Convergence of the WaveHoltz iteration on $${\mathbb {R}}^{d}$$

Abstract 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 $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> . 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 $$1/\sqrt{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> </mml:mrow> </mml:math> 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.