  Interpolation error estimates needed in common finite element applications
using simplicial meshes typically impose restrictions on the both the
smoothness of the interpolated functions and the shape of the simplices. While
the simplest theory can be generalized to admit less smooth functions (e.g.,
functions in H^1(\Omega) rather than H^2(\Omega)) and more general shapes
(e.g., the maximum angle condition rather than the minimum angle condition),
existing theory does not allow these extensions to be performed simultaneously.
By localizing over a well-shaped auxiliary spatial partition, error estimates
are established under minimal function smoothness and mesh regularity. This
construction is especially important in two cases: L^p(\Omega) estimates for
data in W^{1,p}(\Omega) hold for meshes without any restrictions on simplex
shape, and W^{1,p}(\Omega) estimates for data in W^{2,p}(\Omega) hold under a
generalization of the maximum angle condition which previously required p>2 for
standard Lagrange interpolation.
