This article is devoted to the efficient numerical solution of the Helmholtz equation in a two‐ or three‐dimensional (2D or 3D) 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 2D and 3D problems are presented confirming the efficiency of the method discussed.

中文翻译：

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基于快速傅立叶变换的亥姆霍兹问题直接求解器

本文致力于在具有吸收边界条件（ABC）的二维或三维（2D或3D）矩形域中Helmholtz方程的有效数值解。亥姆霍兹问题由正交网格上的标准双线性和三线性有限元离散化，产生了一个可分离的线性方程组。高性能的主要关键是在快速直接求解器中采用快速傅立叶变换（FFT）来解决大型可分离系统。所提出的基于FFT的直接求解器的计算复杂度为$𝒪（ñ\mathrm{日志}ñ）$操作。给出了2D和3D问题的数值结果，证实了所讨论方法的效率。