Partial Differential Equations - Modelling and ... - ResearchGate

10 V. Girault and M.F. Wheeler
Now, we wish to extend this definition to functions u and ϕ that are not
necessarily smooth. Then, we take again a test function v that is sufficiently
smooth in each Ω i , but may not be in H 1 (Ω). Applying Green’s formula to the
last equality in (18) in each Ω i and using the fact that u has zero divergence,
we define:
∫
2∑
(∫
∫
)
(u ·∇c)vdx := (u ·∇c)vdx − c(u · n)vdσ . (19)
Ω
Ω i ∂Ω i
i=1
In order to introduce an upwinding into this formula, we consider each Ω i and
the portion of its boundary where the flow driven by u enters Ω i , i.e., where
{u}·n i < 0. We set
(∂Ω i ) − = {x ∈ ∂Ω i ; {u}·n i (x) < 0}. (20)
Then we replace (19) by
∫
(
2∑ ∫ )
(u ·∇c)vdx := (u ·∇c)vdx−∫
{u}·n i (c int −c ext )v int dσ ,
Ω
i=1 Ω i (∂Ω i) −
(21)
where the superscript int (resp. ext) refers to the interior (resp. exterior) trace
of the function in Ω i , and on the part of (∂Ω i ) − that lies on ∂Ω, c ext =0and
{u} = u. This is a straightforward extension of the Lesaint–Raviart upwind
scheme.
Finally, we wish to extend (21) to the case where u satisfies (14) instead
of (16), while preserving some property analogous to (17). Keeping in mind
the identity:
∫
·∇c)cdx +
Ω(u 1 ∫
u)c
2 Ω(div 2 dx − 1 ∫
(u · n)c 2 dσ =0, (22)
2 ∂Ω
that holds if c and u are sufficiently smooth, we replace (21) by:
∫
Ω
(u ·∇c)vdx :=
2∑
i=1
(∫Ω i
(u ·∇c + 1 2 (div u)c )
vdx
− 1 ∫
∫
)
(u · n Ω )cv dσ − {u}·n i (c int − c ext )v int dσ
2 ∂Ω i\Γ 12 (∂Ω i) −
− 1 ∫
[u] e · n e {cv} e dσ. (23)
2 Γ 12
This is the upwind formulation proposed and analyzed by Rivière et al.
[GRW05].