Preconditioning of a pollution-free discretization of the Helmholtz equation

TL;DR

This paper introduces a pollution-free first order system least squares formulation for the Helmholtz equation, utilizing a block preconditioner that ensures efficient iterative solutions with iteration counts linearly dependent on the wave number.

Contribution

It presents a novel preconditioning strategy for the Helmholtz equation that is easy to implement, applicable to general domains, and reduces iteration counts through an algebraic error estimation approach.

Findings

Number of MINRES iterations scales linearly with wave number.

Preconditioner remains Hermitian positive definite via subspace correction.

Method is applicable to scattering problems and general domains.

Abstract

We present a pollution-free first order system least squares (FOSLS) formulation for the Helmholtz equation, solved iteratively using a block preconditioner. This preconditioner consists of two components: one for the Schur complement, which corresponds to a preconditioner on $L_2(\Omega)$, and another defined on the test space, which we ensure remains Hermitian positive definite using subspace correction techniques. The proposed method is easy to implement and is directly applicable to general domains, including scattering problems. Numerical experiments demonstrate a linear dependence of the number of MINRES iterations on the wave number $\kappa$. We also introduce an approach to estimate algebraic errors which prevents unnecessary iterations.