On the numerical solution of an elliptic problem with nonlocal boundary conditions

In this paper we consider a class of non-standard elliptic transmission problems in disjoint domains. As a model example, we consider an area consisting of two non-adjacent rectangles. In each subarea, a boundary-value problem of elliptic type is considered, where the interaction between their solutions is described by nonlocal integral conjugation conditions. An a priori estimate for its weak solution in an appropriate Sobolev-like space is proved. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate, compatible with the smoothness of the input data, up to a slowly increasing logarithmic factor of the mesh size, is obtained.