On the efficiency of nested GMRES preconditioners for 3D acoustic and elastodynamic H-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 by the acceleration of Hierarchical matrix (H-matrix) based iterative solvers. If 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.

An Introduction to Operator Preconditioning for the Fast Iterative Integral Equation Solution of Time-Harmonic Scattering Problems

“…Other fast algorithms exists like e.g. H-matrix compression techniques [26,72,84] or high-order solvers developed in [33,34,36]. This drastic reduction both in computational cost and memory storage for one iteration of the GMRES gives expectations for solving high frequency problems.…”

“…The resulting linear system is next solved by an iterative Krylov solver [69,106] coupled e.g. with a Fast Multipole Method (FMM) [51,55,62,63,70,82,105,111] or H-matrix [26,72,84]. However, for example in acoustic, the Helmholtz operator for scattering problems is a highly indefinite complex-valued linear operator.…”

The fast multipole method–accelerated line integration boundary element method for 3D heat conduction analysis with heat source

“…Two parts need to be computed for x, the domain integrals and the boundary integrals. During the computed procedure, the domain integral is computed with the help of the FMM-accelerated LIM, the boundary integrals are computed by the FMM FM-LIBEM for 3D heat conduction analysis accelerated BEM, and the result of the domain integral is added into the right vector, then the unknowns are computed by the generalized minimal residual method (Kpadonou et al, 2020;Liegeois et al, 2020). In FM-LIBEM, all the integrals are divided into two kinds for a node x, the near integrals (integral points in cubes that are the neighborhood of C P and itself) and the far integrals (integral points in cubes that are in the interaction list of C P and those wellseparated from itself), the previous one is computed by the tradition LIBEM directly, while the computation of the latter one is done by the FMM.…”

A high-performance boundary element method and its applications in engineering

“…And the domain integral in Eq. (3.4) can be transformed into the following form is added to the right vector, then the unknowns are computed by the generalized minimal residual method (GMRES) [132,133] . In FM-LIBEM, all the integrals are divided into two kinds for a node x , the near integrals (integral points in cubes that are the neighbourhood of P C and itself) and the far integrals (integral points in cubes that are in the interaction list of P C and those well-separated from itself), the previous one is computed by the tradition LIBEM directly, while the computation of the latter one is done by the FMM.…”