To CG or to HDG: A Comparative Study
- 744 Downloads
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.
KeywordsHigh-order finite elements Spectral/hp elements Discontinuous Galerkin method Hybridization Domain decomposition
Unable to display preview. Download preview PDF.
- 7.Boost c++ libraries Google Scholar
- 16.Cockburn, B.: Discontinuous Galerkin methods for computational fluid dynamics. In: Borst, R., Stein, E., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3, pp. 91–123. Wiley, New York (2004) Google Scholar
- 32.Fraejis de Veubeke, B.M.: Displacement and equilibrium models in the finite element method. In: Zienkiewicz, O.C., Holister, G. (eds.) Stress Analysis, pp. 145–197. Wiley, New York (1977) Google Scholar
- 39.Koornwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: Theory and Applications of Special Functions. Academic Press, San Diego (1975) Google Scholar
- 42.Schwab, Ch.: p- and hp- Finite Element Methods: Theory and Applications to Solid and Fluid Mechanics. Oxford University Press, London (1999) Google Scholar
- 51.Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method: Solid Mechanics, vol. 2, 5th edn. Butterworth-Heinemann, Oxford (2000) Google Scholar