In this paper we prove that for stable semi-discretizations of the wave equation for the WaveHoltz iteration is guaranteed to converge to an approximate solution of the corresponding frequency domain problem, if it exists. We show that for certain classes of frequency domain problems, the WaveHoltz iteration without acceleration converges in $O({\omega})$ iterations with the constant factor depending logarithmically on the desired tolerance. We conjecture that the Helmholtz problem in open domains with no trapping waves is one such class of problems and we provide numerical examples in one and two dimensions using finite differences and discontinuous Galerkin discretizations which demonstrate these converge results.

翻译：本文证明了，对于波动方程的稳定半离散化，WaveHoltz迭代保证收敛到相应频域问题的近似解（如果该解存在）。我们证明，对于特定类别的频域问题，未经加速的WaveHoltz迭代在$O({\omega})$次迭代内收敛，其常数因子与期望容差的对数相关。我们推测，在无俘获波的开域中的Helmholtz问题即属于此类问题，并利用有限差分和不连续伽辽金离散化提供了一维和二维数值算例，以验证这些收敛结果。