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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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Stable broken ${\boldsymbol H}(\boldsymbol {\operatorname {curl}})$ polynomial extensions and $p$-robust a posteriori error estimates by broken patchwise equilibration for the curl–curl problem
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by T. Chaumont-Frelet, A. Ern and M. Vohralík HTML | PDF
Math. Comp. 91 (2022), 37-74 Request permission

Abstract:

We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken $\boldsymbol H(\boldsymbol {\operatorname {curl}})$ space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl–curl problem discretized with Nédélec finite elements of arbitrary order. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and inexpensive. They are constructed by a broken patchwise equilibration which, in particular, does not produce a globally $\boldsymbol H(\boldsymbol {\operatorname {curl}})$-conforming flux. The equilibration is only related to edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The error estimates become guaranteed when the regularity pick-up constant is explicitly known. Numerical experiments illustrate the theoretical findings.
References
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Additional Information
  • T. Chaumont-Frelet
  • Affiliation: Inria, 2004 Route des Lucioles, 06902 Valbonne, France; and Laboratoire J.A. Dieudonné, Parc Valrose, 28 Avenue Valrose, 06108 Nice, France
  • MR Author ID: 999028
  • Email: theophile.chaumont@inria.fr
  • A. Ern
  • Affiliation: CERMICS, Ecole des Ponts, 77455 Marne-la-Valle, France; and Inria, 2 rue Simone Iff, 75589 Paris, France
  • MR Author ID: 349433
  • Email: alexandre.ern@enpc.fr
  • M. Vohralík
  • Affiliation: Inria, 2 rue Simone Iff, 75589 Paris, France; and CERMICS, Ecole des Ponts, 77455 Marne-la-Valle, France
  • ORCID: 0000-0002-8838-7689
  • Email: martin.vohralik@inria.fr
  • Received by editor(s): May 28, 2020
  • Received by editor(s) in revised form: May 3, 2021
  • Published electronically: September 28, 2021
  • Additional Notes: This project had received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 647134 GATIPOR)
  • © Copyright 2021 American Mathematical Society
  • Journal: Math. Comp. 91 (2022), 37-74
  • MSC (2020): Primary 65N30, 78M10, 65N15
  • DOI: https://doi.org/10.1090/mcom/3673
  • MathSciNet review: 4350532