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

Source abstract

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 $$\mathbb {R}^d$$ 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 $$1/\sqrt{n}$$ 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.