A finite element method for singular solutions of the Navier-Stokes equations on a non-convex polygon

It is shown in Choi and Kweon (2013) that a solution of the Navier-Stokes equations with no-slip boundary condition on a non-convex polygon can be written as u , p = C 1 ¿ 1 , ¿ 1 + C 2 ¿ 2 , ¿ 2 + u r , p r near each non-convex vertex, where u r , p r ¿ H 2 × H 1 , ¿ i , ¿ i are corner singularity functions for the Stokes problem with no-slip condition, and C i ¿ R are coefficients which are called the stress intensity factors. We design a finite element method to approximate the coefficients C i and the regular part u r , p r , show the unique existence of the approximations, and derive their error estimates. Some numerical examples are given, confirming convergence rates for the approximations.