Explicit Error Bounds for a Nonconforming Finite Element Method

Abstract

Let u be the solution of the following model: $$ \left\{\begin{array}{l} \mbox{find $u \in H^1_0(\Omega)$ such that} \\ [3pt] -\Delta u = f\quad\mbox{in } \Omega, \end{array}\right. $$ where f is a given function in L2 (\Omega)$ and $\Omega$ a bounded open set in $\mathbb{R}^{2}$ with a polygonal boundary. Let uh be a finite element approximation of u. The goal of this paper is to suggest an explicit bound of the error (u - uh ). Furthermore, this bound is obtained without complex computations. The basic idea is to define locally an admissible vector field for the dual model.