A fast Fourier transform based direct solver for the Helmholtz problem

TL;DR

This paper introduces a fast Fourier transform-based direct solver for the Helmholtz equation in rectangular domains, achieving high efficiency with O(N log N) complexity and validated by numerical experiments.

Contribution

The paper presents a novel FFT-based direct solver for Helmholtz problems that significantly reduces computational complexity compared to traditional methods.

Findings

Achieves O(N log N) computational complexity.

Demonstrates efficiency through numerical experiments.

Applicable to 2D and 3D Helmholtz problems.

Abstract

This paper is devoted to the efficient numerical solution of the Helmholtz equation in a two- or three-dimensional rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and trilinear finite elements on an orthogonal mesh yielding a separable system of linear equations. The main key to high performance is to employ the Fast Fourier transform (FFT) within a fast direct solver to solve the large separable systems. The computational complexity of the proposed FFT based direct solver is O(N log N) operations. Numerical results for both two- and three-dimensional problems are presented confirming the efficiency of the method discussed.