Stochastic Homogenization of Linear Elliptic Equations: Higher-Order Error Estimates in Weak Norms Via Second-Order Correctors
Abstract
We are concerned with the homogenization of second-order linear elliptic equations with random coefficient fields. For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for the ensemble, we prove that when measured in weak spatial norms, the solution to the homogenized equation provides a higher-order approximation of the solution to the equation with oscillating coefficients. In the case of nonsymmetric coefficient fields, we provide a higher-order approximation (in weak spatial norms) of the solution to the equation with oscillating coefficients in terms of solutions to constant-coefficient equations. In both settings, we also provide optimal error estimates for the two-scale expansion truncated at second order. Our results rely on novel estimates on the second-order homogenization corrector, which we establish via sensitivity estimates for the second-order corrector and a large-scale $L^p$ theory for elliptic equations with random coefficients. Our results also cover the case of elliptic systems.
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Information & Authors
Information
Published In
SIAM Journal on Mathematical Analysis
Pages: 4658 - 4703
DOI: 10.1137/16M110229X
ISSN (online): 1095-7154
Copyright
© 2017, Society for Industrial and Applied Mathematics.
History
Submitted: 7 November 2016
Accepted: 15 May 2017
Published online: 21 November 2017
Keywords
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Funding Information
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : BE 5922/1.1
Funding Information
National Science Foundation https://doi.org/10.13039/100000001 : 1502731
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