Two-scale homogenization of abstract linear time-dependent PDEs

Neukamm, Stefan and Varga, Mario and Waurick, Marcus (2021) Two-scale homogenization of abstract linear time-dependent PDEs. Asymptotic Analysis, 125 (3-4). pp. 247-287. ISSN 1875-8576 (https://doi.org/10.3233/asy-201654)

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Abstract

Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell’s equations, and the wave equation.

ORCID iDs

Neukamm, Stefan, Varga, Mario and Waurick, Marcus ORCID logoORCID: https://orcid.org/0000-0003-4498-3574;
CORE (COnnecting REpositories)