Optimal Interior and Local Error Estimates of a Recovered Gradient of Linear Elements on Nonuniform Triangulations

Abstract

We examine a simple averaging formula for the gradient of linear finite elements in $R^d$ whose interpolation order in the $L^q$-norm is $O(h^2)$ for $d<2q$ and nonuniform triangulations. For elliptic problems in $R^2$ we derive an interior superconvergence for the averaged gradient over quasiuniform triangulations. Local error estimates up to a regular part of the boundary and the effect of numerical integration are also investigated.