Partial Differential Equations - Modelling and ... - ResearchGate

18 V. Girault and M.F. Wheeler
5 DG Approximation of a Convection-Diffusion Equation
Consider the convection-diffusion equation combining (24) and (15) in the
domain Ω of the previous sections:
− div(K∇c)+u ·∇c = f, in Ω, (51)
K∇c · n Ω =0, on ∂Ω, (52)
where f belongs to L 2 0(Ω), the tensor K satisfies the assumptions listed in
Section 3 and u satisfies (16):
div u =0 inΩ,
u · n Ω =0 on∂Ω.
This problem has a solution c ∈ H 1 (Ω), unique up to an additive constant
under mild restrictions on the velocity u, for instance, when u belongs to
H 1 (Ω) d . We propose to discretize it with a DG method when u is replaced
by the solution u h ∈ X h of a flow problem that satisfies b h (u h ,q h ) = 0 for all
q h ∈ M h :
∑
q h div u h dx −
∑ ∫
{q h } e [u h ] e · n e dσ =0.
E∈E h
∫E
For an integer l ≥ 1, we define
e∈Γ h ∪∂Ω
e
Y h = {c ∈ L 2 (Ω) :∀E ∈E h , c| E ∈ P l (E)}. (53)
In view of (23) and (32), we discretize (51)–(52) by: Find c h ∈ Y h such that
for all v h ∈ Y h :
∑
E∈E h
∫E
−
+ ∑
∑
e∈Γ h ∪∂Ω
E∈E h
∫E
K∇c h ·∇v h dx
∫
( )
{K∇ch · n e } e [v h ] e + ε{K∇v h · n e } e [c h ] e dσ + J0 (c h ,v h )
e
(
uh ·∇c h + 1 2 (div u )
h)c h vh dx − 1 2
− ∑ ∫
{u h }·n E (c int
h
E∈E
(∂E) − h
where (∂E) − is defined by (20)
∑
e∈Γ h ∪∂Ω
∫
[u h ] e · n e {c h v h } e dσ
e
− c ext
h )vh int dσ =
(∂E) − = {x ∈ ∂E : {u h }·n E (x) < 0},
∫
Ω
fv h dx, (54)
and the parameters ε and σ e are the same as previously.
To simplify, we introduce the form t h with the upwind approximation of
the transport term in (54):