Comparison of Geometrical and Algebraic Multigrid Preconditioners for Data-Sparse Boundary Element Matrices

摘要

We present geometric (GMG) and algebraic multigrid (AMG) preconditioners for data-sparse boundary element matrices. Data-sparse approximation schemes such as adaptive cross approximation (ACA) yield an almost linear behavior in N_h, where N_h is the number of (boundary) unknowns. The treated system matrix represents the discretized single layer potential operator (SLP) resulting from the interior Dirichlet boundary value problem for the Laplace equation. It is well known, that the SLP has converse spectral properties compared to usual finite element matrices. Therefore, multigrid components have to be adapted properly. In the case of GMG we present convergence rate estimates for the data-sparse ACA version. Again, uniform convergence can be shown for the V-cycle. Iterative solvers dramatically suffer from the ill-conditioness of the underlying system matrix for growing N_h. Our multigrid-preconditioners avoid the increase of the iteration numbers and result in almost optimal solvers with respect to the total complexity. The corresponding numerical 3D experiments are confirming the superior preconditioning properties for the GMG as well as for the AMG approach.