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Spectral stochastic two‐scale convergence method for parabolic PDEs
Author(s) -
Jardak M.,
Navon I. M.
Publication year - 2011
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2997
Subject(s) - mathematics , convergence (economics) , scale (ratio) , partial differential equation , parabolic partial differential equation , gaussian , stochastic process , spectral method , stochastic differential equation , mathematical analysis , statistical physics , physics , statistics , quantum mechanics , economics , economic growth
Following the theory of two‐scale convergence method introduced by Nguetseng ( SIAM J. Math. Anal. 1989; 20 :608–623) and further developed by Allaire ( SIAM J. Math. Anal. 1992; 23 :1482–1518), we introduce the chaos two‐scale method as a spectral stochastic tool to tackle parabolic partial differential equations where the material properties are stochastic processes σ ε ( t, x , ω) of the form σ( t, x, t/ ε γ , x /ε, ω), oscillating in both space and time variables with different speeds. Periodicity with respect to the fast or local variables is assumed, and, stationary Gaussian material properties processes are considered. Copyright © 2010 John Wiley & Sons, Ltd.