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.

在本工作中，提出了一种基于泰勒级数逼近的无网格方法广义有限差分法（GFDM）来求解平稳的2D和3D Stokes方程。为了克服斯托克斯问题中麻烦的压力波动，提出了斯托克斯问题边界条件的一种新的简单表示法。这种数值方法仅将混合边界条件（动量方程在边界向外法向矢量上的投影）添加到Stokes方程，而对控制方程没有任何其他更改。GFDM可以轻松离散建议的配方。GFDM是从泰勒级数展开和最小二乘逼近演化而来的，未知变量的导数表示为相邻节点函数值的线性组合。通过数值算例验证了所提出的GFDM方案的可行性不仅适用于Stokes问题，还适用于更多涉及的通用问题，如Poiseuille流，Couette流和低雷诺数的Navier-Stokes方程。政权。此外，数值结果和比较结果表明，使用GFDM来解决拟议的Stokes方程式比使用压力Poisson方程的经典式更为精确。