1549301742-The_Theory_of_Difference_Schemes__Samarskii

272 Difference Schemes for Elliptic Equations

4.4 SOME PROPERTIES OF DIFFERENCE ELLIPTIC OPERATORS

In this section we reveal some properties of difference operators approxi-

mating the Laplace operator in a rectangle and derive several estimates for

difference approximations to elliptic second-order operators with variable

coefficients and mixed derivatives.

1. Eigenvalue problems for the Laplace difference operator in a rectan-
   gle. Let Go = {O < Xcx < IQ J Cl: = 1, 2} be a rectangle, wh = {x =
   (i 1 h 1 , i 2 h 2 ), i°' = 0, 1, 2, ... , Ncx, Na h°' =I°'} be a grid in Go and lh be
   the set of all boundary grid nodes. The grid w h is taken to be equidistant
   in each direction x °' with step ha.
   The eigenvalue problem for the Laplace operator in the rectangle Go
   subject to the first kind boundary conditions

6.v+ Av= 0, xEGo, vlr=O, v(x):,i:O,

has an infinite set of the eigenvalues such as

ka = 1,2,. .. , CY= 1,2,

associated with the orthonormal system of eigenfunctions

so that

where

I 1 I 2

(u:v)= j dx 1 j dx 2 u(x 1 ,x 2 )v(x 1 ,x 2 ).

0 0

This problem can be solved by the method of separation of variables. The
eigenvalue problem for the difference Laplace operator Ay = Y:c,x, + Yx 2 x 2
supplied by the first kind boundary conditions may be set up in a quite
similar manner as follows: it is required to find the values of the parameter
A (eigenvalues) associated with nontrivial solutions of the homogeneous
equation subject to the homogeneous boundary conditions

(1) Au+ Av= 0, vi -Yh = 0, v(x):,i:O.