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

Abstract

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[Phi(1), phi(1)]+C-2[Phi(2), phi(2)]+[u(R), p(R)] near each non-convex vertex, where [u(R), p(R)] is an element of H-2 x H-1, [Phi(i), phi(i)] are corner singularity functions for the Stokes problem with no-slip condition, and C-i is an element of 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. (C) 2015 Elsevier B.V. All rights reserved.