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Numerical Analysis of a Finite Element/Volume Penalty Method 173
Proposition 5. Under the assumptions of Proposition 4 (the assumptions (1)
and B(V ) is closed), we introduce
λ ε = 1 ε Buε .
Then |λ ε − λ| = O(ε), where λ is the unique solution of (5) in B(V ).
Proof. Subtracting the variational formulations for u and u ε , we get
(λ ε − λ, Bv) =a(u ε − u, v) ∀v ∈ V.
Now, as the range of B is closed, and λ ε − λ ∈ B(V )=(kerB ⋆ ) ⊥ ,wehave
the inf-sup condition (see, e.g., [BF91])
so that
sup
v∈V
(λ ε − λ, Bv)
|v|
β |λ ε − λ| ≤sup (λε − λ, Bv)
|v|
≥ β |λ ε − λ| ,
=sup a(uε − u, v)
|v|
≤‖a‖|u ε − u| ,
which ensures the first order convergence thanks to Corollary 1.
2.2 Discretized Problem
We consider now a family (V h ) h of inner approximation spaces (V h ⊂ V )and
the associated penalized/discretized problems
⎧
⎪⎨ Find u ε h ∈ V h such that Jh(u ε ε h) = inf J h(v),
ε
v∈V
⎪⎩ Jh(v ε h )= 1 2 a(v h,v h )+ 1 (6)
2ε b(v h,v h ) −〈ϕ, v h 〉.
As far as we know, there does not exist any general theory which would
give an upper bound for the error |u − u ε h
| as the sum of a discretization error
(typically h of h 1/2 for volume penalty, depending on whether the mesh
is boundary-fitted or not), and a penalty error (typically ε for closed-range
penalty terms). We propose here two general properties which are direct consequences
of standard arguments. They are suboptimal in the sense that neither
of them is optimal from both standpoints (discretization and penalty), but,
at least, they make it possible to recover the behavior in extreme situations
(when ε goes to 0 much quicker than h, and the opposite situation).
We shall need the following lemma:
Lemma 1. Under the assumptions (1), there exists C>0 such that
b(u ε ,u ε ) ≤ Cε|u − u ε | .