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To CG or to HDG: A Comparative Study in 3D

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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 made 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 direct 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.

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Acknowledgments

The work was supported by the Department of Energy (DOE NETL DE-EE004449) and under NSF OCI-1148291. DM acknowledges support from the EU FP7 project IDIHOM under Grant No. 265780. SJS additionally acknowledges Royal Academy of Engineering support under their research chair scheme.

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Authors and Affiliations

  1. Scientific Computing and Imaging (SCI) Institute, University of Utah, Salt Lake City, UT, USA

    Sergey Yakovlev

  2. Department of Aeronautics, Imperial College London, London, UK

    David Moxey

  3. School of Computing and Scientific Computing and Imaging (SCI) Institute, University of Utah, Salt Lake City, UT, USA

    Robert M. Kirby

  4. Department of Aeronautics, Imperial College London, London, UK

    Spencer J. Sherwin

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  1. Sergey Yakovlev
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  2. David Moxey
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  4. Spencer J. Sherwin
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Correspondence to Sergey Yakovlev.

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Yakovlev, S., Moxey, D., Kirby, R.M. et al. To CG or to HDG: A Comparative Study in 3D. J Sci Comput 67, 192–220 (2016). https://doi.org/10.1007/s10915-015-0076-6

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  • DOI: https://doi.org/10.1007/s10915-015-0076-6

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