# Analysis of the heterogeneous multiscale method for elliptic homogenization problems

By Weinan E, Pingbing Ming, Pingwen Zhang

## Abstract

A comprehensive analysis is presented for the heterogeneous multiscale method (HMM for short) applied to various elliptic homogenization problems. These problems can be either linear or nonlinear, with deterministic or random coefficients. In most cases considered, optimal estimates are proved for the error between the HMM solutions and the homogenized solutions. Strategies for retrieving the microstructural information from the HMM solutions are discussed and analyzed.

## 1. Introduction and main results

### 1.1. General methodology

Consider the classical elliptic problem

Here is a small parameter that signifies explicitly the multiscale nature of the coefficient . Several classical multiscale methodologies have been developed for the numerical solution of this elliptic problem, the most well known among which is the multigrid technique Reference8. These classical multiscale methods are designed to resolve the details of the fine scale problem Equation1.1 and are applicable for general problems, i.e., no special assumptions are required for the coefficient . In contrast modern multiscale methods are designed specifically for recovering partial information about at a sublinear cost, i.e., the total cost grows sublinearly with the cost of solving the fine scale problem Reference18. This is only possible by exploring the special features that might have, such as scale separation. The simplest example is when

where can either be periodic in , in which case we assume the period to be , or random but stationary under shifts in , for each fixed . In both cases, it has been shown that Reference5Reference36

where is the solution of a homogenized equation:

The homogenized coefficient can be obtained from the solutions of the so-called cell problem. In general, there are no explicit formulas for , except in one dimension.

Several numerical methods have been developed to deal specifically with the case when is periodic in . References Reference3Reference4Reference7 propose to solve the homogenized equations as well as the equations for the correctors. Schwab et al. Reference29Reference38 use multiscale test functions of the form where is periodic in to extract the leading order behavior of , extending an idea that was used analytically in the work of Reference2Reference15Reference34Reference44 for the homogenization problems. These methods have the feature that their cost is independent of , hence sublinear as , but so far they are restricted to the periodic homogenization problem. An alternative proposal for more general problems but with much higher cost is found in Reference20Reference25.

### 1.2. Heterogeneous multiscale method

HMM Reference16Reference17Reference18 is a general methodology for designing sublinear algorithms by exploiting scale separation and other special features of the problem. It consists of two components: selection of a macroscopic solver and estimating the missing macroscale data by solving locally the fine scale problem.

For Equation1.1 the macroscopic solver can be chosen as a conventional finite element method on a triangulation of element size which should resolve the macroscale features of . The missing data is the effective stiffness matrix at this scale. This stiffness matrix can be estimated as follows. Assuming that the effective coefficient at this scale is , if we knew explicitly, we could have evaluated the quadratic form

by numerical quadrature: For any , the finite element space,

where and are the quadrature points and weights in , is the volume of . In the absence of explicit knowledge of , we approximate by solving the problem:

where is a cube of size centered at , and is the linear approximation of at . We then let

Equation1.5 and Equation1.7 together give the needed approximate stiffness matrix at the scale . For convenience, we will define the corresponding bilinear form: For any

where is defined for in the same way that in Equation1.6 was defined for .

In order to reduce the effect of the imposed boundary condition on , we may replace Equation1.7 by

where . For example, we may choose . In Equation1.6, we used the Dirichlet boundary condition. Other boundary conditions are possible, such as Neumann and periodic boundary conditions. In the case when and is periodic in , one can take to be , i.e., and use the boundary condition that is periodic on .

So far the algorithm is completely general. The savings compared with solving the full fine scale problem comes from the fact that we can choose to be smaller than . The size of is determined by many factors, including the accuracy and cost requirement, the degree of scale separation, and the microstructure in . One purpose for the error estimates that we present below is to give guidelines on how to select . As mentioned already, if and is periodic in , we can simply choose to be , i.e., . If is random, then should be a few times larger than the local correlation length of . In the former case, the total cost is independent of . In the latter case, the total cost depends only weakly on (see Reference31).

