Volume 8, pp. 46-87, 1999.
On the convergence of multigrid methods for flow problems
Ingemar Persson, Klas Samuelsson, and Anders Szepessy
Abstract
We prove two theorems on the
residual damping in multigrid methods when solving
convection dominated diffusion equations and
shock wave problems, discretized by the streamline diffusion
finite element method. The first theorem shows that
a V-cycle, including sufficiently many pre and post smoothing steps,
damps the residual in for a
constant coefficient convection problem with small diffusion in
two space dimensions, without the assumption that the coarse
grid is sufficiently fine. The proof is based on
discrete Green's functions for the smoothing and
correction operators
on a uniform unbounded mesh aligned with the characteristic.
The second theorem proves a similar result
for a certain continuous
version of a two grid method, with isotropic artificial
diffusion, applied to
a two dimensional Burgers shock wave problem.
We also present
numerical experiments that verify the residual damping dependence on
the equation, the choice of artificial diffusion
and the number of smoothing steps.
In particular numerical experiments show improved convergence of the multigrid
method, with damped Jacobi smoothing steps,
for the compressible Navier-Stokes equations in two space dimensions by
using the theoretically suggested exponential increase of the
number of smoothing steps on coarser meshes,
as compared to the same amount of work with constant
number of smoothing steps on each level.
Full Text (PDF) [531 KB],
BibTeX
Key words
multigrid methods, convergence, convection-diffusion, conservation laws, Green's function, shock waves.
AMS subject classifications
65N55, 65N30, 35L65.
