On the solution of second-order linear elliptic equations

A method for solving interior boundary value problems for second-order linear elliptic equations by introducing ray variables is described. The region is divided into cells, within which the coefficients and sources of the equations have the smoothness and continuity properties necessary for the existence of regular solutions in the cell. The finite discontinuities of the coefficients (if any) pass along the cell boundaries. The regular solution in a cell is sought in the form of a superposition of the contributions made by volume and boundary sources placed on rays arriving at a given point from the cell boundaries. Next, a finite-analytic scheme for the numerical solution of boundary value problems in a domain with discontinuous coefficients and sources is constructed by matching the regular solutions emerging from cells at the cell boundaries. The scheme does not exhibit the rigid dependence of the accuracy of approximation on the sizes and shape of the cells, which is inherent in finite-difference schemes.