Generalized Hierarchical Basis Multigrid Methods for Convection-Diffusion Problems
|convection-diffusion, Petrov-Galerkin multiscale methods, hierarchical basis, wavelets,
robustness, algebraic multigrid
In this project we consider the efficient solution of discrete convection-diffusion problems with
dominating convection. It is well known that the standard hierarchical basis multigrid
method (HBMG) leads neither to optimal (w.r.t. mesh size) nor to robust solvers for
discrete operators arising from singularly perturbed convection-diffusion problems. Its
performance is strongly dependent on the coefficients in the differential equation
(e.g. strength of convection) and the mesh size. (Pre-)Wavelet splittings of the
underlying function spaces allow for efficient algorithms which can be viewed as
generalized HBMG methods. They can be interpreted as ordinary multigrid methods which
employ a special kind of multiscale smoother and show an optimal convergence behavior for
the respective non-perturbed equations similar to classical multigrid. Here, we present a
general Petrov-Galerkin multiscale approach which makes use of problem-dependent
coarsening strategies known from robust multigrid techniques (matrix-dependent
prolongations, algebraic coarsening) together with certain (pre-)wavelet-like and
hierarchical multiscale decompositions of the trial and test spaces on the finest grid.
The presented numerical results show that by this choice generalized HBMG methods can be
constructed which result in robust yet efficient solvers.
||Generalized Hierarchical Basis Multigrid Methods for Convection-Diffusion
Problems (with M. Griebel). SFB Preprint 720, Sonderforschungsbereich
256, Institut für Angewandte Mathematik, Universität Bonn, 2001.
||Multiskalen-Verfahren für Konvektions-Diffusions Probleme.
Dissertation, Universität Bonn, July 2001.