安徽大学学报（自然科学版）2026，Vol.50Issue(1)：25-33,9.DOI:10.3969/j.issn.1000-2162.2026.01.004
不规则区域上泊松方程的快速求解器
A fast solver for the numerical solution of the Poisson equation on the irregular domain
摘要
Abstract
Based on the Fourier spectral method,a fast solver was established for the numerical solution of the Poisson equation on the irregular domain.Considering the computation on the unit disc,by the polar coordinate transform and Fourier series expansion,a series of two point boundary value problems satisified by Fourier coefficients were obtained.We constructed the Chebyshev polynomial approximation of the right hand side of these boundary value problems and derived the generalized expressions of Fourier coefficients,solved the undetermined coefficients in the expressions using the boundary conditions.For the computation on the irregular domain,we embedded the irregular domain in the unit disc.With the help of the solving process on the unit disc,combined the generalized expressions of Fourier coefficients and discrete Fourier inverse transform,we constructed the system of linear equations which was satisfied by the boundary conditions and the undetermined coefficients.The advantage of the new scheme was by using the FFT and IFFT algorithm,it could improve the computational efficiency,reduced the scale of computation and attained the exponential convergence accuracy for the numerial solution on the irregular domain.These implied that the proposed scheme provided an effective way for the high precision numerical computation of the Poisson equation.关键词
不规则区域/快速傅里叶变换/泊松方程/切比雪夫多项式逼近/指数阶收敛精度Key words
irregular domain/fast Fourier transform/Poisson equation/Chebyshev polynomial approximation/exponential convergence accuracy分类
数理科学引用本文复制引用
邵文婷..不规则区域上泊松方程的快速求解器[J].安徽大学学报（自然科学版）,2026,50(1):25-33,9.基金项目
国家自然科学基金资助项目(12001362,11526132) （12001362,11526132）
上海市自然科学基金资助项目(16ZR1412700) （16ZR1412700）
