1 Introduction and preliminaries
In an inhomogeneous material that obeys the linear elasticity law, the fourthorder stiffness tensor
varies at different . The central goal of computational homogenization methods is determining an effective constitute law, which may serve as a surrogate model for downstream applications Milton (2002); Zohdi and Wriggers (2008).Let be a Representative Volume Element (RVE) where through the whole paper, and be the fourthorder stiffness tensor for any . We denote by the closure of smooth periodic functions with respect to ordinary norm Cioranescu and Donato (1999), and define accordingly. The notation will be replaced as or
for vectorvalued functions, while
(, ) should be always understood as norm (norm, norm) of on the domain regardless of whether is vectorvalued or not. For a discrete set , let be the linear space of discrete functions which take inputs in and return values of , where may be , , , and (the set of symmetric matrices). The double dot operator “” and the single dot operator “” are defined conventionally for any order tensors (including vectors and matrices), and note that now the matrixvector product is rewritten as . A popular method to derive the effective coefficients of is solving a periodic boundary value problem: find with such that for any(1) 
where stands for , belongs to and . The effective stiffness tensor can be obtained by
(2) 
via choosing different .
The starting point of FFTbased homogenization is rewriting the above variational form into the LippmannSchwinger equation. Taking as a linear elastic reference medium, which is a constant fourthorder tensor satisfying certain coercive conditions, we can introduce a Green function such that for any , the convolution is the solution to the variational form
By setting and taking the Fourier series expansion of as , we can derive that for
where is the identity matrix and is the Kronecker product. Moreover, it is easy to show that eq. 1 is equivalent to
where and vanishes due to that is a constant matrix. If taking and , we can immediately read the above equation as
(3) 
the socalled LippmannSchwinger equation Lippmann and Schwinger (1950). The alternative form of eq. 3 in the Fourier space is
Specifically, when , we can derive a closed form of as
Now it comes to discretization schemes. For simplicity, we equally split into parts in every dimension. For any dimensional index with , we denote by
an element in the language of Finite Element Methods (FEM), which could be interpreted as a pixel (for 2D problems) or voxel (for 3D problems). Correspondingly, the frequency domain is denoted by
. We always postulate that an imagelike (i.e., is constant in every ) rather that the ground truth is provided, and introduce a notation where is the geometric center of . The basic scheme of FFTbased homogenization was proposed by Moulinec and Suquent in Moulinec and Suquet (1995, 1998), and can be stated as algorithm 1. Depending on the type of a function’s domain, the Fourier coefficients are defined differently, i.e. if , then is determined by the formulaand if , then satisfies
Note that if is labeled with a superscript “”, the function should be understood as a discrete one with as its domain.
MoulinecSuquent’s scheme (we also call it the basic scheme) gains considerable popularity. The reasons can be summarized as follows: 1) microstructures in heterogeneous materials may be rather complex, which causes that generating high quality meshes dominates overall computational overheads Hughes et al. (2005); 2) the primary information of microstructures is usually provided by modern digital volume imaging techniques Larson et al. (2002); Poulsen (2004); Landis and Keane (2010), and we may not be capable to retrieve the original geometric descriptions and to perform a preprocessing for FEMs; 3) the easy implementation of those schemes and highly optimized FFT packages (e.g., Intel®MKL, FFTW Frigo and Johnson (2005)) secure the global efficiency; 4) generally, it is the effective coefficients we are more interested in rather than the local displacement or the local stress, and unfitted structured meshes may have less influence on desired comparing to and . Moreover, those methods are not limited in linear elasticity, versatile applications such as nonlinear elasticity Moulinec and Suquet (1998); Michel et al. (2001), piezoelectricity Brenner (2009), damage and fracture mechanics Zhu and Yvonnet (2015); Chai et al. (2020), polycrystalline materials Segurado et al. (2018) can be found in literature. We strongly recommend a recent review article Schneider (2021) as an exhaustive reference for historical developments and the current state of the art of FFTbase homogenization methods.
Willot’s scheme is another popular discretization method of the LippmannSchwinger equation Willot (2015). The main idea in Willot’s scheme is replacing the gradient operator in the Fourier space with
(4) 
that is derived from a finite difference stencil. If is isotropic with the Lamé coefficients , the closed form of is
and this is the major difference from the basic scheme. Take
and Willot’s scheme states as algorithm 2.
