A nonsingular-kernel Dirichlet-to-Dirichlet mapping method for the exterior Stokes problem

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.