The final problem is to solve

To summarize, HMM has the following features:

(1)

It gives a framework that allows us to maximally take advantage of the special features of the problem such as scale separation. For periodic homogenization problems, the cost of HMM is comparable to the special techniques discussed in Reference3Reference7Reference29Reference35. However HMM is also applicable for random problems and for problems whose coefficient does not has the structure of . For problems without scale separation, we may consider other possible special features of the problem such as local self-similarity, which is considered in Reference19.

(2)

For problems without any special features, HMM becomes a fine scale solver by choosing an that resolves the fine scales and letting .

Some related ideas exist in the literature. Durlofsky Reference14 proposed an up-scaling method, which directly solves some local problems for obtaining the effective coefficients Reference33Reference40Reference41. Oden and Vemaganti Reference35 proposed a method that aims at recovering the oscillations in locally by solving a local problem with some given approximation to the macroscopic state as the boundary condition. This idea is sometimes used in HMM to recover the microstructural information. Other numerical methods that use local microscale solvers to help extract macroscale behavior are found in Reference26Reference27.

The numerical performance of HMM including comparison with other methods is discussed in Reference31.

This paper will focus on the analysis of HMM. We will estimate the error between the numerical solutions of HMM and the solutions of Equation1.4. We will also discuss how to construct better approximations of from the HMM solutions. Our basic strategy is as follows. First we will prove a general statement that the error between the HMM solution and the solution of Equation1.4 is controlled by the standard error in the macroscale solver plus a new term, called , due to the error in estimating the stiffness matrix. We then estimate . This second part is only done for either periodic or random homogenization problems, since concrete results are only possible if the behavior of is well understood. We believe that this overall strategy will be useful for analyzing other multiscale methods.

We will always assume that is smooth, symmetric and uniformly elliptic:

for some . We will use the summation convention and standard notation for Sobolev spaces (see Reference1). We will use to denote the absolute value of a scalar quantity, the Euclidean norm of a vector and the volume of a set . For the quadrature formula Equation1.5, we will assume the following accuracy conditions for th-order numerical quadrature scheme Reference11:

Here , for . For , we assume the above formula to be exact for .

### 1.3. Main results

Our main results for the linear problem are as follows.

#### Theorem 1.1

Denote by and the solution of Equation1.4 and the solution, respectively. Let

where is the Euclidean norm. If is sufficiently smooth and Equation1.10 holds, then there exists a constant independent of and , such that

If there exits a constant such that , then there exists a constant such that for all ,

At this stage, no assumption on the form of is necessary. can be the solution of an arbitrary macroscopic equation with the same right-hand side as in Equation1.1. Of course for to converge to , i.e., , must be chosen as the solution of the homogenized equation, which we now assume exists. To obtain quantitative estimates on , we must restrict ourselves to more specific cases.

#### Theorem 1.2

For the periodic homogenization problem, we have

In the first case, we replace the boundary condition in Equation1.6 by a periodic boundary condition: is periodic with period . For the second result we do not need to assume that the period of is a cube: In fact it can be of arbitrary shape as long as its translation tiles up the whole space.

Another important case for which a specific estimate on can be obtained is the random homogenization. In this case, using results in Reference43, we have

#### Theorem 1.3

For the random homogenization problem, assuming that in the Appendix holds (see Reference43), we have

where

for any . By choosing small, can be arbitrarily close to .

The probabilistic set-up will be given in the Appendix. To prove this result, we assume that Equation1.8a is used with .

### 1.4. Recovering the microstructural information

In many applications, the microstructure information in is very important. by itself does not give this information. However, this information can be recovered using a simple post-processing technique. For the general case, having , one can obtain locally the microstructural information using an idea in Reference35. Assume that we are interested in recovering and only in the subdomain . Consider the following auxiliary problem:

where satisfies and . We then have

#### Theorem 1.4

There exists a constant such that

For the random problem, the last term was estimated in Reference43.

A much simpler procedure exists for the periodic homogenization problem. Consider the case when and choose , where is the barycenter of . Here we have assumed that the quadrature point is at .

