Partial Differential Equations - Modelling and ... - ResearchGate

so that
|u| 2 1,ω h
≤ 2
Numerical Analysis of a Finite Element/Volume Penalty Method 179
∫ 2π ∫ 1+2h
0
∫
≤ Ch
∂O
∫ 2π ∫ 1+2h
∫
|∂ i u(1,θ)| 2 r
2
rdrdθ+2
∣ ∂ r ∂ i uds
∣ rdrdθ
1
0 1 1
2 ∫ ∂u
2π ∫ (
1+2h ∫ )
1+2h
∣∂n∣
+2h
|∂ r ∂ i u| 2 ds rdrdθ
≤ Ch|u| 2 2,Ω\O + C′ h 2 |u| 2 2,Ω\O ,
0
1
from which we deduce |u| 1,ωh
≤ Ch 1/2 . A similar computation on u gives
immediately |u| 0,ωh
≤ Ch 3/2 .Asforũ h (the two last terms in (15)), the proof
is less trivial. It relies on three technical lemmas which we give now before
ending the proof.
Lemma 2. There exist constants C and C ′ such that, for any non-degenerated
triangle T , for any function w h affine in T ,
1
C |T |‖w h ‖ 2 L ∞ (T ) ≤‖w h‖ 2 L 2 (T ) ≤ C′ |T |‖w h ‖ 2 L ∞ (T ) . (16)
Proof. It is a consequence of the fact that, when deforming the supporting
triangle T ,theL ∞ -norm is unchanged whereas the L 2 -norm scales like |T | 1/2 .
Lemma 3. There exists a constant C such that, for any non-degenerated triangle
T , for any function w h affine in T ,
|w h | 2 1,T ≤ C |T |
ρ 2 ‖w h ‖ 2 L ∞ (T ) ,
T
where ρ T is the diameter of the inscribed circle.
Proof. Again, it is a straightforward consequence of the fact that, when
deforming the supporting triangle T , L ∞ -norm is unchanged whereas the
gradient (which is constant over the triangle) scales like 1/ρ T , so that the
H 1 -seminorm scales like |T | /ρ T .
The last lemma quantifies how one can control the L 2 -normoftheinterpolate
of a regular function on a triangle, by means of the L 2 -norm and the
H 2 -seminorm of the function.
Lemma 4. There exists a constant C such that, for any regular triangle T
(see below), for any u ∈ H 2 (T ),
)
|I h u| 2 0,T
(|u| ≤ C 2 0,T + h4 |u| 2 2,T
.
By regular, we mean that T runs over a set of triangles such that the flatness
diam T/ρ T is bounded.