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]=C1[Φ1,ϕ1]+C2[Φ2,ϕ2]+[ur,pr] near each non-convex vertex, where [ur,pr]∈H2×H1, [Φi,ϕi][Φi,ϕi] are corner singularity functions for the Stokes problem with no-slip condition, and Ci∈RCi∈R are coefficients which are called the stress intensity factors. We design a finite element method to approximate the coefficients CiCi and the regular part [ur,pr], show the unique existence of the approximations, and derive their error estimates. Some numerical examples are given, confirming convergence rates for the approximations.