数值求解椭圆型微分方程的降阶法

The Method of the Reduction of Order for the Numerical Solution to Elliptic Differential Equations

本文对于含混合导数的变系数椭圆型微分方程Ｎｅｕｍａｎｎ问题提出了一种间接构造有限差分格式的降阶法。首先引进将原问题变成等价的一阶方程组，对此方程组建立差分格式；然后进行变量分离得到仅含原变量的差分格式。证明了这一差分格式是唯一可解的、二阶收敛的、且是稳定的，引进新变量的目的是为了对差分格式作理论分析，这一方法特别适用于数值求解导数边界条件问题，间断系数问题以及内边界问题，给出了一个数值例子。

An indirect constructing-difference-scheme method,called the method of the reduction of order,is proposed for numerical solution to Neumann problem of elliptic differential equations with variable coefficients and mixed derivatives.At first,some new variables axe introduced to reduce the original problem into an equivalent system of one order differential equations and a difference scheme is constructed for the latter.Then,the discrete variables are separated to obtain a difference scheme only containing the original variables.The solvability,stability and convergence are analyzed.The aim of introducing the new variables is for the analysis of the difference scheme.The method specially applies to numerical solution to the problem of derivative boundary conditions and the problem with discontinuous coefficients and with inner boundaries.An examined example is presented.