Locally pointwise superconvergence of the tensor-product finite element in three dimensions

Consider a second-order elliptic boundary value problem in three dimensions with locally smooth coefficients and solution. Discuss local superconvergence estimates for the tensor-product finite element approximation on a regular family of rectangular meshes. It will be shown that, by the estimates for the discrete Green’s function and discrete derivative Green’s function, and the relationship of norms in the finite element space such as L2 -norms, W1,∞ -norms, and negative-norms in locally smooth subsets of the domain Ω, locally pointwise superconvergence occurs in function values and derivatives.