e.inO}, which gives, in addition,
the characterization of u as the solution to (9). Now the restriction of u to
Ω \ O satisfies −△u = f, with regular Dirichlet boundary conditions on the
boundary of Ω \ O which decomposes as ∂O∪∂Ω. AsΩ is a convex polygon
and ∂O is smooth, standard theory ensures u| Ω\O
∈ H 2 (Ω \ O).
(9)
We introduce the penalized version of the problem (9)
⎧
⎪⎨
Find u ε ∈ V such that J ε (u ε ) = inf J ε (v),
v∈V
⎪⎩ J ε (v) = 1 ∫
|∇v| 2 + 1 ∫ ∫
|∇v| 2 − fv,
2
2ε
for which linear convergence can be expected:
Ω
O
Ω
(10)
Proposition 9. Let u be the solution to the problem (9), u ε the solution to
the problem (10). It holds ‖u − u ε ‖ H1 (Ω) = O(ε).
Proof. Let us show that
B : v ∈ H 1 0 (Ω) ↦−→ ∇v ∈ L 2 (O) 2
has a closed range. Consider µ ∈ Λ with µ = ∇v. We define w ∈ H 1 0 (O) as
w = v − m(v), where m(v) isthemeanvalueofv over O. BythePoincaré–
Wirtinger inequality, one has
‖w‖ H1 (O) ≤ C ‖µ‖ L 2 (O) 2 .
Now, as O⊂⊂Ω, there exists a continuous extension operator from H 1 (O)
to H0 1 (Ω), so that we can extend w to obtain ˜w ∈ H0 1 (Ω) with a norm controlled
by ‖µ‖ L 2 (O) 2, which proves the closed character of B(V ). The linear
convergence is then given by Corollary 1.