Partial Differential Equations - Modelling and ... - ResearchGate

174 B. Maury
Proof. By the definition of u ε ,
J ε (u ε )= 1 2 a(uε ,u ε ) −〈ϕ, u ε 〉 + 1 2ε b(uε ,u ε ) ≤ J ε (u) = 1 a(u, u) −〈ϕ, u〉,
2
so that
0 ≤ 1 2ε b(u ε,u ε ) ≤ 1 2 a(u, u) − 1 2 a(uε ,u ε )+〈ϕ, u ε − u〉
≤ 1 2 a(u + uε ,u− u ε )+〈ϕ, u ε − u〉,
which yields the estimate by continuity of a(·, ·) andϕ.
Proposition 6. Under the assumptions (1), we denote by u ε h
the solution to
the problem (6). Then
(
|u ε h − u| ≤C min |v h − u| + √ )
|u ε − u| .
v h ∈V h ∩K
Proof. As u ε h minimizes a(v − uε ,v− u ε )+b(v − u ε ,v− u ε )/ε over V h ,
α |u ε h − u ε | 2 ≤ a(u ε h − u ε ,u ε h − u ε )
≤ a(u ε h − u ε ,u ε h − u ε )+ 1 ε b(uε h − u ε ,u ε h − u ε )
(
≤ min a(v h − u ε ,v h − u ε )+ 1
v h ∈V h ε b(v h − u ε ,v h − u ε )
≤
min
v h ∈V h ∩K
(
a(v h − u ε ,v h − u ε )+ 1 ε b(v h − u ε ,v h − u ε )
As v h is in K, the second term is b(u ε ,u ε )/ε, which is bounded by C |u ε − u|
(by Lemma 1). Finally, we get
(
|u ε h − u ε |≤C min |v h − u ε | + √ )
|u ε − u| ,
v h ∈V h ∩K
from which we conclude.
Proposition 7. Under the assumptions (1), it holds
|u ε h − u| ≤ C √ ε
inf
v h ∈V h
|u ε − v h | + |u ε − u| ,
)
)
.
where u ε h
is the solution to (6).
Proof. One has
|u ε h − u| ≤|u ε h − u ε | + |u ε − u| ,
and we control the first term by Céa’s lemma applied to the bilinear form
a + b/ε, whose ellipticity constant behaves like 1/ε.