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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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A Hybrid High-Order Method for Highly Oscillatory Elliptic Problems

Matteo Cicuttin
  • Université Paris-Est, CERMICS (ENPC), 6–8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2; and Inria Paris, 2 rue Simone Iff, 75012 Paris, France
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Alexandre Ern
  • Université Paris-Est, CERMICS (ENPC), 6–8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2; and Inria Paris, 2 rue Simone Iff, 75012 Paris, France
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Simon Lemaire
  • Corresponding author
  • École Polytechnique Fédérale de Lausanne, FSB-MATH-ANMC, Station 8, 1015 Lausanne, Switzerland; and Inria Lille – Nord Europe, 40 avenue Halley, 59650 Villeneuve d’Ascq, France
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-06-19 | DOI: https://doi.org/10.1515/cmam-2018-0013

Abstract

We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomials attached to the faces and cells of a coarse mesh; those attached to the cells can be eliminated locally using static condensation. The main building ingredient is a reconstruction operator, local to each coarse cell, that maps onto a fine-scale space spanned by oscillatory basis functions. The present HHO method generalizes the ideas of some existing multiscale approaches, while providing the first complete analysis on general meshes. It also improves on those methods, taking advantage of the flexibility granted by the HHO framework. The method handles arbitrary orders of approximation k0. For face unknowns that are polynomials of degree k, we devise two versions of the method, depending on the polynomial degree (k-1) or k of the cell unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form (ε12+Hk+1+(εH)12), and we illustrate our theoretical findings on some test-cases.

Keywords: General Meshes; HHO Methods; Multiscale Methods; Highly Oscillatory Problems

MSC 2010: 65N30; 65N08; 76R50

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About the article

Received: 2018-02-15

Accepted: 2018-06-01

Published Online: 2018-06-19


Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0013.

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