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Convergence of FFT‐based homogenization for strongly heterogeneous media
Author(s) -
Schneider Matti
Publication year - 2014
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3259
Subject(s) - mathematics , homogenization (climate) , discretization , a priori and a posteriori , nonlinear system , curl (programming language) , mathematical analysis , regularization (linguistics) , computer science , biodiversity , ecology , philosophy , physics , epistemology , quantum mechanics , artificial intelligence , biology , programming language
The FFT‐based homogenization method of Moulinec–Suquet has recently attracted attention because of its wide range of applicability and short computational time. In this article, we deduce an optimal a priori error estimate for the homogenization method of Moulinec–Suquet, which can be interpreted as a spectral collocation method. Such methods are well‐known to converge for sufficiently smooth coefficients. We extend this result to rough coefficients. More precisely, we prove convergence of the fields involved for Riemann‐integrable coercive coefficients without the need for an a priori regularization. We show that our L 2 estimates are optimal and extend to mildly nonlinear situations and L p estimates for p in the vicinity of 2. The results carry over to the case of scalar elliptic and curl − curl‐type equations, encountered, for instance, in stationary electromagnetism. Copyright © 2014 John Wiley & Sons, Ltd.