Distributional derivatives and stability of discontinuous Galerkin finite element approximation methods

Abstract

The goal of this article is to explore and motivate stabilization requirements for various types of discontinuous Galerkin (DG) methods. A new approach for the understanding of DG approximation methods for second order elliptic partial differential equations is introduced. The approach explains the weaker stability requirements for local discontinuous Galerkin (LDG) methods when compared to interior-penalty discontinuous Galerkin methods while also motivating the existence of methods such as the minimal dissipation LDG method that are stable without the addition of interior penalization. The main idea is to relate the underlying DG gradient approximation to distributional derivatives instead of the traditional piecewise gradient operator associated with broken Sobolev spaces.

Keywords

• Discontinuous Galerkin methods
• broken Sobolev spaces
• distributional derivatives
• numerical derivatives