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Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems


Authors: Assyr Abdulle and Gilles Vilmart
Journal: Math. Comp. 83 (2014), 513-536
MSC (2010): Primary 65N30, 65M60, 74D10, 74Q05
DOI: https://doi.org/10.1090/S0025-5718-2013-02758-5
Published electronically: August 30, 2013
MathSciNet review: 3143682
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Abstract: An analysis of the finite element heterogeneous multiscale method for a class of quasilinear elliptic homogenization problems of nonmonotone type is proposed. We obtain optimal convergence results for dimension $ d\leq 3$. Our results, which also take into account the microscale discretization, are valid for both simplicial and quadrilateral finite elements. Optimal a priori error estimates are obtained for the $ H^1$ and $ L^2$ norms, error bounds similar to those for linear elliptic problems are derived for the resonance error. Uniqueness of a numerical solution is proved. Moreover, the Newton method used to compute the solution is shown to converge. Numerical experiments confirm the theoretical convergence rates and illustrate the behavior of the numerical method.


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Additional Information

Assyr Abdulle
Affiliation: ANMC, Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Email: Assyr.Abdulle@epfl.ch

Gilles Vilmart
Affiliation: ANMC, Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Address at time of publication: École Normale Supérieure de Cachan, Antenne de Bretagne, INRIA Rennes, IRMAR, CNRS, UEB, av. Robert Schuman, F-35170 Bruz, France
Email: Gilles.Vilmart@bretagne.ens-cachan.fr

DOI: https://doi.org/10.1090/S0025-5718-2013-02758-5
Received by editor(s): February 18, 2011
Received by editor(s) in revised form: March 26, 2012, and July 3, 2012
Published electronically: August 30, 2013
Additional Notes: The work of the first author was supported in part by the Swiss National Science Foundation under Grant 200021 134716/1
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.