A conforming discontinuous Galerkin finite element method

TL;DR

This paper introduces a conforming discontinuous Galerkin finite element method that combines the advantages of conforming and DG methods for solving second order elliptic problems, offering simplicity and flexibility.

Contribution

The paper presents a novel conforming DG method that maintains simple formulation and strong boundary enforcement while allowing discontinuous approximation.

Findings

Optimal error estimates in discrete $H^1$ and $L^2$ norms

Numerical results confirm theoretical error bounds

Method combines features of conforming and DG finite element methods

Abstract

A new finite element method with discontinuous approximation is introduced for solving second order elliptic problem. Since this method combines the features of both conforming finite element method and discontinuous Galerkin (DG) method, we call it conforming DG method. While using DG finite element space, this conforming DG method maintains the features of the conforming finite element method such as simple formulation and strong enforcement of boundary condition. Therefore, this finite element method has the flexibility of using discontinuous approximation and simplicity in formulation of the conforming finite element method. Error estimates of optimal order are established for the corresponding discontinuous finite element approximation in both a discrete $H^1$ norm and the $L^2$ norm. Numerical results are presented to confirm the theory.