Partial Differential Equations - Modelling and ... - ResearchGate

16 V. Girault and M.F. Wheeler
For a given force f ∈ L 2 (Ω) d , this problem has a unique solution u ∈ H 1 0 (Ω) d
and p ∈ L 2 0(Ω) (cf., for instance, [Tem79, GR86]). In fact, the solution is more
regular and the scheme below is consistent (cf. [Gri85, Dau89]).
In view of the operator and boundary condition in (11), the relevant spaces
here are H 1 (E h ) d and L 2 0(Ω), and the set Γ h,N is empty. The definition of J 0 is
extended straightforwardly to vectors and the permeability tensor is replaced
by the identity multiplied by the viscosity. Thus, the semi-norms (28) and
(30) are replaced by
|||∇v||| L2 (E h ) =
[|v|] H 1 (E h ) = µ 1 2
[ ∑
E∈E h
‖∇v‖ 2 L 2 (E)
] 1
2
, (41)
(
) 1
|||∇v||| 2 L 2 (E h ) + J 2
0(v, v) . (42)
Again, we choose an integer k ≥ 1 and we discretize H 1 (E h ) d and L 2 0(Ω)
with the finite element spaces
X h = {v ∈ L 2 (Ω) d : ∀E ∈E h , v| E ∈ P k (E) d }, (43)
M h = {q ∈ L 2 0(Ω) :∀E ∈E h , q| E ∈ P k−1 (E)}. (44)
The choice P k−1 for the discrete pressure, one degree less than the velocity, is
suggested by the fact that L 2 is the natural norm for the pressure. Keeping in
mind (13) and (14), we discretize (11) by the following discrete system: Find
u h ∈ X h and p h ∈ M h satisfying for all v h ∈ X h and q h ∈ M h :
µ ∑
∇u h : ∇v h dx
E∈E h
∫E
− µ
− ∑
∑
e∈Γ h ∪∂Ω
E∈E h
∫E
∑
∫
( )
{∇uh · n e } e [v h ] e + ε{∇v h · n e } e [u h ] e dσ + µJ0 (u h , v h )
e
p h div v h dx +
E∈E h
∫E
∑
e∈Γ h ∪∂Ω
q h div u h dx −
∫
∫
{p h } e [v h ] e · n e dσ =
e
∑
e∈Γ h ∪∂Ω
Ω
f · v h dx,
(45)
∫
{q h } e [u h ] e · n e dσ =0, (46)
with the interpretation for the parameters ε and σ of formula (7).
e