Partial Differential Equations - Modelling and ... - ResearchGate

6 V. Girault and M.F. Wheeler
The discontinuous Galerkin method called IIPG is based on (3). It uses the
regularity of the normal derivative of u. If, in addition, we want to use the
regularity of u and its zero boundary value, then we can add or subtract
the following terms to the left-hand side of (3):
∫
{∇v · n e } e [u] e dσ,
Γ 12
∫
(∇v · n Ω )udσ,
∂Ω i\Γ 12
i=1, 2.
Since these terms are zero, the resulting equation is equivalent to (3). The
discontinuous Galerkin method called SIPG is based on subtraction of these
terms:
(
2∑ ∫ ∫
(
∇u ·∇vdx − (∇u · nΩ )v +(∇v · n Ω )u ) )
dσ
i=1 Ω i ∂Ω i\Γ 12
∫
( )
− {∇u · ne } e [v] e + {∇v · n e } e [u] e dσ = fvdx, (4)
∫Γ 12 Ω
and the discontinuous Galerkin methods called NIPG and OBB-DG are based
on addition of this term:
(
2∑ ∫ ∫
(
∇u ·∇vdx − (∇u · nΩ )v − (∇v · n Ω )u ) )
dσ
i=1 Ω i ∂Ω i\Γ 12
∫
( )
− {∇u · ne } e [v] e −{∇v · n e } e [u] e dσ = fvdx. (5)
∫Γ 12 Ω
In fact, the OBB-DG formulation is precisely (5).
Clearly, the contribution of the surface integrals to the left-hand side of
(5) is anti-symmetric and hence the left-hand side of (5) is non-negative when
v = u. The left-hand side of (4) is symmetric, but there is no reason why it
should be non-negative and the left-hand side of (3) has no symmetry and
no positivity. The left-hand side of (5) can be made positive when v = u by
adding to it the jump terms
∫
1
[u] e [v] e dσ +
|Γ 12 | Γ 12
2∑
i=1
∫
1
uv dσ,
|∂Ω i \ Γ 12 | ∂Ω i\Γ 12
where for any set S, |S| denotes the measure of S. But, of course, this will
not do for (3) and (4). However, considering that all these formulations will
be applied to functions in finite-dimensional spaces, we expect to make (3)
and (4) positive by incorporating into the jump terms adequate parameters.
Thus we add
J 0 (u, v) = σ ∫
12
[u] e [v] e dσ +
|Γ 12 | Γ 12
2∑
i=1
σ i
|∂Ω i \ Γ 12 |
∫
∂Ω i\Γ 12
uv dσ, (6)