WaveHoltz: Iterative Solution of the Helmholtz Equation via the Wave Equation

TL;DR

WaveHoltz introduces an iterative method for solving the Helmholtz equation by leveraging wave equation filtering, resulting in a coercive operator that improves convergence properties.

Contribution

The paper presents the WaveHoltz iterative method that transforms the Helmholtz problem into a wave equation filtering process, ensuring positive definiteness in discretized form.

Findings

WaveHoltz filters solutions through wave equation evolution.

The method yields a coercive operator, enhancing numerical stability.

Demonstrates improved convergence for Helmholtz problems.

Abstract

A new idea for iterative solution of the Helmholtz equation is presented. We show that the iteration which we denote WaveHoltz and which filters the solution to the wave equation with harmonic data evolved over one period, corresponds to a coercive operator or a positive definite matrix in the discretized case.