Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems

Patrick Henning; Patrick Henning

doi:10.3934/nhm.2012.7.503

Networks and Heterogeneous Media

2012, Volume 7, Issue 3: 503-524. doi: 10.3934/nhm.2012.7.503

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Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems

Patrick Henning ^{1
,}

1.
Institut für Numerische und Angewandte Mathematik, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster

Received: 01 November 2011 Revised: 01 May 2012
Primary: 35J15, 65G99, 74Q15; Secondary: 65N30.

In this work, we are concerned with the convergence of the multiscale finite element method (MsFEM) for elliptic homogenization problems, where we do not assume a certain periodic or stochastic structure, but an averaging assumption which in particular covers periodic and ergodic stochastic coefficients. We also give a result on the convergence in the case of an arbitrary coupling between grid size $H$ and a parameter $\epsilon$. $\epsilon$ is an indicator for the size of the fine scale which converges to zero. The findings of this work are based on the homogenization results obtained in [B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization, Netw. Heterog. Media, 6 (4), 2011].
- Elliptic homogenization,
- MsFEM,
- convergence,
- multiscale methods
Citation: Patrick Henning. Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems[J]. Networks and Heterogeneous Media, 2012, 7(3): 503-524. doi: 10.3934/nhm.2012.7.503

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Abstract

In this work, we are concerned with the convergence of the multiscale finite element method (MsFEM) for elliptic homogenization problems, where we do not assume a certain periodic or stochastic structure, but an averaging assumption which in particular covers periodic and ergodic stochastic coefficients. We also give a result on the convergence in the case of an arbitrary coupling between grid size $H$ and a parameter $\epsilon$. $\epsilon$ is an indicator for the size of the fine scale which converges to zero. The findings of this work are based on the homogenization results obtained in [B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization, Netw. Heterog. Media, 6 (4), 2011].

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