Convergence of trigonometric and finite-difference discretization schemes for FFT-based computational micromechanics

This paper studies the convergences of several FFT-based discretization schemes that are widely applied in computational micromechanics for deriving effective coefficients, and "convergence" here means the limiting behaviors as spatial resolutions tending to infinity. Those schemes include Moulinec-Suquet's scheme, Willot's scheme and the FEM scheme. Under some reasonable assumptions, we prove that the effective coefficients obtained by those schemes all converge to the theoretical ones. Moreover, for the FEM scheme, we can present several convergence rate estimates under additional regularity assumptions.