Perturbations of Forms and Error Estimates for the Finite Element Method at a Point, with an Application to Improved Superconvergence Error Estimates for Subspaces that Are Symmetric with Respect to a Point

Abstract

We first derive a variety of local error estimates for u - uh at a point x0, where uh belongs to a finite element space Shr and is an approximation to u satisfying the local equations $A(u-u_h,\varphi) = F(\varphi)$ for all $\varphi$ in Shr with compact support in a neighborhood of x0. Here the $A(\cdot,\cdot)$ are bilinear forms associated with second order elliptic equations and the F are linear functionals. In the case that $F \equiv 0$ our results coincide with those of Schatz [SIAM J. Numer. Anal., 38 (2000), pp. 1269--1293] but are improvements when $F \neq 0$. We apply these results to improve the superconvergence error estimates obtained by Schatz, Sloan, and Wahlbin [SIAM J. Numer. Anal., 33 (1996), pp. 505--521] at points x0 where the subspaces are symmetric with respect to x0.