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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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On the robustness of multiscale hybrid-mixed methods
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by Diego Paredes, Frédéric Valentin and Henrique M. Versieux PDF
Math. Comp. 86 (2017), 525-548 Request permission

Abstract:

In this work we prove uniform convergence of the Multiscale Hybrid-Mixed (MHM for short) finite element method for second-order elliptic problems with rough periodic coefficients. The MHM method is shown to avoid resonance errors without adopting oversampling techniques. In particular, we establish that the discretization error for the primal variable in the broken $H^1$ and $L^2$ norms are $O(h + \varepsilon ^\delta )$ and $O(h^2 + h \varepsilon ^\delta )$, respectively, and for the dual variable it is $O(h + \varepsilon ^\delta )$ in the $H(\operatorname {div};\cdot )$ norm, where $0<\delta \leq 1/2$ (depending on regularity). Such results rely on sharpened asymptotic expansion error estimates for the elliptic models with prescribed Dirichlet, Neumann or mixed boundary conditions.
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Additional Information
  • Diego Paredes
  • Affiliation: Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso - IMA/ PUCV, Chile
  • Email: diego.paredes@ucv.cl
  • Frédéric Valentin
  • Affiliation: Department of Computational and Applied Mathematics, National Laboratory for Scientific Computing - LNCC, Av. Getúlio Vargas, 333, 25651-070 Petrópolis - RJ, Brazil
  • Email: valentin@lncc.br
  • Henrique M. Versieux
  • Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro - UFRJ, Rio de Janeiro - RJ, Brazil
  • Email: henrique@im.ufrj.br
  • Received by editor(s): October 15, 2014
  • Received by editor(s) in revised form: May 31, 2015, and August 21, 2015
  • Published electronically: March 28, 2016
  • Additional Notes: The first author was partially supported by CONICYT/Chile through FONDECYT project 11140699 and PCI-CNPq/Brazil.
    The second author was funded by CNPq/Brazil and CAPES/Brazil.
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 525-548
  • MSC (2010): Primary 35J15, 65N15, 65N30
  • DOI: https://doi.org/10.1090/mcom/3108
  • MathSciNet review: 3584539