Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

296 Appendix B
5. While the piecewise polynomial basis fN 1 1; ...; N n p 1 ; ...; N n 1; ...; N n p n g contains
complete polynomials from degree zero up to p ˆ p min e p e, numerical experiments
indicate that stability demands p 5 2, in general.
6. The DGM is elementwise conservative while the standard ®nite element approximation
is conservative only in element patches. In particular, for any element e,
we always have
…
e
d ^ xe f dx ‡ k ˆ 0 …B:11†
dx xe ÿ 1
This property holds for arbitrarily high-order approximations p e.
The DGM is robust and essentially free of the global spurious oscillations of
continuous Galerkin approximations when applied to convection±di€usion
problems.
We now consider the solution to a convection±di€usion problem with a turning
point in the middle of the domain. The Hemker problem is given as follows:
k d2
‡ x
dx2 d
dx ˆÿk 2 cos… x†ÿ x sin… x† on ‰0; 1Š
with …ÿ1† ˆÿ2, …1† ˆ0. Exact solution for above shows a discontinuity of
p p
…x† ˆcos… x†‡erf…x= 2k†=erf…1=
2k†
Figures B.1 and B.2 show the solutions to the above problem …k ˆ 10 ÿ10 and
h ˆ 1=10† obtained with the continuous and discontinuous Galerkin method,
respectively. Extension to two and three dimensions is discussed in references given
in Chapter 2.
u
2.5
2.0
1.5
1.0
0.5
0
–0.5
–1.0
–1.5
–2.0
Exact
p = 2
p = 3
p = 4
p = 5
–2.5
–1.0 –0.6 –0.2 0
x
0.2 0.6 1.0
Fig. B1. Continuous Galerkin approximation.