Multigrid methods for finite element convection-diffusion problems
A. W. CRAIG
Sima-Sintef, Trondheim, Norway
A. J. PERELLA
Department of Mathematical Sciences, University of Durham, England
December 31, 1994
Several problems arise when applying multigrid methods to find the solutions
of discrete systems arising from the finite element approximation of
convection-diffusion equations. Firstly, of course, if the mesh Peclet number
is high, the standard Galerkin method will not produce a good approximation,
and secondly if we do not use an up winding which is defined by the mesh size
when calculating our coarse mesh corrections then these corrections themselves
will b e inaccurate. Typically these inaccuracies will appear as oscillations
in the numerical approximation.
Recently [1] it has been shown that there exists a class of exponentially
upwinded finite element methods which produce particularly accurate
approximations on the boundaries of the elements in 2- and 3-dimensions. In 1
dimension this method reduces to using the Hemker test functions [2].
Accurate solutions on element boundaries imply oscillation-free
approximations.
In this paper we describe the multigrid method derived from these upwinded
methods and show its rapid convergence. The method can be applied to any
Petrov-Galerkin (including, of course, plain Galerkin) approximation of a
stationary convection diffusion equation.
References
[1] A. W. CRAIG and A. J. PERELLA, `A class of Petrov-Galerkin methods for
stationary convection diffusion problems', In preparation.
[2] P. W. HEMKER, `A numerical study of stiff two-point boundary problems',
Thesis, Mathematisch Centrum, Amsterdam. (1977).