Generalized finite difference method for solving stationary 2D and 3D Stokes equations with a mixed boundary condition

Abstract

In the present work, a generalized finite difference method (GFDM), a meshless method based on Taylor-series approximations, is proposed to solve stationary 2D and 3D Stokes equations. To overcome the troublesome pressure oscillation in the Stokes problem, a new simple formulation of boundary condition for the Stokes problem is proposed. This numerical approach only adds a mixed boundary condition, the projections of the momentum equation on the boundary outward normal vector, to the Stokes equations, without any other change to the governing equations. The proposed formulation can be easily discretized by the GFDM. The GFDM is evolved from the Taylor series expansions and moving-least squares approximation, and the derivative expressed of unknown variables as linear combinations of function values of neighboring nodes. Numerical examples are utilized to verify the feasibility of the proposed GFDM scheme not only for the Stokes problem, but also for more involved and general problems, such as the Poiseuille flow, the Couette flow and the Navier–Stokes equations in low-Reynolds-number regime. Moreover, numerical results and comparisons show that using the GFDM to solve the proposed formulation of the Stokes equations is more accurate than the classical formulation of the pressure Poisson equation.