Averaging algorithms and the support-operator method in elliptic problems with discontinuous coefficients

A method is proposed for solving elliptic boundary value problems with discontinuous coefficients. The method is based on an approximation of the energy integral followed by the construction of a finite-difference scheme by varying the corresponding functionals. It is shown that the solution to the original problem can be approximated by an element of the linear span spanned by basis vectors reflecting the features of the solution: for span elements, the flux component normal to the boundary and the tangent component of the gradient are both continuous across the discontinuity. The expression for the energy functional is exact for span elements and approximates the energy integral for arbitrary solutions. Numerical grids can be structure-fitted (as in the support-operator method) or not structure-fitted (e.g., rectangular, as in the averaging method). The weak convergence of the algorithms is proved. A method is discussed for choosing the control volume associated with a mesh point so as to satisfy the approximation conditions on the faces of the volume. It is shown that such a volume can be constructed for two-dimensional problems, and strong convergence is proved for them.