Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part II. Interior estimates

This part contains new interior pointwise error estimates for the finite el ement method for second order elliptic problems in R-N. Global estimates we re considered in Part I. In the sense to be discussed below, these sharpen known interior quasi-optimal L-infinity and W-infinity(1) estimates in that they indicate a more local dependence of the error at a point on the deriv atives of the solution near the point. The higher the order of the finite e lement the more local the behavior of the finite element approximation. As a consequence of these estimates, new types of local error expansions will be derived which are in the form of inequalities. These expansion inequalit ies are valid for large classes of finite elements defined on irregular gri ds in R-N and have applications to superconvergence, extrapolation, and a p osteriori estimates for both smooth and nonsmooth problems.