AMG-based wavelet-like multiscale solvers for convection-diffusion problems

Michael Griebel
Frank Kiefer

Department of Applied Mathematics, Division for Scientific Computing and Numerical Simulation
Wegelerstr. 6, D-53115 Bonn, Germany
Tel.: +49-228-73-3427, FAX: +49-228-73-7527

We consider the efficient solution of discrete convection dominated convection-diffusion problems. It is well known that the standard hierarchical basis multigrid method (HBMG) leads for discrete operators arising from singularly perturbed convection-diffusion problems neither to optimal (w.r.t. meshsize) 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 of the underlying function spaces allow efficient algorithms that can be viewed as generalized HBMG methods. They can be interpreted as ordinary multigrid methods that use a special kind of multiscale smoother and show for the respective non-perturbed equations an optimal convergence behaviour similar to classical multigrid. In our general Petrov--Galerkin multiscale approach we apply 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. We demonstrate through extensive experiments that by this choice we are able to construct generalized HBMG methods, that result in possibly robust solvers.