Convergence Results for 3D Sparse Grid Approaches. Modelling, Analysis and211 Simulation

摘要

The convergence behavior is investigated of solution algorithms for the211u001eanisotropic Poisson problem on partially ordered, sparse families of regular 211u001egrids in 3D. In order to study multilevel techniques on sparse families of grids, 211u001efirst the authors consider the convergence of a two-level algorithm that applies 211u001esemi-coarsening successively in each of the coordinate directions. This algorithm 211u001eshows good convergence, but recursive application of the successive semi-211u001ecoarsening is not sufficiently efficient. Therefore the authors introduce another 211u001ealgorithm, which uses collective 3D semi-coarsened coarse grid corrections. The 211u001econvergence behavior of this collective version is worse, due to the lack of 211u001ecorrespondence between the solutions on the different grids. By solving for the 211u001etrivial solution the authors demonstrate that a good convergence behavior of the 211u001ecollective version of the algorithm can be retained when the different solutions 211u001eare sufficiently coherent. In order to solve also non-trivial problems, the 211u001eauthors develop a defect correction process. This algorithm makes use of 211u001ehierarchical smoothing in order to deal with the problems related to the lack of 211u001ecoherence between the solutions on the different grids. Now good convergence 211u001erates are obtained also for non-trival solutions. All convergence results are 211u001eobtained for two-level processes. The results show convergence rates which are 211u001ebounded, independent of the discretization level and of the anisotropy in the 211u001eproblem.