M. Griebel and F. Kiefer.
Generalized hierarchical basis multigrid methods for
SFB Preprint 720, Sonderforschungsbereich 256, Institut für
Angewandte Mathematik, Universität Bonn, 2001.
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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.