Some properties of difference elliptic operators

In this section we reveal some properties of difference operators approximating the Laplace operator in a rectangle and derive several estimates for difference approximations to elliptic second-order operators with variable coefficients and mixed derivatives. 

Eigenvalue problems for the Laplace difference operator in a rectan- 

The eigenvalue problem for the Laplace operator in the rectangle Go subject to the first kind boundary conditions 

This problem can be solved by the method of separation of variables. The eigenvalue problem for the difference Laplace operator i. y = - - 

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This problem can be solved by the method of separation of variables. The eigenvalue problem for the difference Laplace operator Ay = ySlXl + Vx2x2 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 

The conditions // = jjNi =0 follow immediately from the relations fi(0)r](x2) = 0, 11(1 r](x2) = 0 and r](x2) 0. As we learn from Chapter 2, Section 3, a solution of this problem acquires the form 

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