Multiscale Asymptotic Analysis and Parallel Algorithm of Parabolic Equation in Composite Materials

An efficient parallel multiscale numerical algorithm is proposed for a parabolic equation with rapidly oscillating coefficients representing heat conduction in composite material with periodic configuration. Instead of following the classical multiscale asymptotic expansion method, the Fourier transform in time is first applied to obtain a set of complex-valued elliptic problems in frequency domain. The multiscale asymptotic analysis is presented and multiscale asymptotic solutions are obtained in frequency domain which can be solved in parallel essentially without data communications. The inverse Fourier transform will then recover the approximate solution in time domain. Convergence result is established. Finally, a novel parallel multiscale FEM algorithm is proposed and some numerical examples are reported.

In (1), the rapid spatial oscillations in the coefficients translate the periodic structure of the body which comes from assembling -scaled versions of the reference cell .This body has to be thought of as being made of a composite material.
An efficient analysis approach to such a problem induces a need for homogenization.The mathematical theory has been presented in a lot of works; see, for example, [3][4][5][6].Several efficient numerical methods have been proposed and analyzed; for instance, see [7][8][9].From these literatures , we found that the homogenized equation happens to be a system of the same type with constant coefficients.So the normal step-by-step time-marching algorithms such as the backward-Euler, Crank-Nicolson methods, or any higher order methods need to be applied to solve the homogenized equation step by step in time.Although these algorithms are effective in solving many practical problems, one of major drawbacks is that they are not easily parallelizable along the time axis, since they require the information of the solutions at previous time steps in order to advance to the next time step.
In this work, in order to propose a parallelizable multiscale numerical algorithm, we apply the Fourier transform and inverse Fourier transform to present the multiscale asymptotic analysis for (1).Numerical methods for timedependent problems based on the use of Fourier transform have been considered in [10,11], providing us with the starting point for using Fourier transform in multiscale asymptotic analysis.In [10,11], Douglas et al. introduced an efficient parallel method for solving wave equations in the space-time domain after taking the Fourier transforms.The parallel method had been further developed and extended to viscoelasticity, parabolic problems, and Navier-Stokes equations; see [12,13].Later Sheen et al. also applied Laplace transform to solve parabolic problems; see [14,15].In addition, the early study of homogenization by employing means of integral transformation is [16], where Fourier and Laplace transforms were applied to prove the weak convergence theorems of the homogenization procedure for the timedependent equations.
The main procedures in this paper are now briefly described.First, we take the Fourier transform with respect to the time variable in the parabolic equation and then present the multiscale asymptotic expansions of the solutions for the resulting equations in space-frequency domain at specific frequency points of interest, which are independent and may therefore be done in parallel.Finally, the approximate solution of parabolic equation in space-time domain is retrieved by the discrete inverse Fourier transform.It should be stated that our purpose of using Fourier transform is not just to prove the homogenization convergence theorems as done in [16], but to propose a parallelizable multiscale numerical algorithm.The new contributions presented in this study are the following.The strong convergence theorem of the second-order multiscale asymptotic solution in spacefrequency domain is derived where we carefully estimate the bounds associated with frequency variable  as it has a singularity for both zero and infinity in inverse Fourier transform.The convergence result for the approximate solution in time domain is obtained.A novel parallel multiscale FEM algorithm and numerical examples in 3D composite materials are given.
The remainder of this paper is organized as follows.In Section 2, the original parabolic equation is transformed to a set of complex-valued elliptic problems.The multiscale asymptotic expansions in space-frequency domain are then discussed.In Section 3, the multiscale truncation error estimates in space-frequency domain are derived under some assumptions.The approximate solution in space-time domain is inverted back by discrete inverse Fourier transform and the convergence result is proved in Section 4. Finally, a multiscale algorithm is proposed and numerical simulations are carried out to validate the presented method.
Throughout the paper, let  denote a positive constant independent of  and .All complex-valued functions are assumed to have values in the complex field C and standard notations for function spaces and their associated norms will be used in this paper.