Let be defined piecewise as follows:

(1)

, where is the solution of Equation1.6 with the boundary condition that is periodic with period and .

(2)

is periodic with period .

For this case, we can prove

#### Theorem 1.5

Let be defined as above. Then

Similar results with some modification hold for nonlinear problems. The details are given in §5.

### 1.5. Some technical background

In this subsection, we will list some general results that will be frequently referred to later on.

Given a triangulation , it is called regular if there is a constant such that

and if the quantity

approaches zero, where is the diameter of and is the diameter of the largest ball inscribed in . satisfies an inverse assumption if there exists a constant such that

A regular family of triangulation of satisfying the inverse assumption is called quasi-uniform.

The following interpolation result for the Lagrange finite element is adapted from Reference10. Here and in what follows, for any is understood in a piecewise manner.

#### Theorem 1.6 (Reference10).

Let be th-order Lagrange interpolate operator, and assume that the following inclusions hold:

Then

If is regular, we have the global estimate

Inequality Equation1.18 is proven in Reference10, Theorem 3.1.6, and Equation1.19 is a direct consequence of Equation1.18 and the inverse inequality below.

Using Equation1.19 with and , we have . Hence

We will also need the following form of the inverse inequality.

#### Theorem 1.7 (Reference10, Theorem 3.2.6).

Assume that is regular, and assume also that the two pairs and with and satisfy

Then there exists a constant such that

for any .

If in addition satisfies the inverse assumption, then there exists a constant such that

for any and , with

The following simple result will be used repeatedly.

#### Lemma 1.8

Let and be symmetric matrices satisfying Equation1.9. Let be the solution of

with either the Dirichlet or periodic boundary condition on . Let be a solution of Equation1.23 with and replaced by and , respectively, and let satisfy the same boundary condition as . Then

#### Proof.

Inequality Equation1.24 is a direct consequence of

The following simple result underlies the stability of HMM for problem Equation1.1.

#### Lemma 1.9

Let be the solution of

where is a linear function and satisfies

Then we have

#### Proof.

Notice that on the edges of , using the fact that is a constant in , and integration by parts leads to

which implies

This gives the first result in Equation1.26. Multiplying Equation1.25 by and integrating by parts, we obtain

This gives the second part of Equation1.26.

#### Remark 1.10

For this result, the coefficient may depend on the solution, i.e., Equation1.25 may be nonlinear.

#### Remark 1.11

The same result holds if we use instead a periodic boundary condition: is periodic with period .

## 2. Generalities

Here we prove Theorem 1.1. We will let for convenience.

Since is the numerical solution associated with the quadratic form , is the exact solution associated with the quadratic form , defined for any as

To estimate , we view as an approximation to , and we use Strang’s first lemma Reference10.

Using Equation1.26 with and Equation1.9, for any , we have

Similarly, for any , we obtain

The existence and the uniqueness of the solutions to Equation1.8 follow from Equation2.1 and Equation2.2 via the Lax-Milgram lemma and the Poincaré inequality.

To streamline the proof of Theorem 1.1, we introduce the following auxiliary bilinear form .

Classical results on numerical integration Reference11, Theorem 6 give for any ,

Moreover, for any , if and are bounded, we have Reference11, Theorem 8,

### Proof of Theorem 1.1.

Using the first Strang lemma Reference10, Theorem 4.1.1, we have

Let and using Equation1.19 with , we have

It remains to estimate for and . Using Equation2.3, we get

This gives Equation1.11

To get the L estimate, we use the Aubin-Nitsche dual argument Reference10. To this end, consider the following auxiliary problem: Find such that

The standard regularity result reads Reference24

Putting into the right-hand side of Equation2.7, we obtain

Using Equation2.6 with , we bound the first two terms in the right-hand side of the above identity as

and

The last term in the right-hand side of Equation2.9 may be decomposed into

It follows from Equation2.4 that

By definition of and using Equation1.20, we get

Combining the above estimates and using Equation2.8 lead to Equation1.12.

