American Mathematical Society

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

StartLayout 1st Row with Label left-parenthesis 1.1 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column minus d i v left-parenthesis a Superscript epsilon Baseline left-parenthesis x right-parenthesis nabla u Superscript epsilon Baseline left-parenthesis x right-parenthesis right-parenthesis 2nd Column equals f left-parenthesis x right-parenthesis 3rd Column Blank 4th Column x element-of upper D subset-of double-struck upper R Superscript d Baseline comma 2nd Row 1st Column u Superscript epsilon Baseline left-parenthesis x right-parenthesis 2nd Column equals 0 3rd Column Blank 4th Column x element-of partial-differential upper D period EndLayout EndLayout

Here epsilon is a small parameter that signifies explicitly the multiscale nature of the coefficient a Superscript epsilon Baseline left-parenthesis x right-parenthesis . 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 a Superscript epsilon Baseline left-parenthesis x right-parenthesis . In contrast modern multiscale methods are designed specifically for recovering partial information about u Superscript epsilon 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 a Superscript epsilon Baseline left-parenthesis x right-parenthesis might have, such as scale separation. The simplest example is when

StartLayout 1st Row with Label left-parenthesis 1.2 right-parenthesis EndLabel a Superscript epsilon Baseline left-parenthesis x right-parenthesis equals a left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis comma EndLayout

where a left-parenthesis x comma y right-parenthesis can either be periodic in y , in which case we assume the period to be upper I equals left-bracket negative 1 slash 2 comma 1 slash 2 right-bracket Superscript d , or random but stationary under shifts in y , for each fixed x element-of upper D . In both cases, it has been shown that Reference5Reference36

StartLayout 1st Row with Label left-parenthesis 1.3 right-parenthesis EndLabel double-vertical-bar u Superscript epsilon Baseline left-parenthesis x right-parenthesis minus upper U left-parenthesis x right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis upper D right-parenthesis Baseline right-arrow 0 comma EndLayout

where upper U left-parenthesis x right-parenthesis is the solution of a homogenized equation:

StartLayout 1st Row with Label left-parenthesis 1.4 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column minus d i v left-parenthesis script upper A left-parenthesis x right-parenthesis nabla upper U left-parenthesis x right-parenthesis right-parenthesis 2nd Column equals f left-parenthesis x right-parenthesis 3rd Column Blank 4th Column x element-of upper D comma 2nd Row 1st Column upper U left-parenthesis x right-parenthesis 2nd Column equals 0 3rd Column Blank 4th Column x element-of partial-differential upper D period EndLayout EndLayout

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

Several numerical methods have been developed to deal specifically with the case when a left-parenthesis x comma y right-parenthesis is periodic in y . 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 phi left-parenthesis x comma x slash epsilon right-parenthesis where phi left-parenthesis x comma y right-parenthesis is periodic in y to extract the leading order behavior of u Superscript epsilon Baseline left-parenthesis x right-parenthesis , 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 epsilon , hence sublinear as epsilon right-arrow 0 , 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 script upper P Subscript k finite element method on a triangulation script upper T Subscript upper H of element size upper H which should resolve the macroscale features of a Superscript epsilon Baseline left-parenthesis x right-parenthesis . 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 script upper A Subscript upper H Baseline left-parenthesis x right-parenthesis , if we knew script upper A Subscript upper H Baseline left-parenthesis x right-parenthesis explicitly, we could have evaluated the quadratic form

integral Underscript upper D Endscripts nabla upper V left-parenthesis x right-parenthesis dot script upper A Subscript upper H Baseline left-parenthesis x right-parenthesis nabla upper V left-parenthesis x right-parenthesis d x

by numerical quadrature: For any upper V element-of upper X Subscript upper H , the finite element space,

StartLayout 1st Row with Label left-parenthesis 1.5 right-parenthesis EndLabel upper A Subscript upper H Baseline left-parenthesis upper V comma upper V right-parenthesis asymptotically-equals sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts StartAbsoluteValue upper K EndAbsoluteValue sigma-summation Underscript x Subscript script l Baseline element-of upper K Endscripts omega Subscript script l Baseline left-parenthesis nabla upper V dot script upper A Subscript upper H Baseline nabla upper V right-parenthesis left-parenthesis x Subscript script l Baseline right-parenthesis comma EndLayout

where StartSet x Subscript script l Baseline EndSet and StartSet omega Subscript script l Baseline EndSet are the quadrature points and weights in upper K , StartAbsoluteValue upper K EndAbsoluteValue is the volume of upper K . In the absence of explicit knowledge of script upper A Subscript upper H Baseline left-parenthesis x right-parenthesis , we approximate left-parenthesis nabla upper V dot script upper A Subscript upper H Baseline nabla upper V right-parenthesis left-parenthesis x Subscript script l Baseline right-parenthesis by solving the problem:

