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