Interface problems are important in many fields of science and technology including physics, engineering, life science, and etc. It is therefore necessary to develop efficient and accurate numerical approaches to simulate various interface problems. This dissertation focuses on some partial differential equations (PDEs) based numerical methods in interface problems. It is composed of two parts. In first part of the work, based on an elliptic solver in a weak formulation, we have developed a numerical solver for a multiphase Stokes flow which decomposes the equations into three elliptic equations and tracking the moving interface. Numerically, our method is efficient and has the second order convergence in space. In the second half of the work, we develop a regularized least square radial basis function (RLS-RBF) method for constructing differential operators on meshless domains. With this method, we can construct differential operators on manifolds. Numerically, our method is stable and has the second order convergence in space when constructing the Laplace-Beltrami operator on a unit sphere. Finally, we extend the regularized least square (RLS) method to solving the PDEs on evolving surface by coupling with the grid based particle method (GBPM).