StartLayout 1st Row with Label left-parenthesis 1.6 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column minus d i v left-parenthesis a Superscript epsilon Baseline left-parenthesis x right-parenthesis nabla v Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis right-parenthesis 2nd Column equals 0 3rd Column Blank 4th Column x element-of upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis comma 2nd Row 1st Column v Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis 2nd Column equals upper V Subscript script l Baseline left-parenthesis x right-parenthesis 3rd Column Blank 4th Column x element-of partial-differential upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis comma EndLayout EndLayout

where upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis is a cube of size delta centered at x Subscript script l , and upper V Subscript script l is the linear approximation of upper V at x Subscript script l . We then let

StartLayout 1st Row with Label left-parenthesis 1.7 right-parenthesis EndLabel left-parenthesis nabla upper V dot script upper A Subscript upper H Baseline nabla upper V right-parenthesis left-parenthesis x Subscript script l Baseline right-parenthesis asymptotically-equals StartFraction 1 Over delta Superscript d Baseline EndFraction integral Underscript upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis Endscripts nabla v Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis dot a Superscript epsilon Baseline left-parenthesis x right-parenthesis nabla v Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis d x period EndLayout

Equation1.5 and Equation1.7 together give the needed approximate stiffness matrix at the scale upper H . For convenience, we will define the corresponding bilinear form: For any upper V comma upper W element-of upper X Subscript upper H Baseline

upper A Subscript upper H Baseline left-parenthesis upper V comma upper W right-parenthesis colon equals sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts StartFraction vertical-bar Over upper K EndFraction vertical-bar delta Superscript d Baseline sigma-summation Underscript x Subscript script l Baseline element-of upper K Endscripts omega Subscript script l Baseline integral Underscript upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis Endscripts nabla v Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis dot a Superscript epsilon Baseline left-parenthesis x right-parenthesis nabla w Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis d x comma

where w Subscript script l Superscript epsilon is defined for upper W element-of upper X Subscript upper H in the same way that v Subscript script l Superscript epsilon in Equation1.6 was defined for upper V .

In order to reduce the effect of the imposed boundary condition on partial-differential upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis , we may replace Equation1.7 by

StartLayout 1st Row with Label left-parenthesis 1.8 a right-parenthesis EndLabel left-parenthesis nabla upper V dot script upper A Subscript upper H Baseline nabla upper V right-parenthesis left-parenthesis x Subscript script l Baseline right-parenthesis asymptotically-equals StartFraction 1 Over left-parenthesis delta Superscript Super Superscript prime Superscript Baseline right-parenthesis Superscript d Baseline EndFraction integral Underscript upper I Subscript delta Sub Superscript Sub Super Superscript prime Sub Superscript Subscript Baseline left-parenthesis x Subscript script l Baseline right-parenthesis Endscripts nabla v Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis dot a Superscript epsilon Baseline left-parenthesis x right-parenthesis nabla v Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis d x comma EndLayout

where delta prime less-than delta . For example, we may choose delta Superscript prime Baseline equals delta slash 2 . 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 a Superscript epsilon Baseline left-parenthesis x right-parenthesis equals a left-parenthesis x comma x slash epsilon right-parenthesis and a left-parenthesis x comma y right-parenthesis is periodic in y , one can take upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis to be x Subscript script l Baseline plus epsilon upper I , i.e., delta equals epsilon and use the boundary condition that v Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis minus upper V Subscript script l Baseline left-parenthesis x right-parenthesis is periodic on upper I Subscript delta .

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 upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis to be smaller than upper K . The size of upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis is determined by many factors, including the accuracy and cost requirement, the degree of scale separation, and the microstructure in a Superscript epsilon Baseline left-parenthesis x right-parenthesis . One purpose for the error estimates that we present below is to give guidelines on how to select upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis . As mentioned already, if a Superscript epsilon Baseline left-parenthesis x right-parenthesis equals a left-parenthesis x comma x slash epsilon right-parenthesis and a left-parenthesis x comma y right-parenthesis is periodic in y , we can simply choose upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis to be x Subscript script l Baseline plus epsilon upper I , i.e., delta equals epsilon . If a left-parenthesis x comma y right-parenthesis is random, then delta should be a few times larger than the local correlation length of a Superscript epsilon . In the former case, the total cost is independent of epsilon . In the latter case, the total cost depends only weakly on epsilon (see Reference31).

