AMG-based wavelet-like multiscale solvers for convection-diffusion problems
Department of Applied Mathematics,
Division for Scientific Computing and Numerical Simulation
Wegelerstr. 6, D-53115 Bonn, Germany
We consider the efficient solution of discrete convection dominated convection-diffusion
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
(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.