On the Compact Discontinuous Galerkin method for polytopal meshes
For researchers in numerical methods for elliptic PDEs, this work provides theoretical foundations and practical algorithms for CDG on polytopal meshes, but the improvements are incremental over existing DG methods.
The paper presents stability and convergence analysis for the hp-version of the Compact Discontinuous Galerkin method on polytopal meshes, introduces unified algorithms for CDG, LDG, and BR2 methods, and shows numerically that CDG yields faster assembly and solving times than LDG and BR2.
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.