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Second order accurate Cartesian grid methods have been well developed for interface problems in the literature. However, it is challenging to develop third or higher order accurate methods for problems with curved interfaces and internal boundaries. In this paper, alternative approaches based on two different grids are developed for some interface and internal layer problems, which are different from adaptive mesh refinement (AMR) techniques. For one dimensional, or two-dimensional problems with straight interfaces or boundary layers 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 difference scheme at boarder grid points that connect two meshes. A two-grid method that employs a second order discretization near the interface in the fine mesh and a fourth order discretization away from the interface in the coarse and boarder grid points is proposed. For two dimensional problems with a curved interface or an internal layer, a level set representation is utilized for which we can build a fine mesh within a tube $|φ({\bf x}) | \le δh$ of the interface. A new super-third seven-point discretization that can guarantee the discrete maximum principle has been developed at hanging nodes. The coefficient matrices of the finite difference equations developed in this paper are M-matrices, which leads to the convergence of the finite difference schemes. Non-trivial numerical examples presented in this paper have confirmed the desired accuracy and convergence.