A finite element method for computing accurate solutions for Poisson equations with corner singularities using the stress intensity factor

Abstract

In this article, we consider the Poisson equation with homogeneous Dirichlet boundary conditions, on a polygonal domain with one reentrant corner. The solution of the Poisson equation with a concave corner yields a singular decomposition, u=w+ληs, where w is regular, s is a singular function, and the coefficient λ is the so called stress intensity factor. This stress intensity factor can be computed using the extraction formula. We introduce a new non-homogeneous boundary value problem, which has ‘zero’ stress intensity factor. Using the solution of this new partial differential equation, we can compute an accurate solution of the original problem, simply by adding singular part. We obtain an optimal convergence rate with smaller errors when compared with others.