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Theoretical analysis demonstrates a $1/\sqrt{n}$ error decay for WaveHoltz iterates in $d$-dimensional space, proving convergence to outgoing Helmholtz solutions.
In this paper, we analyse the WaveHoltz method, a time-domain iterative method for solving the Helmholtz equation, in the constant-coefficient case in all of Rᵈ 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 difference as 1/√n 1 / n in terms of the iteration number n . This guarantees the convergence of the real part of the WaveHoltz iterates to the real part of the outgoing solution of the Helmholtz equation.
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Backman et al. (2026) studied this question.
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