An Optimal O(N) Helmholtz Solver for Complex Geometry using WaveHoltz and Overset Grids

Abstract: We develop efficient and high-order accurate solvers for the Helmholtz equation on complex geometry. The schemes are based on the WaveHoltz algorithm which computes solutions of the Helmholtz equation by time-filtering solutions of the wave equation. The approach avoids the need to invert an indefinite matrix which can cause convergence difficulties for many iterative solvers for indefinite Helmholtz problems. Complex geometry is treated with overset grids which use Cartesian grids throughout most of the domain together with curvilinear grids near boundaries. The basic WaveHoltz fixed-point iteration is accelerated using GMRES and also by a deflation technique using a set of precomputed eigenmodes. The solution of the wave equation is solved efficiently with implicit time-stepping using as few as five time-steps per period, independent of the mesh size. The time-domain solver is adjusted to remove dispersion errors in time and this enables the use of such large time-steps without degrading the accuracy. When multigrid is used to solve the implicit time-stepping equations, the cost of the resulting WaveHoltz scheme scales linearly with the total number of grid points N (at fixed frequency) and is thus optimal in CPU-time and memory usage as the mesh is refined. A simple rule-of-thumb formula is provided to estimate the number of points-per-wavelength required for a p-th order accurate scheme which accounts for pollution (dispersion) errors. Numerical results are given for problems in two and three space dimensions, to second and fourth-order accuracy, and they show the potential of the approach to solve a wide range of large-scale problems.