Two-scale homogenization of abstract linear time-dependent PDEs

Stefan Neukamm, Mario Varga, Marcus Waurick

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2 Citations (Scopus)
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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.
Original languageEnglish
Pages (from-to)247-287
Number of pages41
JournalAsymptotic Analysis
Volume125
Issue number3-4
DOIs
Publication statusPublished - 6 Oct 2021

Keywords

  • periodic and stochastic homogenization
  • unfolding
  • abstract evolutionary equations
  • Maxwell's equations

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