Sweeping preconditioners for the iterative solution of quasiperiodic Helmholtz transmission problems in layered media

TL;DR

This paper introduces a sweeping preconditioner for quasi-optimal domain decomposition methods, improving iterative solutions of Helmholtz transmission problems in layered media through high-order boundary integral discretizations.

Contribution

It develops a novel sweeping preconditioner based on shape perturbation series and boundary integral equations for efficient Helmholtz problem solutions in layered media.

Findings

Significant reduction in iteration counts for Helmholtz problems.

High-order boundary integral discretizations improve accuracy.

Preconditioner demonstrates robustness across various configurations.

Abstract

We present a sweeping preconditioner for quasi-optimal Domain Decomposition Methods (DDM) applied to Helmholtz transmission problems in periodic layered media. Quasi-optimal DD (QO DD) for Helmholtz equations rely on transmission operators that are approximations of Dirichlet-to-Neumann (DtN) operators. Employing shape perturbation series, we construct approximations of DtN operators corresponding to periodic domains, which we then use as transmission operators in a non-overlapping DD framework. The Robin-to-Robin (RtR) operators that are the building blocks of DDM are expressed via robust boundary integral equation formulations. We use Nystr\"om discretizations of quasi-periodic boundary integral operators to construct high-order approximations of RtR. Based on the premise that the quasi-optimal transmission operators should act like perfect transparent boundary conditions, we construct an approximate LU factorization of the tridiagonal QO DD matrix associated with periodic layered media, which is then used as a double sweep preconditioner. We present a variety of numerical results that showcase the effectiveness of the sweeping preconditioners applied to QO DD for the iterative solution of Helmholtz transmission problems in periodic layered media.