A Constructive A Priori Error Estimation for Finite Element Discretizations in a Non-Convex Domain Using Singular Functions

Abstract

In solving elliptic problems by the finite element method in a bounded domain which has a re-entrant corner, the rate of convergence can be improved by adding a singular function to the usual interpolating basis. When the domain is enclosed by line segments which form a corner of $\pi/2$ or $3\pi/2$, we have obtained an explicit a priori $H^{1}_{0}$ error estimation of $O(h)$ and an $L^{2}$ error estimation of $O(h^{2})$ for such a finite element solution of the Poisson equation. Particularly, we emphasize that all constants in our error estimates are numerically determined, which plays an essential role in the numerical verification of solutions to non-linear elliptic problems.