Stable discontinuous Galerkin FEM without penalty parameters

TL;DR

This paper introduces a modified local discontinuous Galerkin method for second-order elliptic problems that guarantees stability without penalty parameters by employing a novel discrete Poincaré--Friedrichs inequality and jump lifting technique.

Contribution

The paper presents a stable LDG method that eliminates the need for extrinsic penalization, applicable to general meshes with hanging nodes.

Findings

Achieves stability without penalty parameters.

Applicable to general simplicial meshes with hanging nodes.

Provides a theoretical stability proof via a new inequality.

Abstract

We propose a modified local discontinuous Galerkin (LDG) method for second--order elliptic problems that does not require extrinsic penalization to ensure stability. Stability is instead achieved by showing a discrete Poincar\'e--Friedrichs inequality for the discrete gradient that employs a lifting of the jumps with one polynomial degree higher than the scalar approximation space. Our analysis covers rather general simplicial meshes with the possibility of hanging nodes.