Convergence of the Waveholtz Iteration on $\mathbb{R}^d$

Olof Runborg; Elliot Backman

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arXiv · 2510.15606

Convergence of the Waveholtz Iteration on $\mathbb{R}^d$

Olof Runborg·Elliot Backman

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Abstract

In this paper we analyse the Waveholtz method, a time-domain iterative method for solving the Helmholtz iteration, in the constant-coefficient case in all of $\mathbb{R}^d$. We show that the difference between a Waveholtz iterate and the outgoing Helmholtz solution satisfies a Helmholtz equation with a particular kind of forcing. For this forcing, we prove a frequency-explicit estimate in weighted Sobolev norms, that shows a decrease of the differences as $1/\sqrt{n}$ in terms of the iteration number $n$. This guarantees the convergence of the real parts of the Waveholtz iterates to the real part of the outgoing solution of the Helmholtz equation.

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Olof Runborg, Elliot Backman. 2025-10-17. Convergence of the Waveholtz Iteration on $\mathbb{R}^d$. https://arxiv.org/abs/2510.15606

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