Achieving Superconvergence by One-Dimensional Discontinuous Finite Elements: The CDG Method

Abstract

Novelty of this work is the development of a finite element method using discontinuous $P_k$ element, which has two-order higher convergence rate than the optimal order. The method is used to solve a one-dimensional second order elliptic problem. A totally new approach is developed for error analysis. Superconvergence of order two for the CDG finite element solution is obtained. The $P_k$ solution is lifted to an optimal order $P_{k+2}$ solution elementwise. The numerical results confirm the theory.