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

摘要

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] are corner singularity functions for the Stokes problem with no-slip condition, and Ci∈R are coefficients which are called the stress intensity factors. We design a finite element method to approximate the coefficients Ci 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.