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On the robustness of multiscale hybrid-mixed methods


Authors: Diego Paredes, Frédéric Valentin and Henrique M. Versieux
Journal: Math. Comp. 86 (2017), 525-548
MSC (2010): Primary 35J15, 65N15, 65N30
DOI: https://doi.org/10.1090/mcom/3108
Published electronically: March 28, 2016
MathSciNet review: 3584539
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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(\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

DOI: https://doi.org/10.1090/mcom/3108
Keywords: Asymptotic expansion, homogenization, elliptic equation, multiscale method, hybridization, finite element
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.
Article copyright: © Copyright 2016 American Mathematical Society