2 DGM for elliptic problems

38 2 Elliptic problems
∑
∫
Γ ∈F ID
h
Γ
σ −1 (n · 〈∇e h 〉) 2 dS (2.174) AB.9
≤ C HC M
C W
˜C(1 + CI )(1 + C A )h 2µ−2 |u| 2 H µ (Ω) .
This and ( 2.168) AB.4 yield the estimate
{
‖e h ‖ 2 1,σ ≤ ˜C 1 + C }
HC M
(1 + C I )(1 + C A ) h 2µ−2 |u| 2 H
C (Ω). W
(2.175) AB.11
It follows from ( AB.3 2.167), ( AB.4a 2.169), ( AB.6 2.171) and ( AB.11 2.175) that
B h (e h ,ψ − Π h1 ψ) ≤ c 4 h µ |ψ| H 2 (Ω) |u| H µ (Ω), (2.176) AB.12
{
1/2
where c 4 = c ˜C1/2 2 1 + C hC M
C W
(1 + C I )(1 + C A )}
+
2 ˜C(C A + CJ 2)1/2 .
Finally, by ( 2.176), AB.12 ( 2.166) AB.2 and ( 2.162),
A8.4
exerAB1
‖e h ‖ 2 L 2 (Ω) ≤ cc 4h µ |u| H µ (Ω)‖e h ‖ L 2 (Ω), (2.177) AB.13
which already implies estimate ( 2.164). A8.20
⊓⊔
Exercise 2.34. Verify identity ( AB.1 2.165).
Remark 2.35. In RWG01 [RWG01] the Neumann problem (i.e., ∂Ω = ∂Ω N ) was considered.
The penalty coefficient σ was chosen according
σ| Γ = C W
h β , Γ ∈ F h , (2.178) A3.3a
Γ
insteand of ( 2.72), A3.3 where β ≥ 1/2 is a real number. If triangular grids do
not contain any hanging node (i.e., T h are conforming) then the optimal error
estimation in the L 2 -norm of the NIPG method was proved provided that
β ≥ 3 for d = 2 and β ≥ 3/2 for d = 3. In this case the interior penalty
is so strong that DGFE methods behave likes the standard conforming (i.e.,
continuous) finite element schemes. For more details see [RWG01].
RWG01
sec:num_sc
2.7 Numerical examples
In this section, we shall justify the a priori error estimates ( 2.131), A4.1 ( 2.141) A4.32 and
( 2.164). A8.20 In the first example, we assume that the exact solution is sufficiently
regular. We show that the use of a higher degree of polynomial approximation
increases the rate of convergence of the method. In the second example, the exact
solution has a singularity. Then the order of convergence does not increase
with the increasing degree of the used polynomial approximation. The computational
results are in agreement with theory and show that the accuracy
of the method is determined by the degree of the polynomial approximation
as well the regularity of the solution.