Finite difference approximations for some interface problems with variable coefficients

General elliptic interface problem with variable coefficients and curvilinear interface is transformed into analogous problem with rectilinear interface. For the numerical solution of transformed problem a finite difference scheme with averaged right-hand side is proposed. Convergence rate estimate in discrete norm, compatible with the smoothness of data, is obtained. Analogous parabolic problem with the singularity of “concentrated capacity” type is considered. Two finite difference schemes for its solution are proposed and investigated.