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

Abdulle, Assyr; Vilmart, Gilles

doi:10.1090/S0025-5718-2013-02758-5

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems
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by Assyr Abdulle and Gilles Vilmart;

Math. Comp. 83 (2014), 513-536

DOI: https://doi.org/10.1090/S0025-5718-2013-02758-5

Published electronically: August 30, 2013

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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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