Fourier Transform and Multiscale Asymptotic Expansions
Denote the Fourier transform of a function (⋅, ) by ρ(⋅, ), where In order to take the Fourier transform of (1), we first extend   and  by zero when  < 0 and then we transfer (1) into the following complex-valued elliptic problem in spacefrequency domain, where  is some fixed parameter in frequency domain.It is not difficult to check the existence and uniqueness of the problem (4) with an application of the Lax-Milgram lemma; see [17].
The method requires solving a finite set of problem (4) at specific frequency point .It is natural to treat them simultaneously because they are independent.Moreover, observe that ( 4) is an equation with rapidly oscillating coefficients in spatial domain.The direct accurate numerical computation of the solution is difficult especially in 3D problems because it would require huge computational cost.We will apply the method of multiscale asymptotic expansion.Let the derivative with respect to  be where  denotes the macroscopic variable and  describes the microscopic one.Two scales describe the model:  gives the position of a point in domain Ω while  gives its position in the reference cell .On expanding û in asymptotic expansion of the form where û (, , ) is -periodic in , the space-frequency (4) becomes On equating orders at ( −2 ), with the well-known theorem of the existence and uniqueness of the solutions in quotient space, we deduce that û0 (, , ) = û0 (, ) . ( Equating terms at ( −1 ), we have with the method of separation of variables and we can obtain where   () ( = 1, . . ., ) are the solutions of the following problems: At ( 0 ), we have On averaging (12) over , we get the homogenized equation where the homogenized coefficients are defined by Remark 2. Under the assumptions ( 1 )-( 2 ) of the coefficients for the parabolic equation, one can prove that the homogenized coefficients â are also symmetric and uniform elliptic for some constants  1 ,  2 > 0; see, for example, [3,5].Therefore the existence and uniqueness of the solution û0 for the homogenized (13) can be established.
Therefore, we can obtain the first-order and second-order multiscale asymptotic solution for the space-frequency (4) as Remark 4. It is worthwhile to note that all the cell problems are real valued, while the homogenized equation is complex valued.Therefore, when we do numerical computation, we need to repeatedly solve the homogenized problem at a set of specific frequency points but solve all the cell problems only once.

Multiscale Truncation Error Estimate in Frequency Domain
In order to derive the truncation error estimate for multiscale asymptotic solutions, we first discuss the regularity for the homogenized solution û0 .The key point is to derive precise estimates for the regularity bounds associated with  since the behavior of these bounds as  approaches zero and infinity is of critical importance in obtaining the multiscale truncation error estimate and when inverting the multiscale asymptotic solution back to time domain by the inverse Fourier transform.
For  = 1, by means of Theorem 1.2 in [6], we can also derive the estimate (43).Therefore the proof is complete.
Remark 7. It is worth noting that, for any Ω * ⊂⊂ Ω, it is well known that homogenized solution û0 ∈  3 (Ω * ) if f(⋅, ), and  ∈  1 (Ω).If û0 does not have the local  3 regularity near the boundary Ω, follow the way in [19]; we can use the asymptotic expansion in the interior subdomain and construct the boundary layer near Ω.

Inverse Fourier Transform
In this section, we will construct the approximate solution of the parabolic equation ( 1) in space-time domain by the inverse Fourier transform.Recall the formula of the inverse Fourier transform as provided that V(⋅, ) is a real function.Fix a sufficiently large  * > 0 such that   (⋅, ) is negligible for || >  * .Set Δ =  * /, where  ∈ Z + is a selected segment number for the interval (0,  * ) and the space-time solution   of (1) can be approximated by where  −1/2 = ( − 1/2)Δ,  = 1, . . ., .
In order to get an estimate for   −   , * ,Δ , let For the estimate of  2 , the following lemma was proved in [17].

