On Solving Second-Order Linear Elliptic Equations

— A method is presented for solving interior boundary-value problems for second-order elliptic equations by transition to ray variables. The domain is divided into cells within which the coefficients and sources have the smoothness and continuity properties necessary for the existence of a regular classical solution in the cell. The finite discontinuities of the coefficients (if any) are located on the cell boundaries. The regular solution in the cell is sought in the form of a superposition of the contributions made by volume and boundary sources placed on the rays arriving at the given point from the cell boundaries. Next, a finite analytic scheme for the numerical solution of the boundary value problem in a domain with discontinuous coefficients and sources is constructed by matching the regular solutions emerging from cells at the cell boundaries. The scheme exhibits no hard dependence of the accuracy of approximation on the sizes and shape of the cells, which is inherent in finite-difference schemes.