Interior penalty discontinuous Galerkin method on very general polygonal and polyhedral meshes

TL;DR

This paper develops a mathematical foundation for interior penalty discontinuous Galerkin methods applied to second order elliptic equations on highly general polygonal and polyhedral meshes, enabling practical and robust computations.

Contribution

It extends interior penalty discontinuous Galerkin methods to very general meshes satisfying shape regularity, broadening their applicability in computational sciences.

Findings

Provides a rigorous mathematical foundation for the method.

Demonstrates robustness and efficiency on complex meshes.

Applicable to a wide class of polygonal and polyhedral meshes.

Abstract

This paper focuses on interior penalty discontinuous Galerkin methods for second order elliptic equations on very general polygonal or polyhedral meshes. The mesh can be composed of any polygons or polyhedra which satisfies certain shape regularity conditions characterized in a recent paper by two of the authors in [17]. Such general meshes have important application in computational sciences. The usual $H^1$ conforming finite element methods on such meshes are either very complicated or impossible to implement in practical computation. However, the interior penalty discontinuous Galerkin method provides a simple and effective alternative approach which is efficient and robust. This article provides a mathematical foundation for the use of interior penalty discontinuous Galerkin methods in general meshes.