Constructing a CDG Finite Element with Order Two Superconvergence on Rectangular Meshes

Abstract

A novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation on rectangular meshes. This CDG method with discontinuous $P_k$ ($k\geqslant 1$) elements converges to the true solution two orders above the continuous finite element counterpart. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and the $L^2$ norm. A local post-process is defined which lifts a $P_k$ CDG solution to a discontinuous $P_{k+2}$ solution. It is proved that the lifted $P_{k+2}$ solution converges at the optimal order. The numerical tests illustrate the theoretic findings.

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.