The problem of constructing variational-difference schemes for elliptic equations

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In [l] variational difference schemes were constructed and the question of the rate of convergence of these schemes was studied, for boundary value problems for elliptic equations of order 2m with natural boundary conditions. However, for domains of arbitrary form the problem of the method of solving the system of grid equations obtained remains open, since the matrix of the system may be arbitrarily ill conditioned (it is easy to be convinced of this by the example of a onedimensional boundary value problem on the segment [a, bl, if the grid is so arranged that the size of the intersection of the segment [XC,, ri] with [a, b] is sufficiently small in comparison with h = xi zi-+; here x,, x1, . . . are the nodes of the grid).