The final problem is to solve

StartLayout 1st Row with Label left-parenthesis 1.8 right-parenthesis EndLabel min Underscript upper V element-of upper X Subscript upper H Baseline Endscripts one-half upper A Subscript upper H Baseline left-parenthesis upper V comma upper V right-parenthesis minus left-parenthesis f comma upper V right-parenthesis period EndLayout

Figure 1.

Illustration of HMM for solving Equation1.1. The dots are the quadrature points. The little squares are the microcell upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis .

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 a Superscript epsilon Baseline left-parenthesis x right-parenthesis does not has the structure of a left-parenthesis x comma x slash epsilon right-parenthesis . 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 upper H that resolves the fine scales and letting script upper A Subscript upper H Baseline left-parenthesis x right-parenthesis equals a Superscript epsilon Baseline left-parenthesis x right-parenthesis .

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 nabla u Superscript epsilon locally by solving a local problem with some given approximation to the macroscopic state upper U 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 u Superscript epsilon 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 e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis , due to the error in estimating the stiffness matrix. We then estimate e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis . This second part is only done for either periodic or random homogenization problems, since concrete results are only possible if the behavior of u Superscript epsilon is well understood. We believe that this overall strategy will be useful for analyzing other multiscale methods.

We will always assume that a Superscript epsilon Baseline left-parenthesis x right-parenthesis is smooth, symmetric and uniformly elliptic:

StartLayout 1st Row with Label left-parenthesis 1.9 right-parenthesis EndLabel lamda upper I less-than-or-equal-to a Superscript epsilon Baseline less-than-or-equal-to upper Lamda upper I EndLayout

for some lamda comma upper Lamda greater-than 0 . We will use the summation convention and standard notation for Sobolev spaces (see Reference1). We will use StartAbsoluteValue dot EndAbsoluteValue to denote the absolute value of a scalar quantity, the Euclidean norm of a vector and the volume of a set upper K . For the quadrature formula Equation1.5, we will assume the following accuracy conditions for k th-order numerical quadrature scheme Reference11:

StartLayout 1st Row with Label left-parenthesis 1.10 right-parenthesis EndLabel integral minus Subscript upper K Baseline p left-parenthesis x right-parenthesis d x colon StartFraction 1 Over StartAbsoluteValue upper K EndAbsoluteValue EndFraction integral Underscript upper K Endscripts p left-parenthesis x right-parenthesis d x equals sigma-summation Underscript script l equals 1 Overscript upper L Endscripts omega Subscript script l Baseline p left-parenthesis x Subscript script l Baseline right-parenthesis for all p left-parenthesis x right-parenthesis element-of script upper P Subscript 2 k minus 2 Baseline period EndLayout

Here omega Subscript script l Baseline greater-than 0 , for script l equals 1 comma ellipsis comma upper L . For k equals 1 , we assume the above formula to be exact for p element-of script upper P 1 .

1.3. Main results

Our main results for the linear problem are as follows.

Theorem 1.1

Denote by upper U 0 and upper U Subscript normal upper H normal upper M normal upper M the solution of Equation1.4 and the normal upper H normal upper M normal upper M solution, respectively. Let

e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis equals max Underscript StartLayout 1st Row x Subscript script l Baseline element-of upper K 2nd Row upper K element-of script upper T Subscript upper H Baseline EndLayout Endscripts double-vertical-bar script upper A left-parenthesis x Subscript script l Baseline right-parenthesis minus script upper A Subscript upper H Baseline left-parenthesis x Subscript script l Baseline right-parenthesis double-vertical-bar comma

where double-vertical-bar dot double-vertical-bar is the Euclidean norm. If upper U 0 is sufficiently smooth and Equation1.10 holds, then there exists a constant upper C independent of epsilon comma delta and upper H , such that

StartLayout 1st Row with Label left-parenthesis 1.11 right-parenthesis EndLabel 1st Column double-vertical-bar upper U 0 minus upper U Subscript normal upper H normal upper M normal upper M Baseline double-vertical-bar Subscript 1 2nd Column upper C left-parenthesis upper H Superscript k Baseline plus e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis right-parenthesis comma 2nd Row with Label left-parenthesis 1.12 right-parenthesis EndLabel 1st Column double-vertical-bar upper U 0 minus upper U Subscript normal upper H normal upper M normal upper M Baseline double-vertical-bar Subscript 0 2nd Column upper C left-parenthesis upper H Superscript k plus 1 Baseline plus e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis right-parenthesis period EndLayout

