A Finite Element Method Using Singular Functions for the Poisson Equation: Corner Singularities

Abstract

Consider the Poisson equation with homogeneous Dirichlet boundary conditions on a polygonal domain with one reentrant corner. In this paper, we develop a new finite element method for the accurate computation of the solution and stress intensity factors. It is well known that the solution of such a problem has a singular function representation: $u=w+\lambda \eta s$, where $w\in H^2(\Omega) \cap H^1_0(\Omega)$, $\lambda\in {\cal R}$ and $\eta$ are the stress intensity factor and cut-off function, respectively, and s is a known singular function depending only on the reentrant angle. By using the dual singular and an extra cut-off function, we are able to derive a new extraction formula for $\lambda$ in terms of w and, hence, deduce a well-posed variational problem for w. Standard continuous piecewise linear finite element approximation yields $O(h)$ optimal accuracy for w, which, in turn, implies the same accuracy for u in the $H^1$ norm. We are able only to prove $O(h^{1+\frac\pi\omega})$ error bounds for w and u in the $L^2$ norm and for $\lambda$ in the absolute value, where $\omega$ is the internal angle.