An Error Analysis of the Finite Element Method Overcoming Corner Singularities for the Stationary Stokes Problem

In this paper, we introduce a mixed finite element method to overcome corner singularities of the stationary Stokes system on a non-convex polygon. The reference Choi and Kweon (2013) says that the velocity field and the pressure function are composed of singular parts and regular remainders near a non-convex vertex, and furthermore the regular parts are sufficiently smooth on the polygon. We use the corner singularity expansion and propose new extraction formulae for coefficients of singularities. For the proposed numerical method, we first try to find finite element solutions for the regular parts and then calculate approximations of the coefficients by the derived formulae expressed by the remainders and given functions. We show error estimates of the approximations for the regular parts and coefficients, and give some numerical experiments to confirm the efficiency and reliability of the proposed method.