It remains to prove Equation1.13. As in Reference37, for any point , we define the regularized Green’s function and the discrete Green’s function as

where is the regularized Dirac- function defined in Reference37. It is well known that

A proof for Equation2.11 can be obtained by using the weighted-norm technique Reference37. We refer to Reference9, Chapter 7 for details. Using the definition of and , a simple manipulation gives

Using Equation2.11, we obtain

Using Equation2.6, we get

where we have used the inverse inequality Equation1.21.

Similarly, we have

A combination of the above three estimates yields

If , then there exits a constant such that for all ,

We thus obtain Equation1.13 and this completes the proof.

Combining the foregoing proof for the L and W estimates, using the Green’s function defined in Reference39, we obtain

### Remark 2.1

Under the same condition for the W estimate in Theorem Equation1.1, we have

## 3. Estimating e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis $e(\mathrm{HMM})$

In this section, we estimate for problems with locally periodic coefficients. The estimate of for problems with random coefficients can be found in the Appendix.

We assume that , where is smooth in and periodic in with period . Define , and we introduce as

where is defined as: For , is periodic in with period and satisfies

Given , the homogenized coefficient is given by

Note that is smooth and bounded in all norms.

First let us consider the case when , and Equation1.6 is solved with the periodic boundary condition. Denote by the solution of Equation1.6 with the coefficients replaced by . may be viewed as a perturbation of . Using Lemma Equation1.8, we get

Observe that . A direct calculation yields

Using Equation3.3, we get

Next we consider the more general case when is a cube of size not necessarily equal to . The following analysis applies equally well to the case when the period of is of general and even nonpolygonal shape. This situation arises in some examples of composite materials Reference30. We will show that if is much larger than , then the averaged energy density for the solution of Equation1.6 closely approximates the energy density of the homogenized problem. We begin with the following observation:

We first establish some estimates on the solution of the cell problem Equation1.6. We will write instead of if there is no risk of confusion.

### Lemma 3.1

There exists a constant independent of and such that for each ,

### Proof.

We still denote by the solution of Equation1.6 with the coefficient replaced by . Using Lemma Equation1.8, we get

Define , which obviously satisfies

Note that is simply the boundary layer correction for the cell problem Equation1.6 Reference5. It is proved in Reference45, (1.51) in §1.4, using the rescaling over and .

This together with Equation3.7 gives

A straightforward calculation gives

which together with Equation3.10 leads to

This gives Equation3.6.

As in Equation3.6, we also have

### Proof.

Note that . We have

where

Using Equation3.7 and Equation2.2, we bound as

Using the symmetry of , and

where , and

Using Equation1.6 and , integrating by parts makes the first term in the right-hand side of vanish; therefore we write as

Using Equation3.12, we bound as

Using Equation3.2 and integrating by parts, we obtain

which together with Equation3.5 gives

The last term of is bounded by

where we have used . Using Equation3.11, we get

Consequently, we obtain

Combining the estimates for and gives the desired result Equation3.13.

### Remark 3.3

An explicit expression for is available in one dimension, from which we may show that the bound for is sharp.

## 4. Reconstruction and compression

### 4.1. Reconstruction procedure

Next we consider how to construct better approximations to from . We will restrict ourselves to the case when .

#### Proof of Theorem 1.4.

Subtracting Equation1.1 from Equation1.14, we obtain

Using classical interior estimates for elliptic equation Reference24, we have

Using the Hopf maximum principle, we get

A combination of the above two results implies Theorem 1.4.

#### Proof of Theorem 1.5.

Denote and define as the solution of

with the boundary condition that is periodic on and

where is the barycenter of .

It is easy to verify that takes the explicit form

Note that the periodic extension of is still . This means that is also well defined for the whole of and takes the same explicit form as Equation4.2.

Using for and that is a piecewise constant on , we obtain

As in Equation3.7, we have

From the construction of , we have for any ,

Since is constant over , we get

Adding up for all and using the a priori estimate , we obtain