Topological Problems Arising when Solving Boundary Value Problems for Elliptic Partial Differential Equations by the Method of Finite Differences

摘要

When the method of finite differences is used to approximately solve a boundary value problem for an elliptic partial differential equation over a two-dimensional domain R, the first step is to choose a set of netpoints, N. Next, a system of algebraic equations connecting the values of the approximate solutions at the netpoints is set up. Finally, the system of algebraic equations is solved. Usually, N is taken to be the set of points belonging to a rectangular grid, together with the points of intersection of gridlines with the boundary of R. When a computer is used, one or more of the following assumptions are often made in order to simplify the programming: (1) All the points of N are gridpoints. (2) The 'interior netpoints' are 'gridconnected'. (3) The number of 'irregular' netpoints is much smaller than the number of 'regular' netpoints. In the present paper these three assumptions are analyzed. (Author)