矩形域上离散解二维Poisson方程的推广Fourier分析法

本文考虑在矩形域上解二维Ｐｏｉｓｓｏｎ方程边值问题的推广Ｆｏｕｒｉｅｒ分析法，边值问题的边界条件为一对边周期条件，另两边为Ｄｉｒｉｃｈｌｅｔ条件，或一边Ｄｉｒｉｃｈｌｅｔ条件、一边Ｎｅｕｍａｎｎ条件．这两个边值问题分别用五点差分格式和九点差分格式逼近，而在解离散的线性方程组时采用推广的Ｆｏｕｒｉｅｒ分析法．理论分析和数值试验表明：（ｉ）九点差分格式比五点差分格式精确，而计算量增加不多；（ｉｉ）推广的Ｆｏｕｒｉｅｒ分析法比通常的Ｆｏｕｒｌｅｒ分析法［１］优越，无论在计算时间或存贮量均较节省。

THE METHOD OF GENERALIZED FOURIER ANALYSIS FOR THE DISCRETE SOLUTION OF TWO DIMENSIONAL POISSON'S EQUATION ON A RECTANGULAR REGION

In this paper, the generalized Fourier analysis method for solving two dimensional Poisson's equation on a rectangular region is considered. The boundary conditions of this boundary value problem are a pair of edge periodic conditions, with the other two edges of Dirichlet conditions or one edge Diricc hlet, and another Neumann condition. These boundary value problem are appr- oximated by five-point scheme and nine-point scheme, while the discrete linear systems are solved by genecalized Fourier analysis meshod. Theorical analysis and numerical experiments show that: (i) the nine point scheme is more accurate than the five point scheme, while the computational work is increased not so much: (ii) generalized Fourier analysis method is more efficient than the usual Fourier analysis method, whether in the time spent or in the storage require- ment.