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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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$hp$-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements
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by Andrea Cangiani, Zhaonan Dong and Emmanuil H. Georgoulis HTML | PDF
Math. Comp. 91 (2022), 1-35 Request permission

Abstract:

We extend the applicability of the popular interior penalty discontinuous Galerkin method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for curved element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical trace and Markov-type inverse estimates to arbitrary element shapes. A further new $H^1-L_2$-type inverse estimate on essentially arbitrary element shapes enables the proof of inf-sup stability of the method in a streamline-diffusion-like norm. These inverse estimates may be of independent interest. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach.
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Additional Information
  • Andrea Cangiani
  • Affiliation: SISSA, via Bonomea 265, 34136 Trieste, Italy
  • MR Author ID: 757643
  • Email: Andrea.Cangiani@sissa.it
  • Zhaonan Dong
  • Affiliation: Inria, 2 rue Simone Iff, 75589 Paris, France; and CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée 2, France
  • MR Author ID: 1163337
  • ORCID: 0000-0003-4083-6593
  • Email: Zhaonan.Dong@inria.fr
  • Emmanuil H. Georgoulis
  • Affiliation: School of Mathematics and Actuarial Science, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom; Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou 15780, Greece; and IACM-FORTH, Crete, Greece
  • MR Author ID: 750860
  • Email: Emmanuil.Georgoulis@le.ac.uk
  • Received by editor(s): June 3, 2019
  • Received by editor(s) in revised form: May 20, 2020, December 22, 2020, April 19, 2021, and April 30, 2021
  • Published electronically: August 5, 2021
  • Additional Notes: The first author was supported by the MRC (MR/T017988/1), the second author by IACM-FORTH, Greece, and the third author by The Leverhulme Trust (RPG-2015- 306). This work was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant” (Proj. no. 3270)
  • © Copyright 2021 American Mathematical Society
  • Journal: Math. Comp. 91 (2022), 1-35
  • MSC (2020): Primary 65J10, 65M60
  • DOI: https://doi.org/10.1090/mcom/3667
  • MathSciNet review: 4350531