Two-scale homogenization of abstract linear time-dependent PDEs

Publikationstyp
Journal Article
Date Issued
2021
Sprache
English
Author(s)
Neukamm, Stefan  
Varga, Mario  
Waurick, Marcus  
Institut
Mathematik E-10  
TORE-URI
http://hdl.handle.net/11420/10810
Journal
Asymptotic analysis  
Volume
125
Issue
3-4
Start Page
247
End Page
287
Citation
Asymptotic Analysis 125 (3-4): 247-287 (2021)
Publisher DOI
10.3233/ASY-201654
Scopus ID
2-s2.0-85117919952
ArXiv ID
1905.02945
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.
Subjects
abstract evolutionary equations
Maxwell's equations
Periodic and stochastic homogenization
unfolding