1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference Schemes

for Elliptic Equations

This chapter is devoted to various difference approximations of second-order

elliptic eqnations. In Sections 1-3 we present results of more a detailed ex-

ploration of the Dirichlet difference problem for Poisson's equation. The

approximation technique for the Laplace operator and formulations of dif-

ference boundary conditions are described for regions of arbitrary shape.

The maximum principle (Section 2) and all of its corollaries are established

for grid equations of common structure. These tools are aimed at establish-

ing the uniform convergence with the rate O(I h 12 ) for the difference scheme

constructed in Section 1 for the case of an arbitrary d01nain. In Section 4

we study the properties of the difference Laplace operator and develop the

difference operators corresponding to elliptic operators of general form with

variable coefficients~ In Section 5 higher-accuracy sche1nes are designed for

Poisson's equation in a rectangle.

4.1 THE DIRICHLET DIFFERENCE PROBLEM

FOR POISSON'S EQUATION

We now turn to the design of difference sche1nes for solving the Dirichlet
problem in which it is required to find a continuous in G + r function u( x)

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