Superconvergence of High Order Finite Difference Schemes Based on Variational Formulation for Elliptic Equations

The classical continuous finite element method with Lagrangian Qk basis reduces to a finite difference scheme when all the integrals are replaced by the (k+1)x(k+1)Gauss-Lobatto quadrature. We prove that this finite difference scheme is (k+2) order accurate in the discrete 2-norm for an elliptic equation with Dirichlet boundary conditions, which is a superconvergence result of function values. We also give a convenient implementation for the case k=2, which is a simple fourth order accurate elliptic solver on a rectangular domain.