Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors

Journal article

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.

Authors: Bella, P; Fehrman, B; Fischer, J; Otto, F

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Mathematical Analysis
• Volume: 49
• Issue: 6
• Pages: 4658-4703
• Publication date: 2017-11-21
• Acceptance date: 2017-05-15
• DOI: 10.1137/16M110229X
• EISSN: 1095-7154
• ISSN: 0036-1410
• Keywords: elliptic equation; second-order corrector; homogenization error; two-scale expansion; stochastic homogenization