二维和三维椭圆型偏微分方程的局部傅里叶配置方法:理论和MATLAB代码

IF 3.4 Q1 ENGINEERING, MECHANICAL 国际机械系统动力学学报(英文) Pub Date : 2022-12-13 DOI:10.1002/msd2.12061
Yan Gu, Zhuojia Fu, Mikhail V. Golub
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引用次数: 3

摘要

针对椭圆型边值问题,提出了一种局部傅里叶配点法。该方法首先将整个域离散为一组重叠的小子域,然后在每个子域中,使用伪谱傅里叶配置法对未知函数及其导数进行近似。该方法的核心思想是将伪谱法的快速收敛性和局部离散化技术的高稀疏性相结合,形成一种适用于大规模模拟的新框架。该方法是求解具有复杂几何形状的数值大尺度边值问题的一种有竞争力的替代方法。通过二维和三维泊松方程、亥姆霍兹方程和修正亥姆霍兹方程的初步数值实验,验证了该方法的准确性和有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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A localized Fourier collocation method for 2D and 3D elliptic partial differential equations: Theory and MATLAB code

A localized Fourier collocation method is proposed for solving certain types of elliptic boundary value problems. The method first discretizes the entire domain into a set of overlapping small subdomains, and then in each of the subdomains, the unknown functions and their derivatives are approximated using the pseudo-spectral Fourier collocation method. The key idea of the present method is to combine the merits of the quick convergence of the pseudo-spectral method and the high sparsity of the localized discretization technique to yield a new framework that may be suitable for large-scale simulations. The present method can be viewed as a competitive alternative for solving numerically large-scale boundary value problems with complex-shape geometries. Preliminary numerical experiments involving Poisson, Helmholtz, and modified-Helmholtz equations in both two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.

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