Partial Differential Equations - Modelling and ... - ResearchGate

Discontinuous Galerkin Methods 5
2 An Elementary Derivation of Some Simple DG
Methods
In this section, we use very simple examples to derive the equations that are
at the basis of IIPG, SIPG, NIPG, OBB-DG methods and the upwind DG
method of Lesaint–Raviart. In each example, we work out the equations on
a plane domain Ω, with boundary ∂Ω, partitioned into two non-overlapping
subdomains Ω 1 and Ω 2 with interface Γ 12 , and to fix ideas we assume that
each subdomain has part of its boundary on ∂Ω.
2.1 The General Idea for Elliptic Problems
Consider the Laplace equation with a homogeneous Dirichlet boundary condition
in Ω and with data in L 2 (Ω):
−∆u = f in Ω, u =0 on∂Ω. (1)
Let v be a test function that is sufficiently smooth in each Ω i , but does not
belong necessarily to H 1 (Ω). If we multiply both sides of the first equation in
(1) by v, apply Green’s formula in each Ω i , and assume that the solution u is
smooth enough, we obtain:
2∑
(∫
∫
) ∫
∇u ·∇vdx − (∇u · n i )| Ωi v| Ωi dσ = fvdx, (2)
Ω i ∂Ω i Ω
i=1
where n i denotes the unit normal to ∂Ω i , exterior to Ω i .Ifu has sufficient
smoothness, then the trace of ∇u · n i on the interface has the same absolute
value, but opposite signs, on Γ 12 when coming either from Ω 1 or from Ω 2 .
As the change in sign comes from the normal vector, we choose once and for
all the normal’s orientation on Γ 12 ; for example, we choose the orientation of
n 1 . Therefore, setting n e = n 1 , denoting by n Ω the exterior normal to ∂Ω,
denoting by [v] e and {v} e the jump and average of the trace of v across Γ 12 :
and using the identity
[v] e = v| Ω1 − v| Ω2 , {v} e = 1 2 (v| Ω 1
+ v| Ω2 ) ,
∀a 1 ,a 2 ,b 1 ,b 2 ∈ R, a 1 b 1 −a 2 b 2 = 1 2 [(a 1 + a 2 )(b 1 − b 2 )+(a 1 − a 2 )(b 1 + b 2 )] ,
(2) becomes
(
2∑ ∫ ∫
∇u ·∇vdx −
Ω i
i=1
∂Ω i\Γ 12
(∇u · n Ω )vdσ
)
∫
− {∇u · n e } e [v] e dσ
Γ 12
∫
= fvdx. (3)
Ω