On the Compact Discontinuous Galerkin method for polytopal meshes

TL;DR

This paper analyzes the stability and convergence of the Compact Discontinuous Galerkin (CDG) method on polytopal meshes, introduces efficient algorithms for implementation, and compares its performance with related methods through numerical experiments.

Contribution

It provides the first stability and convergence analysis for the hp-version of the CDG method on polytopal meshes and introduces a unified framework for implementing CDG, LDG, and BR2 methods.

Findings

CDG yields a compact stencil with faster assembly and solve times.

Performance depends on the number of facets per mesh element.

Correct flux directions are crucial for variable polynomial degrees.

Abstract

The Compact Discontinuous Galerkin method was introduced by Peraire and Persson in (SIAM J. Sci. Comput., 30, 1806--1824, 2008). In this work, we present the stability and convergence analysis for the $hp$-version of this method applied to elliptic problems on polytopal meshes. Moreover, we introduce fast and practical algorithms that allow the CDG, LDG, and BR2 methods to be implemented within a unified framework. Our numerical experiments show that the CDG method yields a compact stencil for the stiffness matrix, with faster assembly and solving times compared to the LDG and BR2 methods. We numerically study how coercivity depends on the method parameters for various mesh types, with particular focus on the number of facets per mesh element. Finally, we demonstrate the importance of choosing the correct directions for the numerical fluxes when using variable polynomial degrees.