Partial Differential Equations - Modelling and ... - ResearchGate

182 B. Maury
⎧
⎪⎨
Find u ε h ∈ V h such that Jh(u ε ε ) = inf Jh(v ε h ),
v h ∈V h
⎪⎩ Jh(v ε h )= 1 ∫
|∇v h | 2 + 1 ∫
∫
|∇v h | 2 − fv h .
2 Ω 2ε O h Ω
(18)
We may now state the primal/dual estimate.
Proposition 13 (Primal/dual error estimate for (8)). Let u be the weak
solution to (8), u ε h
the solution to (11), λ the Lagrange multiplier (see Proposition
12), and λ ε h = B hu ε h
/ε (see Definition 3). We have the following error
estimate:
|u − u ε h| + |λ − λ ε h|≤C(h 1/2 + ε 1/2 ). (19)
Proof. The proof of this estimate is quite technical (in particular, the discrete
inf-sup condition, see below), and we shall detail it on a forthcoming paper.
Let us simply say here that it relies on the following ingredients:
1. some general properties of the continuous penalty method which we established
in the beginning of this section,
2. an abstract stability estimate for saddle point-like problems with stabilization,
in the spirit of Theorem 1.2 in [BF91],
3. a uniform discrete inf-sup condition for B h :
(B h v h , λ h )
sup
≥ β ‖λ h ‖
v h ∈V h
|v h |
Λh
, (20)
4. some approximation properties for V h (Proposition 11 and a similar property
for the Lagrange multiplier).
Remark 2 (Optimal estimate, role of η in the definition of O h ). The estimate
we establish is still suboptimal in ε: the order 1/2 is obtained, whereas the
continuous method converges linearly. It is due to the fact that we had to
introduce a discrete operator B h , and the difference leads to an extra term
which scales like ε 1/2 . It calls for some comments on the parameter η which
appears in the definitions of O h and B h (see Definitions 2 and 3). The smaller
η is, the closer B h approaches B, which reduces the ε 1/2 term in the estimate.
This observation may suggest to have η go to zero in the theoretical
estimate. But, on the other hand, when η goes to 0, so does the inf-sup constant
β (see (20)), so that 1/β, which is hidden in the constant C in the error
estimate (19), blows up.
Remark 3 (Boundary fitted meshes). Although it is somewhat in contradiction
with its original purpose, the penalty method can be used together with
a discretization based on a boundary fitted mesh. In that case, the approximation
error behaves no longer like h 1/2 , but like h. More important, it is not
necessary to get rid of the tiny triangles which were incompatible, in case of a
Cartesian mesh, with the uniform discrete inf-sup condition. Now considering
that the half order in ε was lost because of the fact we introduced a reduced
obstacle, one can expect to recover the optimal order of convergence, both in
h and in ε.