Generalized hierarchical basis multigrid-methods for convection dominated problems
by
Frank Kiefer
University of Bonn, Institut für Angewandte Mathematik, Abteilung für Wissenschaftliches Rechnen und Numerische Simulation, Bonn, Germany

We consider the efficient solution of discrete convection dominated convection-diffusion problems.

It is well known that the standard hierarchical basis multigrid- method (HBMG) for discrete operators arising from singularly perturbed convection-diffusion problems is leading neither to optimal (w.r.t. mesh-size) nor to robust solvers, i.e., the performance still depends strongly on the coefficients in the differential equation (e.g. strength of convection).

(Pre-)wavelet splittings allow efficient algorithms that can be viewed as generalized HBMG methods. For the underlying non-perturbed equations they show an optimal convergence behaviour similar to classical multigrid-methods. This is because of the enhanced stability properties of the corresponding splittings.

In our approach we apply problem depending coarsening strategies known from robust multigrid-techniques together with certain (pre-)wavelet-like multiscale decompositions of the underlying fine-grid ansatz-spaces. We demonstrate by extensive experiments that it is possible to construct generalized HBMG methods, which result in overall robust solvers for some classes of problems.

http://wwwwissrech.iam.uni-bonn.de/people/kiefer.html

Date received: February 23, 2000