To CG or to HDG: A Comparative Study

Abstract

Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called static condensation in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement our HDG approach within a spectral/hp element framework so that comparisons can be done between HDG and the traditional CG approach.We demonstrate that the HDG approach generates a global trace space system for the unknown that although larger in rank than the traditional static condensation system in CG, has significantly smaller bandwidth at moderate polynomial orders. We show that if one ignores set-up costs, above approximately fourth-degree polynomial expansions on triangles and quadrilaterals the HDG method can be made to be as efficient as the CG approach, making it competitive for time-dependent problems even before taking into consideration other properties of DG schemes such as their superconvergence properties and their ability to handle hp-adaptivity.

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

“…1 for the range of polynomial orders often used in NUMA. We should also note here that there exist DG methods, such as Hybridizable Discontinuous Galerkin (HDG), that are computationally competitive with CG at moderate polynomial orders [23], although a thorough analysis of the computational cost of HDG is required.…”

Efficient computation of the minimum of shape quality measures on curvilinear finite elements

“…With recent developments in the field of high-order finite element methods [3], such as discontinuous Galerkin [4] or spectral [5,6] methods, there is a renewed interest for high-order (curved) mesh generation. The classical finite element method, a.k.a.…”

Optimization of a regularized distortion measure to generate curved high‐order unstructured tetrahedral meshes

“…It is well known that for problems with smooth solutions, the approximation obtained with high-order methods converges exponentially with the order of the approximating polynomial. More generally, high-order methods have been shown to deliver higher accuracy with a lower computational cost than low-order methods in many practical applications [6,7,8,9,10,11,12,13,14,15]. In addition, the accurate approximation of the domain geometry eliminates the spurious effects in the solution that can arise from a piecewise linear representation of the curved domain boundaries [16,17,18,19,20].…”