Partial Differential Equations - Modelling and ... - ResearchGate

Discontinuous Galerkin Methods 9
(
2∑ ∫ ∫
(
µ ∇u : ∇v dx − (∇u · nΩ )v + ε(∇v · n Ω )u ) )
dσ
Ω i ∂Ω i\Γ 12
∫
− µ ( )
{∇u · n e } e [v] e + ε{∇v · n e } e [u] e dσ + µJ0 (u, v)
Γ 12
(
2∑
∫
∫
) ∫
+ − p div v dx + p(v · n Ω ) dσ + {p} e [v] e · n e dσ
i=1 Ω i ∂Ω i\Γ 12 Γ 12
∫
= f · v dx, (13)
Ω
(
2∑ ∫ ∫
) ∫
q div u dx − q(u · n Ω ) dσ − {q} e [u] e · n e dσ =0,
Ω i ∂Ω i\Γ 12 Γ 12
i=1
i=1
with the interpretation for the parameters ε and σ of the formula (7).
(14)
2.3 Upwinding in a Transport Problem: General Idea
Consider the simple transport problem in Ω:
c + u ·∇c = f in Ω, (15)
where f belongs to L 2 (Ω)andu is a sufficiently smooth vector-valued function
that satisfies
div u =0 inΩ, u · n Ω =0 on∂Ω. (16)
Recall the notation
2∑ ∂c
u ·∇c = u i ,
∂x
i=1 i
and note that when the functions involved are sufficiently smooth, Green’s
formula and (16) yield ∫
(u ·∇c)cdx =0. (17)
Ω
For the applications we have in mind, let us assume that c is sufficiently
smooth in each Ω i , but is not necessarily in H 1 (Ω).Then,wemustgivea
meaning to the product u ·∇c. From the following identity and the fact that
the divergence of u is zero:
div(cu) =c(div u)+u ·∇c = u ·∇c,
and we derive for any smooth function ϕ with compact support in Ω
∫
〈u ·∇c, ϕ〉 = 〈div(cu),ϕ〉 = −〈cu, ∇ϕ〉 = − (cu) ·∇ϕdx
2∑
∫
= − (cu) ·∇ϕdx. (18)
i=1 Ω i
We use the last equality to define u ·∇c in the sense of distributions.
Ω