On the Compact Discontinuous Galerkin method for polytopal meshes

TL;DR

This work analyzes the hp-version of the Compact Discontinuous Galerkin (CDG) method for elliptic problems on polytopal meshes, establishing well-posedness and hp-a priori error estimates via a reduced primal formulation that uses lifting operators and fluxes. It introduces fast, unified algorithms to implement CDG, LDG, and BR2 within a single framework, enabling efficient assembly despite variable polynomial degrees. Numerical experiments show that CDG achieves a compact stiffness stencil with faster assembly times and competitive solving times, while highlighting the critical roles of flux directions and parameters (such as $\alpha_F$ and $\chi_F$) for stability on complex meshes. The results demonstrate practical benefits of hp-CDG on polytopal meshes and point to future work in hp-adaptivity and a posteriori error estimation for further efficiency gains.

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.

Figures (7)

• Figure 1: Schematical representation of the sets $\mathcal{F}_K^\mathrm{out}$ (purple on the left) and $\mathcal{N}_K^\mathrm{out}$ (light blue on the right) assuming $\alpha_F=1$ for $\beta_F = \boldsymbol{n}_F\cdot[1,0]^{\top} \geq 1$. Internal element case (left panel) and Dirichlet boundary element case (right panel).
• Figure 2: Computed errors in $L^2$ norm (a) and CDG norm (b) w.r.t. the mesh size $h$ and computed errors in $L^2$ norm and CDG norm (c) w.r.t. the polynomial order $p$.
• Figure 3: Comparison of computational times for the different methods (CDG, BR2, $\mathrm{LDG}_{\mathsf{w}}$, and $\mathrm{LDG}_{\mathsf{f}}$) and different polynomial orders ($p=2,4$). The considered times are divided in assembly time (a--b) and solving time (c--d).
• Figure 4: Comparison of sparsity patterns for the different methods (CDG, BR2, $\mathrm{LDG}_{\mathsf{w}}$, and $\mathrm{LDG}_{\mathsf{f}}$) with a mesh of 100 elements and polynomial degree $p=1$.
• Figure 5: Computed errors in $L^2(\Omega)$ norm (a-c) and CDG norm (d-f) w.r.t. the mesh size $h$ and for different mesh types. The errors have been computed both considering $\chi_F$ as in Assumption \ref{['asm:xiF']} (left), $\chi_F=1$ (center), and $\chi_F=0.1$ (right).

Theorems & Definitions (21)

• Remark 2.3: Relation with the LDG method
• Remark 2.4: Relation with the BR2 method
• Remark 2.5: The CDG2 version
• Remark 2.6: Absence of a stabilization term
• Lemma 2.7
• proof
• Lemma 2.8: Coercivity of $\mathcal{A}_h$
• proof
• Lemma 2.9: Continuity of $\mathcal{A}_h(\cdot, \cdot)$
• Lemma 3.1: A priori error bound