This paper studies the finite element method for solving the exterior Stokes problem in two dimensions. A nonlocal boundary condition is defined using a nonsingular-kernel Dirichlet-to-Dirichlet (DtD) mapping, which maps the Dirichlet data on an interior circle to the Dirichlet data on another circular artificial boundary based on the Poisson integral formula of the Stokes problem. The truncated problem is then solved using the MINI-element method and a simple DtD iteration strategy, resulting into a sequence of unique and geometrically (h- independent) convergent solutions. The quasi-optimal error estimate is proved for the iterative solution at the end of the iteration process. Numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed method.

本文研究了求解二维外 Stokes 问题的有限元方法。非局部边界条件是使用非奇异核狄利克雷到狄利克雷 （DtD） 映射定义的，该映射基于斯托克斯问题的泊松积分公式，将内圆上的狄利克雷数据映射到另一个圆形人工边界上的狄利克雷数据。然后使用 MINI-element 方法和简单的 DtD 迭代策略求解截断问题，从而得到一系列独特的几何 （h- 无关） 收敛解。在迭代过程结束时，为迭代解证明了准最优误差估计。通过数值实验验证了所提方法的准确性和效率。