Adaptive Sparse Grid Multilevel Method for Elliptic PDEs Based on Finite Differences

摘要

We present a multilevel approach for the solution of partial differential equations. It is based on a multiscale basis which is constructed from a one-dimenisonal multiscale basis by the tensory product approach. Together with the use of has tables as data structure, this allows in a simple way for adaptive refinement and is, due to the tensor product approach, well suited for higher dimensional problems. Also, the adaptive treatment of partial differential equations, the discretization (involving finite differences) and the solution (here by preconditioned BiCG) can be programmed easily. We describe the basic features of the method, discuss the discretization, the solution and the refinement procedures and report on the results of different numerical experiments.