Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Saturday, January 25th, 1997

A robust multigrid solver for convection-dominated problems

Standard multigrid solvers are based on an appropriate smoother damping high-oscillatory modes and certain restriction and prolongation operators. Applied to elliptic problems, multigrid algorithms render excellent convergence rates. But those algorithms fail to work, if the convection dominates the flow too much.

I want to present a new strategy how to extend standard multigrid methods to convection-dominated problems. It is based on adapting the restriction and prolongation operators to the differential operator at hand. The construction of the operators can be written in terms of a multiresolution analysis. This tool might be helpful for analytical examinations.

Up to now, test calculations have been performed for

with d=1,2. In the one-dimensional case, linear ( const., ) as well as nonlinear problems () have been examined, whereas for d=2 only the linear coefficient case has been considered. For all computations we obtain convergence rates comparable to those of standard multigrid schemes applied to elliptic problems as long as the grid Peclet number is less than one.