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Local energy error estimates for the finite element method for ellipticproblems were originally proved in 1974 by Nitsche and Schatz. These estimatesshow that the local energy error may be bounded by a local approximation term,plus a global "pollution" term that measures the influence of solution qualityfrom outside the domain of interest and is heuristically of higher order.However, the original analysis of Nitsche and Schatz is restricted toquasi-uniform grids. We present local a priori energy estimates that are validon shape regular grids, an assumption which allows for highly graded meshes andwhich much more closely matches the typical practical situation. Our chieftechnical innovation is an improved superapproximation result.

Local energy estimates for the finite element method on sharply varying grids