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The classical continuous finite element method with Lagrangian $Q^k$ basisreduces to a finite difference scheme when all the integrals are replaced bythe $(k+1)\times (k+1)$ Gauss-Lobatto quadrature. We prove that this finitedifference scheme is $(k+2)$-th order accurate in the discrete 2-norm for anelliptic equation with Dirichlet boundary conditions, which is asuperconvergence result of function values.

Superconvergence of high order finite difference schemes based on variational formulation for elliptic equations