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

Olof Runborg, Elliot Backman

TL;DR

This work proves the convergence of the Waveholtz time-domain iteration for the full-space Helmholtz problem with constant coefficients by recasting the iteration error as an outgoing Helmholtz solution with a forcing that depends on iterates through a Fourier multiplier. By extending the wave operator ${\mathcal S}$ and the fixed-point operator ${\Pi}$ to weighted Sobolev spaces and employing a frequency-dependent Helmholtz estimate (via the limiting absorption principle), the authors establish a concrete bound on the real part of the error that decays like $n^{-1/2}$ and scales with frequency as $\omega^{2s-2}$ (or $\omega^{2s-1}$ for $H^1$-norms). The analysis avoids eigenfunction expansions and instead uses Fourier-analytic techniques to obtain a rigorous convergence guarantee for the real parts of the iterates to the outgoing Helmholtz solution, with explicit dependence on data norms. These results clarify the theoretical behavior of Waveholtz on unbounded domains and provide guidance on iteration counts for high-frequency problems, while leaving open the extension to variable wave speeds.

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.

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

TL;DR

This work proves the convergence of the Waveholtz time-domain iteration for the full-space Helmholtz problem with constant coefficients by recasting the iteration error as an outgoing Helmholtz solution with a forcing that depends on iterates through a Fourier multiplier. By extending the wave operator and the fixed-point operator to weighted Sobolev spaces and employing a frequency-dependent Helmholtz estimate (via the limiting absorption principle), the authors establish a concrete bound on the real part of the error that decays like and scales with frequency as (or for -norms). The analysis avoids eigenfunction expansions and instead uses Fourier-analytic techniques to obtain a rigorous convergence guarantee for the real parts of the iterates to the outgoing Helmholtz solution, with explicit dependence on data norms. These results clarify the theoretical behavior of Waveholtz on unbounded domains and provide guidance on iteration counts for high-frequency problems, while leaving open the extension to variable wave speeds.

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 . 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 in terms of the iteration number . This guarantees the convergence of the real parts of the Waveholtz iterates to the real part of the outgoing solution of the Helmholtz equation.
Paper Structure (14 sections, 10 theorems, 157 equations)

This paper contains 14 sections, 10 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convergence of Waveholtz
  4. Proof of Theorem 3.1
  5. The error equation
  6. Estimate of the error
  7. Estimate of the Helmholtz Equation
  8. Proof of Theorem \ref{['thm:HEBound']}
  9. An estimate for the case $\omega = 1$
  10. Extension to the $H^1_s(\mathbb{R}^d)$-norm
  11. The case of $\omega \geq 1$
  12. The case of outgoing solutions
  13. Conclusion
  14. Extension and boundedness of $\mathcal{S}$ and $\Pi$

Key Result

Proposition 1

Suppose $\Omega \subset \mathbb{R}^d$ is an open bounded subset such that its boundary $\partial \Omega$ is a $(d-1)$-dimensional smooth manifold. Then if $s > \frac{1}{2}$ the trace operator$\tau: H^s(\Omega) \longrightarrow H^{s-\frac{1}{2}}(\partial\Omega)$ is bounded, satisfying for some $K \in \mathbb{R}^+$ and every $v\in H^s(\Omega)$.

Theorems & Definitions (20)

  • Definition 1
  • Proposition 1: Theorem 9.4 in LionsMagenes
  • Proposition 2: Theorem 4.1 (i) in Agmon
  • Theorem 3
  • Remark 1
  • Remark 2
  • Lemma 4
  • proof
  • Proposition 5
  • Lemma 6
  • ...and 10 more