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

The classical continuous finite element method with Lagrangian $$Q^k$$ Q k basis reduces to a finite difference scheme when all the integrals are replaced by the $$(k+1)\times (k+1)$$ ( k + 1 ) × ( k + 1 ) Gauss–Lobatto quadrature. We prove that this finite difference scheme is $$(k+2)$$ ( k + 2 ) th 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 $$k=2$$ k = 2 , which is a simple fourth order accurate elliptic solver on a rectangular domain.