Superconvergence of $$C^0$$-$$Q^k$$ Finite Element Method for Elliptic Equations with Approximated Coefficients

Abstract

We prove that the superconvergence of $$C^0$$-$$Q^k$$ finite element method at the Gauss–Lobatto quadrature points still holds if variable coefficients in an elliptic problem are replaced by their piecewise $$Q^k$$ Lagrange interpolants at the Gauss–Lobatto points in each rectangular cell. In particular, a fourth order finite difference type scheme can be constructed using $$C^0$$-$$Q^2$$ finite element method with $$Q^2$$ approximated coefficients.