Higher-accuracy schemes for Poissons equation

In this section we consider higher-accuracy schemes for the Dirichlet problem (1) of Section 1 in a rectangle. 

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 

Without loss of generality we may restrict ourselves to the careful analysis of the two-dimensional case (p = 2) where 

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We substitute here Lu = —/ and replace LiL u by the difference operator. The outcome of this is 

This operator is defined on the nine-point pattern shown in Fig. 16. 

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