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

2018/11/29 by X. I. Yuanzhe, Liu, Xiao, Yousef Saad +4
Engineering · Physics and Astronomy · #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1811.12378

openalex publication_date 2018/11/29 · openalex created_date 2018/12/11 · openalex updated_date 2026/07/28

Abstract

We propose an iterative solution method for the 3D 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 3D 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.

Citations

• A Flexible Inner-Outer Preconditioned GMRES Algorithm

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