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A second order finite difference method has been developed for elliptic interface problems that involve discontinuous coefficients, singular source terms, non-smooth or even discontinuous coefficients across an arbitrary interface. The method is based on Cartesian grids and coupled with a multigrid solver. The new method will preserve the discrete maximum principle. The convergence of the new method has been proved. Two different finite element spaces and corresponding Galerkin methods for elliptic interface problems using Cartesian grids have been developed. Some theoretical results about these finite element methods are also obtained. We believe that we are the first to develop these new finite element spaces over Cartesian grids. Some public subroutines of fast solvers for Helmholtz and Poisson equations on irregular domains either exterior or interior with various boundary conditions have been developed and made public In collaboration with J. K. Hunter of UC Davis and H. K. Zhao of UC Irvine, we formulate a model for the spreading on a surface of a drop that deposits an autophobic monolayer of surfactant We present numerical solutions of the model equations using an immersed interface method and a level set method.