Parallel Multiscale FEM Algorithm and Numerical Tests
In simulation, we will apply the following first-order and second-order difference quotients of some functions () given by where (  ), the set of elements with the node   ; (  ), is the number of elements of (  );  is the number of nodes on the element ;   are the nodes of the element ;   () are the Lagrangian shape functions;  ℎ denotes the FEM solution of ; [/  ]  (  ) is the value of /  at the node   associated with the element .
The following parallel multiscale FEM algorithm are presented to solve (1).
Step 1. Solve the first-order cell problems of   defined in (18) for  = 1, . . .,  and the second-order cell problems of   defined in (19) for ,  = 1, . . .,  in local fine grid of the periodic cell  by FEM.
Step 2. Compute the homogenized coefficients â defined in (14) in the local fine grid of the periodic cell  by numerical integration.
Step 4. Extend all the cell functions by periodicity from the local fine grid of the periodic cell  to the gird of the whole domain Ω.Then compute the multiscale asymptotic solutions given by (20) at frequency points  −1/2 ,  = 1, . . ., .
Step 5. Calculate the approximation solutions for any fixed time  =  * for the original equation (1) according to the formula (57).
Remark 10.Notice that cell problems are independent of the time variable  or the frequency variable , so they only need to be solved once. is in computing the time-dependent homogenized equation step by step.In our presented algorithm, we can simultaneously solve the homogenized equations at different frequent points, the main computation the algorithm is parallel.Also, notice that each step be done simultaneously.this respect, different from other multiscale algorithms for we called it parallel algorithm.
Remark 12.If some conditions of the geometric are satisfied (see [19]), then we can employ the homogeneous Dirichlet boundary conditions on the boundary  for the cell problems (18) and (19).Following the way in [19], one can also a similar convergence result to that of Theorem 6.Now we discuss the finite element computation for the homogenized Equation (13) in Ω.Let û0, = û0, + û0, and f = f + f , where the subscript  the real part of a function and the subscript  denotes the imaginary part.
Example.Consider the parabolic equation ( 1) in 3D composite materials, where whole domain Ω and the periodic cell  are illustrated in Figures 1(a the inclusion and other parts are the matrix.The parameters of the composite materials for two cases will be considered, where  =  −1 .Let () =  − and () = 0.
To assess the validity of the presented method, the exact solution   of the original parabolic equation ( 1) must be available.It is extremely difficult and even impossible to find the exact solution; we replace it by the numerical solution fully resolved by FEM in space and FDM in time using fine meshes (FD-FE method).It should be noted that more considerable computing time will be required to solve (1) using fully resolved FD-FE method when  is small, so we only present numerical results for  = 1/8.Here, the linear tetrahedral elements are employed for the semidiscrete problem of (1) using fine meshes, and the solution is computed by the Euler midpoint scheme.
Without confusion, let   denote the numerical solution for the original problem (1) using the fully resolved FD-FE method.Let   1 ,   2 denote the numerical solutions inverted back by the first-order and second-order multiscale asymptotic solutions in accordance with (57), respectively.Also, let  0 denote the numerical solution inverted back by the homogenized solutions according to the following formula: All the codes are implemented on parallel hierarchical grid (PHG), which is a toolbox for developing parallel adaptive finite element programs.PHG is currently under active development at State Key Laboratory of Scientific and Engineering Computing of CAS.We use linear tetrahedron elements to solve the related problems numerically.
The numerical results in the cross-section  = 0.5625 and at time  = 0.1 for Case 1 and Case 2 are displayed in Figures 2 and 3, respectively.Although the first-order and second-order multiscale asymptotic solutions have the same convergence rate as stated in Theorem 9, numerical results displayed in Figure 2 confirm that the second-order corrector terms are crucial in some cases.
The relative errors vary with the segment number  in the numerical integration formula (57) for homogenized method (HM), first-order multiscale method (1st MsM), and second-order multiscale method (2nd MsM) corresponding to  0 ,   1 , and   2 that are listed in Figure 4. We observe that the relative errors become stable after  = 80.Also, the relative errors evolution over time is listed in Figure 5.The results show that the accuracy of the proposed method especially the 2nd MsM is in good agreement with the fully resolved case.It is worthwhile to note that the relative errors do not grow rapidly as time increases.The costs required by the developed multiscale finite element method are not sensitive to the time variable but lie on the number of space-frequency equations which is determined by the segment number , and each spacefrequency equation is solved by multiscale finite element method with the computational cost listed in Table 1; for the fully resolved case, we need to compute /Δ steps and the cost for each time step is also listed in Table   This clearly demonstrates that the present algorithm is very effective and tremendous saving in computing time is achieved.

Table 1 :
Computational cost for Case 1 and Case 2.