Partial Differential Equations - Modelling and ... - ResearchGate

Numerical Analysis of a Finite Element/Volume Penalty Method 169
we present the model problem, describe how we penalize and discretize it,
and we show how the abstract framework applies to this situation. We finish
by presenting an error estimate for the primal and dual components of the
solutions, in terms of the quantities ε (the penalty parameter) and h (the
mesh step size), whose proof is postponed to another paper.
2 Preliminaries, Abstract Framework
2.1 Continuous Problem
We recall here some standard properties concerning the penalty method applied
to infinite dimensional problems. Most of those properties are established
in [BF91], with a slightly different formalism. We shall consider the following
set of assumptions:
⎫
V is a Hilbert space, ϕ ∈ V ′ ,
a(·, ·) bilinear, symmetric, continuous, elliptic (a(v, v) ≥ α |v| 2 ),
b(·, ·) bilinear, symmetric, continuous, non-negative,
⎪⎬
K = {u ∈ V | b(u, u) =0} =kerb,
(1)
J(v) = 1 a(v, v) −〈ϕ, v〉,
2 u
=argmin
K J,
J ε (v) = 1 2 a(v, v)+ 1 b(v, v) −〈ϕ, v〉,
2ε uε
=argmin
V J ε.
⎪⎭
Proposition 1. Under the assumptions (1), the solution u ε to the penalized
problem converges to u.
Proof. We write the variational formulation for the penalized problem:
Taking v = u ε , we get
a(u ε ,v)+ 1 ε b(uε ,v)=〈ϕ, v〉 ∀v ∈ V. (2)
α |u ε | 2 ≤ a(u ε ,u ε ) ≤‖ϕ‖|u ε |
so that |u| ε is bounded. We extract a subsequence, still denoted by (u ε ), which
converges weakly to some z ∈ V .AsJ ε ≥ J and b(u, u) =0,wehave
J(u ε ) ≤ J ε (u ε ) ≤ J ε (u) =J(u) ∀ε >0, (3)
so that (J is convex and continuous) J(z) ≤ lim inf J(u ε ) ≤ J(u). As
J(u ε )+ 1 2ε b(uε ,u ε ) ≤ J(u),