Convergence of the Semi-Discrete WaveHoltz Iteration

TL;DR

This paper proves convergence of the WaveHoltz iteration for stable semi-discretizations of the wave equation, showing it converges in O(ω) iterations for certain frequency domain problems, with numerical validation.

Contribution

It establishes theoretical convergence guarantees for the WaveHoltz iteration in semi-discrete settings and identifies classes of problems where convergence is assured.

Findings

WaveHoltz iteration converges for stable semi-discretizations.

Convergence rate is O(ω) iterations for certain problems.

Numerical examples confirm theoretical convergence results.

Abstract

In this paper we prove that for stable semi-discretizations of the wave equation for the WaveHoltz iteration is guaranteed to converge to an approximate solution of the corresponding frequency domain problem, if it exists. We show that for certain classes of frequency domain problems, the WaveHoltz iteration without acceleration converges in $O({\omega})$ iterations with the constant factor depending logarithmically on the desired tolerance. We conjecture that the Helmholtz problem in open domains with no trapping waves is one such class of problems and we provide numerical examples in one and two dimensions using finite differences and discontinuous Galerkin discretizations which demonstrate these converge results.