Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors
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 no...
| Main Authors: | , , , |
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| Format: | Journal article |
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Society for Industrial and Applied Mathematics
2017
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| _version_ | 1797097602905276416 |
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| author | Bella, P Fehrman, B Fischer, J Otto, F |
| author_facet | Bella, P Fehrman, B Fischer, J Otto, F |
| author_sort | Bella, P |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T04:57:51Z |
| format | Journal article |
| id | oxford-uuid:d736a16b-8ecb-4949-a5ab-51756a7028f3 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T04:57:51Z |
| publishDate | 2017 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:d736a16b-8ecb-4949-a5ab-51756a7028f32022-03-27T08:39:32ZStochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctorsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d736a16b-8ecb-4949-a5ab-51756a7028f3Symplectic Elements at OxfordSociety for Industrial and Applied Mathematics2017Bella, PFehrman, BFischer, JOtto, FWe 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. |
| spellingShingle | Bella, P Fehrman, B Fischer, J Otto, F Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors |
| title | Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors |
| title_full | Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors |
| title_fullStr | Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors |
| title_full_unstemmed | Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors |
| title_short | Stochastic homogenization of linear elliptic equations: higher-order error estimates in weak norms via second-order correctors |
| title_sort | stochastic homogenization of linear elliptic equations higher order error estimates in weak norms via second order correctors |
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