Solving the Three-Dimensional High-frequency Helmholtz Equation Using Contour Integration and Polynomial Preconditioning

We propose an iterative solution method for the three-dimensional high-frequency Helmholtz equation that exploits a contour integral formulation of spectral projectors. In this framework, the solution in certain invariant subspaces is approximated by solving complex-shifted linear systems, resulting in faster GMRES iterations due to the restricted spectrum. The shifted systems are solved by exploiting a polynomial fixed-point iteration, which is a robust scheme even if the magnitude of the shift is small. Numerical tests in three dimensions indicate that O(n(1/3)) matrix-vector products are needed to solve a high-frequency problem with a matrix size n with high accuracy. The method has a small storage requirement, can be applied to both dense and sparse linear systems, and is highly parallelizable.