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.

Keywords

Finite element / Conforming discontinuous Galerkin (CDG) method / Stabilizer free / Rectangular mesh / Superconvergent / 65N15 / 65N30

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