2 DGM for elliptic problems

20 2 Elliptic problems
The scheme ( 2.62) A2.22c is called symmetric interior penalty Galerkin (SIPG)
method. It was derived by Arnold ([Arn82]) Arnold and Wheeler ([Whe78]) Whe78 by adding
penalty terms to the form Bh s . This formulation leads to a symmetric bilinear
form which is coercive if the penalty parameter σ is sufficiently large.
Moreover, the Aubin-Nitsche trick can be used for obtaining an optimal error
estimate in the L 2 (Ω)-norm.
The method ( 2.63), A2.22d called nonsymmetric interior penalty Galerkin (NIPG)
method, was proposed by Girault, Riviére and Wheeler in [RWG99]. RWG99 In this
case the bilinear form B n,σ
h
is nonsymmetric and does not allow to obtain
optimal error estimates in the L 2 (Ω)-norm with the aid of the Aubin-Nitsche
trick. However, numerical experiments show that the odd degrees of the polynomial
approximation give the optimal order of convergence. On the other
hand, a favorable property of NIPG method is the coercivity of B n,σ
h
(·, ·) for
any penalty parameter σ > 0.
Finally, the method 2.64), A2.22e called incomplete interior penalty Galerkin
(IIPG) method, was studied in [DSW04], Dawson2004 [Sun03], Sun-PhD [SW05]. Sun05 In this case the
bilinear form B i,σ
h
is nonsymmetric and does not allow to obtain an optimal
error estimate in the L 2 (Ω)-norm. Moreover, the penalty parameter σ should
be chosen sufficiently large in order to guarantee the coercivity of B i,σ
h . The
advantage of the IIPG method is the simplicity of the discrete diffusion operator.
This is particularly advantageous in the case when the diffusion operator
is nonlinear. (See, e.g. [Dol08].
iipg07
It would also be possible to define the scheme Bh i (u,v) = li h (v) ∀v ∈ S hp,
where Bh i (u,v) = ai h (u,v) and li h (v) = F h i (v), but this method does not make
sense, because it does not contain the Dirichlet boundary condition ( 2.2).
A0.2a
sec:2.4
2.4 Properties of the bilinear forms
We are interested in the existence of the numerical solutions of method ( 2.60)
A2.22a
– ( 2.64) A2.22e and in a priori error estimates of the difference of numerical solution
and the exact one. Therefore, we define the following mesh-dependent norm
|||u||| Th
=
(
|u| 2 H 1 (Ω,T h ) + Jσ h(u,u)) 1/2
, (2.68) en
In what follows, if there is no danger of misunderstanding, we shall omit
the subscript T h . This means that we shall simply write ||| · ||| = ||| · ||| Th
.
exer1 Exercise 2.10. Prove that ||| · ||| is a norm on the spaces H 1 (Ω, T h ) and S hp .
Let Γ ∈ Fh I be an inner face and K(L)
Γ
and K (R)
Γ
the elements sharing Γ.
There are several possibilities how to define the penalty weight σ. We can set
σ| Γ =
(
max
C W
h K
(L)
Γ
, h K
(R)
Γ
), (2.69) A3.1