Preconditioners for sparse grid discretizations.
Technical Report No. 746, SFB 256, Universität Bonn, Germany,
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In this paper we deal with preconditioners for sparse grid finite-difference- and Petrov-Galerkin-discretizations of the Poisson equation. We analyse the Jacobi-preconditioner for the simple setting of non-adaptive grids and periodic boundary conditions. The analysis shows that the resulting condition numbers mainly depend on the underlying tensor product Wavelets. For example, high order Lifting-Interpolets lead to l2-condition numbers which are essentially independent of the finest mesh size. Based on this observation we introduce a so-called Lifting-preconditioner for discretizations which use Interpolets as trial-functions. Numerical examples show the efficiency of the preconditioners for cases which are not covered by our analysis, e.g., adaptive grids.