Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d-dimensional Q(k)-type elements with d greater than or equal to 1 and k greater than or equal to 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H-1 norm. For d-dimensional P-k-type elements, we consider the standard Lagrange interpolation-the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d greater than or equal to 2 and k greater than or equal to d + 1 that such interpolation and the finite element solution are not superclose in both H-1 and L-2 norms and that not all such interpolation points are superconvergence points for the finite element approximation. (C) 2003 Wiley Periodicals, Inc.