Convergence of FFT-based homogenization for strongly heterogeneous media

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 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.