We propose an extension of the hybrid difference method called the interface hybrid difference methods for solving elliptic interface equations. The hybrid difference method is composed of two types of approximations: one is the finite difference approximation of PDEs within cells (cell FD) and the other is the intercell finite difference (intercell FD) on edges of cells. The intercell finite difference is derived from continuity of normal fluxes. For the interface problems a new intercell condition is introduced when the cell interface and the problem interface are coincident. The domain is decomposed into cells so that each cell is contained exclusively in one of the subregions. Complete convergence analysis in the discrete energy norm is presented and numerical examples are provided to confirm the theoretical results.
