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We propose an iterative solution method for the 3D high-frequency Helmholtzequation that exploits a contour integral formulation of spectral projectors.In this framework, the solution in certain invariant subspaces is approximatedby solving complex-shifted linear systems, resulting in faster GMRES iterationsdue to the restricted spectrum. The shifted systems are solved by exploiting apolynomial fixed-point iteration, which is a robust scheme even if themagnitude of the shift is small. Numerical tests in 3D indicate that$O(n^{1/3})$ matrix-vector products are needed to solve a high-frequencyproblem with a matrix size $n$ with high accuracy. The method has a smallstorage requirement, can be applied to both dense and sparse linear systems,and is highly parallelizable.

Solving the 3D High-Frequency Helmholtz Equation using Contour Integration and Polynomial Preconditioning