A generalized finite difference method for solving Stokes interface problems

Abstract

In this paper, a new scheme is proposed to solve the Stokes interface problem. The scheme turns the Stokes interface problem into two coupled Stokes non-interface subproblems and adds a mixed boundary condition to overcome the numerical pressure oscillation. Since the interface becomes the boundary of the subproblems, the scheme has the advantage to deal with the interface problem with complex geometry. Furthermore, a generalized finite difference method (GFDM) is adopted to solve the coupled Stokes non-interface subproblems. The GFDM is developed from the Taylor series expansions and moving-least squares approximation. Due to the flexibility of the GFDM, it is convenient to handle the complex boundary conditions that appeared in the proposed scheme. The numerical examples verify the accuracy and stability of the GFDM to solve the Stokes interface problem with the mixed boundary conditions. Moreover, for some given numerical examples, the proposed scheme is more accurate than the classical formula of the pressure Poisson equation, especially in terms of pressure.