Convergence of FFT‐based homogenization for strongly heterogeneous media

Abstract

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 L2 estimates are optimal and extend to mildly nonlinear situations and Lp 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.