  In the current work, a number of algorithms are developed and compared for
the numerical solution of periodic (quasi-static) linear elastic mechanical
boundary-value problems (BVPs) based on two different discretizations of
Fourier series. The first is standard and based on the trapezoidal
approximation of the Fourier mode integral, resulting in trapezoidal
discretization (TD) of the truncated Fourier series. Less standard is the
second discretization based on piecewise-constant approximation of the Fourier
mode integrand, yielding a piecewise-constant discretization (PCD) of this
series. Employing these, fixed-point algorithms are formulated via
Green-function preconditioning (GFP) and finite-difference discretization (of
differential operators; FDD). In particular, in the context of PCD, this
includes an algorithm based on the so-called "discrete Green operator" (DGO)
recently introduced by Eloh et al. (2019), which employs GFP, but not FDD. For
computational comparisons, the (classic) benchmark case of a cubic inclusion
embedded in a matrix (e.g., Suquet, 1997; Willot, 2015) is employed. Both
discontinuous and smooth transitions in elastic stiffness at the
matrix-inclusion (MI) interface are considered. In the context of both TD and
PCD, a number of GFP- and FDD-based algorithms are developed. Among these, one
based on so-called averaged-forward-backward-differencing (AFB) is shown to
result in the greatest improvement in convergence rate. As it turns out, AFB is
equivalent to the "rotated scheme" (R) of Willot (2015) in the context of TD.
In the context of PCD, comparison of the performance and convergence behavior
of AFB/R- and DGO-based algorithms shows that the former is more efficient than
the latter for larger phase contrasts.
