To CG or to HDG: a comparative study in 3D
J. Sci. Comp., vol. 67, pp. 192-220 (2016)
@article{yakovlev-2016,
title = {{To CG or to HDG: a comparative study in 3D}},
author = {Yakovlev, S. and Moxey, D. and Sherwin, S. J. and Kirby, R. M.},
journal = jsc,
volume = {67},
number = {1},
pages = {{192-220}},
year = {2016},
abstract = {Since the inception of discontinuous Galerkin (DG) methods for elliptic problems, there has existed a question of whether DG methods can be made more computationally efficient than continuous Galerkin (CG) methods. Fewer degrees of freedom, approximation properties for elliptic problems together with the number of optimization techniques, such as static condensation, available within CG framework make it challenging for DG methods to be competitive until recently. However, with the introduction of a static-condensation-amenable DG method -- the hybridizable discontinuous Galerkin (HDG) method -- it has become possible to perform a realistic comparison of CG and HDG methods when applied to elliptic problems. In this work, we extend upon an earlier 2D comparative study, providing numerical results and discussion of the CG and HDG method performance in three dimensions. The comparison categories covered include steady-state elliptic and time-dependent parabolic problems, various element types and serial and parallel performance. The postprocessing technique, which allows for superconvergence in the HDG case, is also discussed. Depending on the linear system solver used and the type of the problem (steady-state vs time-dependent) in question the HDG method either outperforms or demonstrates a comparable performance when compared with the CG method. The HDG method however falls behind performance-wise when the iterative solver is used, which indicates the need for an effective preconditioning strategy for the method.},
url = {https://davidmoxey.uk/assets/pubs/2015-hdg.pdf},
doi = {10.1007/s10915-015-0076-6}
}
Discontinuous Galerkin methods carry more degrees of freedom than continuous ones, which long made them hard to justify on cost for elliptic problems. The hybridizable discontinuous Galerkin method changes that comparison, since it can be statically condensed like a continuous discretisation. This paper extends an earlier two-dimensional study into three dimensions, across steady elliptic and time-dependent parabolic problems, several element types, and both serial and parallel runs.
Abstract
Since the inception of discontinuous Galerkin (DG) methods for elliptic problems, there has existed a question of whether DG methods can be made more computationally efficient than continuous Galerkin (CG) methods. Fewer degrees of freedom, approximation properties for elliptic problems together with the number of optimization techniques, such as static condensation, available within CG framework make it challenging for DG methods to be competitive until recently. However, with the introduction of a static-condensation-amenable DG method – the hybridizable discontinuous Galerkin (HDG) method – it has become possible to perform a realistic comparison of CG and HDG methods when applied to elliptic problems. In this work, we extend upon an earlier 2D comparative study, providing numerical results and discussion of the CG and HDG method performance in three dimensions. The comparison categories covered include steady-state elliptic and time-dependent parabolic problems, various element types and serial and parallel performance. The postprocessing technique, which allows for superconvergence in the HDG case, is also discussed. Depending on the linear system solver used and the type of the problem (steady-state vs time-dependent) in question the HDG method either outperforms or demonstrates a comparable performance when compared with the CG method. The HDG method however falls behind performance-wise when the iterative solver is used, which indicates the need for an effective preconditioning strategy for the method.