Difference problem Dirichlet

As a final result problem ( ) is associated with the Dirichlet difference problem relating to the determination of a grid function y x) defined on the grid W , satisfying at the inner nodes, that is, on the equation... 

The Dirichlet difference problem in a domain of rather compHcated configuration. If a solution of the Dirichlet problem needs to be determined in a domain G with a nonlinear boundary, the grid ( f G) is, generally speaking, non-equidistant near the boundary. We describe below such a grid and give the possible classification of its nodes. 

In the preceding section the Dirichlet difference problem was set up in the form (1) 3 Consider as one possible example the so-called scheme with weights for the heat conduction equation... 

Estimation of a solution of the Dirichlet difference problem. We make use of a priori estimates obtained in Section 2 for a grid equation of common structure for constructing a uniform estimate of a solution of the Dirichlet difference problem (24)-(26) arising in Section 1 ... 

Theorem 1 For a solution of the Dirichlet difference problem the estimate... 

This theorem expresses the stability of the Dirichlet difference problem (1) with respect to the boundary data and the right-hand side. 

The statement of the Dirichlet difference problem providing a higher-order approximation. On the basis of the cross scheme it is possible to construct a scheme with the error of approximation 0( h j ) or 0 h ) on a solution in the case of a square (cube) grid. In order to raise the order of approximation, we exploit the fact that u = u x) is a solution of Poisson s equation... 

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