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

摘要

We first derive a variety of local error estimates for u - u(h) at a point x(0), where uh belongs to a finite element space S-r(h) and is an approximation to u satisfying the local equations A(u- u(h), φ) = F(φ) for all φ in S-r(h) with compact support in a neighborhood of x(0). Here the A(.,.) are bilinear forms associated with second order elliptic equations and the F are linear functionals. In the case that F = 0 our results coincide with those of Schatz [SIAM J. Numer. Anal., 38 ( 2000), pp. 1269 - 1293] but are improvements when F &NOTEQUAL; 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 x(0) where the subspaces are symmetric with respect to x(0).