Partial Differential Equations - Modelling and ... - ResearchGate

178 B. Maury
Cartesian triangulation T h of Ω and the associated first order Finite Element
space V h . There exists C>0 such that
inf
ũ h ∈V h
‖u − ũ h ‖ H1 (Ω) ≤ Ch1/2 .
Proof. We shall use in the proof the following notations: given a domain ω
and v a function over ω, wedenoteby|u| 0,ω
the L 2 -norm of v over ω, by
(∫ ) 1/2
|v| 1,ω
= |∇v| 2
ω
the H 1 -seminorm, etc...
We denote by I h is the standard interpolation operator from C(Ω) onto
V h . Notice that u is continuous over Ω (it is piecewise H 2 , and continuous
through the interface ∂O). Let us assume here that the constant value U
on O is 0 (which can be achieved by subtracting a smooth extension of this
constant outside O). We define ũ h as the function in V h which is 0 at every
vertex contained in a triangle which intersects O, and which identifies to I h u
at all other vertices. We introduce a narrow band around O
ω h = {x ∈ Ω | x/∈ O, d(x, O) < 2 √ 2h}.
As u| Ω\O
∈ H 2 (Ω \ O), the standard finite element estimate gives
|u − ũ h | 0,Ω\(O∪ωh ) ≤ Ch2 |u| 2,Ω\O
, (13)
|u − ũ h | 1,Ω\(O∪ωh ) ≤ Ch|u| 2,Ω\O . (14)
By construction, both L 2 -andH 1 -errors in O are zero. There remain to
estimate the error in the band ω h . The principle is the following: ũ h is a poor
approximation of u in ω h , but it is not very harmful because ω h is small.
Similar estimates are proposed in [SMSTT05] or [AR]. We shall give here a
proof more adapted to our situation. First of all, we write
‖u − ũ h ‖≤|u| 0,ωh
+ |u| 1,ωh
+ |u h | 0,ωh
+ |u h | 1,ωh
= A + B + C + D. (15)
We assume here that u is C 2 in Ω \ O (the general case h ∈ H 2 (Ω \ O) is
obtained immediately by density). Using polar coordinates (we assume here
that the radius of O is 1), we write
For i =1,2,onehas
|u| 2 1,ω h
=
∫ 2π ∫ 1+2h
0
∂ i u(r, θ) =∂ i u(1,θ)+
1
|∇u| 2 rdrdθ.
∫ r
1
∂ r ∂ i udr,