Data-sparse algebraic multigrid methods for large scale boundary element equations

Authors:
U. Langer;D. Pusch
Affiliations:
Institute of Computational Mathematics, Johannes Kepler University, Linz, Austria;Institute of Computational Mathematics, Johannes Kepler University, Linz, Austria
Venue:
Applied Numerical Mathematics - Selected papers from the 16th Chemnitz finite element symposium 2003
Year:
2005

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Abstract

This paper presents new algebraic multigrid (AMG) preconditioners for data-sparse boundary element matrices arising from the adaptive cross approximation (ACA) to dense boundary element matrices. Here we mainly consider the single layer potential integral equation, resulting from the direct boundary integral formulation of the interior, or exterior Dirichlet boundary value problems for the Laplace equation in two, or three spatial dimensions, as the most interesting boundary integral equation. The standard collocation, or Galerkin boundary element discretizations lead to fully populated system matrices which require O(N"h^2) of storage units and cause the same complexity for a single matrix-by-vector multiplication, where N"h denotes the number of boundary unknowns. Data-sparse matrix approximation schemes such as ACA reduce this complexity to an almost linear behavior in N"h. Since the single layer potential operator is a pseudo-differential operator of the order -1, the resulting boundary element matrices have large condition numbers on fine meshes. Iterative solvers dramatically suffer from this property for growing N"h. Our AMG-preconditioners avoid the increase of the iteration numbers and result in almost optimal solvers with respect to the total complexity. This behavior is confirmed by our numerical experiments. Let us mention that our AMG-preconditioners use only single grid information provided by the ACA system matrix and by the usual mesh data on the finest level anyway. Similar results are valid for the hypersingular integral operator that is easier to handle since it is a pseudo-differential operator of the order +1.