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Second order accurate Cartesian grid methods have been well developed forinterface problems in the literature. However, it is challenging to developthird or higher order accurate methods for problems with curved interfaces andinternal boundaries. In this paper, alternative approaches based on twodifferent grids are developed for some interface and internal layer problems,which are different from adaptive mesh refinement (AMR) techniques. For onedimensional, or two-dimensional problems with straight interfaces or boundarylayers that are parallel to one of the axes, the discussion is relatively easy.One of challenges is how to construct a fourth order compact finite differencescheme at boarder grid points that connect two meshes. A two-grid method thatemploys a second order discretization near the interface in the fine mesh and afourth order discretization away from the interface in the coarse and boardergrid points is proposed. For two dimensional problems with a curved interfaceor an internal layer, a level set representation is utilized for which we canbuild a fine mesh within a tube $|\varphi({\bf x}) | \le \delta h$ of theinterface. A new super-third seven-point discretization that can guarantee thediscrete maximum principle has been developed at hanging nodes. The coefficientmatrices of the finite difference equations developed in this paper areM-matrices, which leads to the convergence of the finite difference schemes.Non-trivial numerical examples presented in this paper have confirmed thedesired accuracy and convergence.

High order two-grid finite difference methods for interface and internal layer problems