Partial Differential Equations - Modelling and ... - ResearchGate

180 B. Maury
Proof. The interpolation operator I h : H 2 (T ) → L 2 (T ) is continuous, and
|u| 2,T
scales like h/ρ 2 T ≈ 1/h whereas the L2 -norms scale like h.
We may now complete the proof of Proposition 11. The problematic triangles
are those on which ũ h neither identifies to 0 nor to I h u. On such triangles,
ũ h sticks to I h u at 1 or 2 vertices, and vanishes at 2 or 1 vertices. As a consequence,
the L ∞ -norm of ũ h is less than the L ∞ -norm of I h u.LetT be such
a triangle. We write (using Lemma 2, the latter remark, the fact that I h is a
contraction from L ∞ onto L ∞ , Lemma 2 again, and Lemma 4),
|ũ h | 2 0,T ≤ C′ |T |‖ũ h ‖ 2 L ∞ (T ) ≤ C′ |T |‖I h u‖ 2 L ∞
( (T )
)
≤ C′
C ‖I hu‖ 2 L 2 (T ) ≤ C′′ |u| 2 0,T + h4 |u| 2 2,T
.
By summing up all these contributions over all triangles outside O which
intersect ω h (they are all contained in ω 2h ) and using the fact that the L 2 -norm
of u on ω h behaves like h 3/2 |u| 2,ωh
, we obtain
|ũ h | 2 0,ω h
≤ ∑
(
)
|ũ h | 2 0,T ≤ C |u| 2 0,ω 2h
+ h 4 |u| 2 2,ω 2h
≤ Ch 3 |u| 2 2,ω 2h
,
T ∩ω h ≠∅
which gives the expected h 3/2 -estimate for |u h | 0,ωh
. The last term of (15)
is handled straightforwardly: Thanks to Lemmas 2 and 3, which imply the
inverse inequality |ũ h | 1,ωh
≤ Ch −1 |ũ h | 0,ωh
, we obtain the h 1/2 bound for
|ũ h | 1,ωh
.
3.2 Primal/Dual Estimate
Proposition 5 asserts the convergence of λ ε towards λ, the Lagrange multiplier
associated to the constraint. One may wonder whether λ ε h = Buε h
/ε is likely
to approximate λ. In general, a positive answer to that question can be given
as soon as a uniform discrete inf-sup condition for B is fulfilled. This condition
is not verified in the situation we consider. The non-uniformity of the inf-sup
condition is due to the fact that there may exist triangles whose intersection
with O is very small. We propose here a way to overcome this problem by
suppressing those tiny areas in the penalty term, which leads us to introduce a
discrete version B h of B. Let us first give some properties for the continuous
Lagrange multiplier, and we shall give a precise description of the way the
obstacle is lifted.
Proposition 12 (Saddle-point formulation of (9)). Let u be the weak
solution to (8). There exists a unique λ ∈ Λ = L 2 (O) 2 such that λ is a
gradient, and
∫
∫ ∫
∇u ·∇v + λ ·∇v = fv ∀v ∈ V.
Ω
O
Ω
In addition, λ is in H 1 (O) 2 .