We study the convergences of several FFT-based schemes that are widely applied in computational homogenization for deriving effective coefficients, and the term "convergence" here means the limiting behaviors as spatial resolutions going to infinity. Those schemes include Moulinec-Suquent's scheme [Comput Method Appl M, 157 (1998), pp. 69-94], Willot's scheme [Comptes Rendus M\'{e}canique, 343 (2015), pp. 232-245], and the FEM scheme [Int J Numer Meth Eng, 109 (2017), pp. 1461-1489]. Under some reasonable assumptions, we prove that the effective coefficients obtained by those schemes are all convergent to the theoretical ones. Moreover, for the FEM scheme, we can present several convergence rate estimates under additional regularity assumptions.

翻译：我们研究了在计算同质化中广泛用于计算有效系数的若干基于FFT的计划的趋同,这里的“趋同”一词是指空间分辨率无穷的限制性行为。这些计划包括Mourinec-Suquent的计划[Comput方法M,157(1998),第69-94页]、Willot的计划[Comptes Rendus M\{e}canique,343(2015),第232-245页)和FEM计划[Int J Numer Meth Eng,109(2017),第146-1489页]。根据一些合理的假设,我们证明这些计划获得的有效系数都与理论系数一致。此外,对于FEM计划,我们可以在额外的正常假设下提出若干趋同率估计数。