It has been numerically demonstrated that convergences of MoulinecSuquent’s scheme will deteriorate as the contrast ratio of grows, and an extreme example shows that the basic scheme indeed fails to converge for a porous material Schneider et al. (2016), while Willot’s scheme stays a stable convergence history in high contrast settings Willot (2015). Meanwhile, computational overheads of the basic and Willot’s schemes are almost same. Those advantages make Willot’s scheme become another standard method in discretizing the LippmannSchwinger equation eq. 3.
Interestingly, it is pointed in Schneider et al. (2017) that Willot’s scheme is related to a reduced integration variational problem: find with such that for any ,
where is the trilinear Lagrange finite element space of . The reduced integration technique is significantly efficient in constructing stiffness matrices comparing to full integration, while the latter needs evaluations on all the Gaussian quadrature points ( for trilinear elements) in each element. However, due to that the stiffness matrix derived by reduced integration may be singular, there exist socalled “hourglassing” or “zeroenergy” modes Koh and Kikuchi (1987); Pugh et al. (1978). For example when is an even integer, it is easy to find nonzero such that for all , and this is closely related with the frequency , which also implies there does not exist a positive constant independent of satisfying
In Schneider et al. (2017), Schneider et al. proposed a novel scheme by replacing reduced integration in Willot’s scheme with full integration, i.e., find with such that for any ,
where is the Gaussian quadrature point on the element determined by symbols of . The FFT technique is utilized in solving reference problem: find with such that for any ,
where . The motivation is converting into
via the Discrete Fourier Transform (DFT), where
is the symmetric Kronecker product and . Combining the symmetric relations of , we can hence derive an explicit formula of in the Fourier space as for , where and . The scheme is summarized in algorithm 3, we called it “the FEM scheme” for it is essentially a FEM with the FFT acting as a preconditioner Saad (2003).Currently, most researches on FFTbased homogenization are focused on developing fast algorithms to accelerate convergences of iterating processes and designing schemes to stabilize performances on high/infinite contrast materials Eyre and Milton (1999); Michel et al. (2001); Zeman et al. (2010); Eloh et al. (2019); Schneider (2020), while few of them consider the convergences of those methods with respect to the spatial resolution , which could be viewed as an analogy of estimate theories in FEMs Ciarlet (1991); Brenner and Scott (2008). To our knowledge, the work by Schneider Schneider (2015) is only published one which seriously discusses such a question. In his article, a priori error estimate of MoulinecSuquet’s scheme is formulated with the trigonometric interpolation operator Zygmund (1968). However, because of involving pointwise evaluations, the trigonometric interpolation operator is to some certain “incompatible” with Lebesgue integrable functions.
Since the main objective in this article is studying convergences with respect to spatial resolutions, it is necessary to clarify the relation between the provided information and the ground truth . As mentioned previously, the ground truth is hidden from the scheme part, we hence cannot hold an optimistic anticipation on convergence rates like immersed FEMs Li et al. (2003); Chen et al. (2009) or unfitted FEMs Burman et al. (2014); Huang et al. (2017); Chen et al. (2021). The assumptions for and is elucidated as follows.
Assumption A
The RVE domain consists of Lipschitz subdomains that are labeled by , and only takes a constant tensor in each subdomain .
Assumption B
The constant tensors satisfy properties: (symmetricity) for any ,
(coercivity) there exist positive constants and such that for any and ,
Assumption C
For any , if then , else will be arbitrarily chosen from .
In some cases, we will use the notation to represent a positive constant which depends on .
Here are the contributions and structure of this article:

In section 3, we prove the convergence of the effective coefficients obtained by Willot’s scheme.

In section 4, by assuming some suitable regularities and combining several priori estimates in FEM theories, we present convergence rates of the solution and the effective coefficients derived by the FEM scheme.
2 Convergence of MoulinecSuquent’s scheme
Let be a complex trigonometric polynomial space
and be the orthogonal projection onto Conway (1990). Take where the set is the cube with as its center and as its edge length. In the following analysis, for a function , the convolution should be understood as applying to the periodic extension of , which induces that
where
(5) 
It is easy to see that for , and we can hence define an operator that plays an essential role in our analysis:
Definition 1.
For any ,
where and
We can show the following facts of :
Proposition 1.
The following statements hold true for

Let , then , and if then ;

Let , then and

for a series of with , then .
Proof.
From the definition of , it follows that . Moreover, if , we have
Then holds by taking into the definition of .
From the fact for , we have