On the efficiency of nested GMRES preconditioners for 3D acoustic and elastodynamic [formula omitted]-matrix accelerated Boundary Element Methods

Abstract

This article is concerned with the derivation of fast Boundary Element Methods for 3D acoustic and elastodynamic problems. In particular, we are interested in the acceleration of Hierarchical matrix (H-matrix) based iterative solvers. While H-matrix representations allow to reduce the storage requirements and the cost of a matrix–vector product, the number of iterations for an iterative solver, as the frequency or the problem size increases, remains an issue.We consider an inner–outer preconditioning strategy, i.e., the preconditioner is applied through an iterative solver at the inner level. The preconditioner is defined as a H-matrix representation of the system matrix with a given accuracy. We investigate the influence of various parameters of the preconditioner, i.e., the H-matrix accuracy, the GMRES threshold and the maximum number of iterations of the inner solver. Different numerical results are presented to compare the efficiency of the preconditioner with respect to the unpreconditioned reference system. Finally, we propose a way to define the optimal setting for this preconditioner.