Fibre homogenisation |
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Authors: | Shane Cooper Marcus Waurick |
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Institution: | 1. Department of Mathematical Sciences, Durham University, Mountjoy Centre, Stockton Rd, Durham, DH1 3LE, England, United Kingdom;2. Department of Mathematics and Statics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow, G1 1XH, Scotland, United Kingdom |
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Abstract: | In this article we present a novel method for studying the asymptotic behaviour, with order-sharp error estimates, of the resolvents of parameter-dependent operator families. The method is applied to the study of differential equations with rapidly oscillating coefficients in the context of second-order PDE systems and the Maxwell system. This produces a non-standard homogenisation result that is characterised by ‘fibre-wise’ homogenisation of the related Floquet-Bloch PDEs. These fibre-homogenised resolvents are shown to be asymptotically equivalent to a whole class of operator families, including those obtained by standard homogenisation methods. |
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Keywords: | Resolvent estimates Fibre homogenisation Second-order PDE systems Maxwell's equations |
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