Partial Differential Equations - Modelling and ... - ResearchGate

8 V. Girault and M.F. Wheeler
In this case, we suppress from J 0 the boundary term on ∂Ω 2 \ Γ 12 :
J 0 (u, v) = σ ∫
∫
12
σ 1
[u] e [v] e dσ +
uv dσ, (9)
|Γ 12 | Γ 12
|∂Ω 1 \ Γ 12 | ∂Ω 1\Γ 12
and the IIPG, SIPG, NIPG and OBB-DG formulations become:
2∑
∫
∫
(
∇u ·∇vdx − (∇u · nΩ )v + ε(∇v · n Ω )u ) dσ
Ω i ∂Ω 1\Γ 12
∫
( )
− {∇u · ne } e [v] e + ε{∇v · n e } e [u] e dσ + J0 (u, v)
Γ
∫
12
∫
∫
= fvdx + g 2 vdσ− ε g 1 (∇v · n Ω ) dσ
Ω
∂Ω 2\Γ 12 ∂Ω 1\Γ 12
∫
σ 1
+
g 1 vdσ, (10)
|∂Ω 1 \ Γ 12 | ∂Ω 1\Γ 12
i=1
with the same values of ε, σ 1 and σ 12 as in (7).
2.2 The General Idea for the Stokes Problem
Consider the incompressible Stokes problem in Ω with data f in L 2 (Ω) 2 :
−µ∆u + ∇p = f, div u =0 inΩ, u = 0 on ∂Ω, (11)
where the viscosity parameter µ is a given positive constant. This is a typical
problem with a linear constraint (the zero divergence) and a Lagrange
multiplier (the pressure p).
For treating the pressure term and divergence constraint, we take again
a test function v that is not necessarily globally smooth, but has smooth
components in each Ω i , and assuming the pressure p is sufficiently smooth,
we apply Green’s formula in each Ω i :
∫
Ω
(∇p) · v dx =
(
2∑
∫
∫
)
− p div v dx + p(v · n Ω ) dσ
Ω i ∂Ω i\Γ 12
∫
+ {p} e [v] e · n e dσ. (12)
Γ 12
i=1
We apply the same formula to the divergence constraint. Thus combining (12)
with (7), we have the following IIPG, SIPG, NIPG and OBB-DG formulations
for the Stokes problem (11):