If there exits a constant upper C 0 such that e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis StartAbsoluteValue ln upper H EndAbsoluteValue less-than upper C 0 , then there exists a constant upper H 0 such that for all upper H less-than-or-equal-to upper H 0 ,

StartLayout 1st Row with Label left-parenthesis 1.13 right-parenthesis EndLabel double-vertical-bar upper U 0 minus upper U Subscript normal upper H normal upper M normal upper M Baseline double-vertical-bar Subscript 1 comma normal infinity Baseline less-than-or-equal-to upper C left-parenthesis upper H Superscript k Baseline plus e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis right-parenthesis StartAbsoluteValue ln upper H EndAbsoluteValue period EndLayout

At this stage, no assumption on the form of a Superscript epsilon Baseline left-parenthesis x right-parenthesis is necessary. upper U 0 can be the solution of an arbitrary macroscopic equation with the same right-hand side as in Equation1.1. Of course for upper U Subscript normal upper H normal upper M normal upper M to converge to upper U 0 , i.e., e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis right-arrow 0 , upper U 0 must be chosen as the solution of the homogenized equation, which we now assume exists. To obtain quantitative estimates on e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis , we must restrict ourselves to more specific cases.

Theorem 1.2

For the periodic homogenization problem, we have

e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis less-than-or-equal-to StartLayout Enlarged left-brace 1st Row 1st Column upper C epsilon 2nd Column if upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis equals x Subscript script l Baseline plus epsilon upper I comma 2nd Row 1st Column upper C left-parenthesis StartFraction epsilon Over delta EndFraction plus delta right-parenthesis 2nd Column otherwise period EndLayout

In the first case, we replace the boundary condition in Equation1.6 by a periodic boundary condition: v Subscript script l Superscript epsilon Baseline minus upper V Subscript script l is periodic with period epsilon upper I . For the second result we do not need to assume that the period of a left-parenthesis x comma dot right-parenthesis 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 e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis 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 bold left-parenthesis bold upper A bold right-parenthesis in the Appendix holds (see Reference43), we have

double-struck upper E e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis less-than-or-equal-to StartLayout Enlarged left-brace 1st Row 1st Column upper C left-parenthesis kappa right-parenthesis left-parenthesis StartFraction epsilon Over delta EndFraction right-parenthesis Superscript kappa Baseline 2nd Column d equals 3 comma 2nd Row 1st Column remains open 2nd Column d equals 2 comma 3rd Row 1st Column upper C left-parenthesis kappa right-parenthesis left-parenthesis StartFraction epsilon Over delta EndFraction right-parenthesis Superscript 1 slash 2 Baseline 2nd Column d equals 1 comma EndLayout

where

kappa equals StartFraction 6 minus 12 gamma Over 25 minus 8 gamma EndFraction

for any 0 less-than gamma less-than 1 slash 2 . By choosing gamma small, kappa can be arbitrarily close to 6 slash 25 .

The probabilistic set-up will be given in the Appendix. To prove this result, we assume that Equation1.8a is used with delta Superscript prime Baseline equals delta slash 2 .

1.4. Recovering the microstructural information

In many applications, the microstructure information in u Superscript epsilon Baseline left-parenthesis x right-parenthesis is very important. upper U Subscript normal upper H normal upper M normal upper M by itself does not give this information. However, this information can be recovered using a simple post-processing technique. For the general case, having upper U Subscript normal upper H normal upper M normal upper M , one can obtain locally the microstructural information using an idea in Reference35. Assume that we are interested in recovering u Superscript epsilon and nabla u Superscript epsilon only in the subdomain upper Omega subset-of upper D . Consider the following auxiliary problem:

StartLayout 1st Row with Label left-parenthesis 1.14 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column minus d i v left-parenthesis a Superscript epsilon Baseline left-parenthesis x right-parenthesis nabla ModifyingAbove u With tilde Superscript epsilon Baseline left-parenthesis x right-parenthesis right-parenthesis 2nd Column equals f left-parenthesis x right-parenthesis 3rd Column Blank 4th Column x element-of upper Omega Subscript eta Baseline comma 2nd Row 1st Column ModifyingAbove u With tilde Superscript epsilon Baseline left-parenthesis x right-parenthesis 2nd Column equals upper U Subscript normal upper H normal upper M normal upper M Baseline left-parenthesis x right-parenthesis 3rd Column Blank 4th Column x element-of partial-differential upper Omega Subscript eta Baseline comma EndLayout EndLayout

where upper Omega Subscript eta satisfies upper Omega subset-of upper Omega Subscript eta Baseline subset-of upper D and dist left-parenthesis partial-differential upper Omega comma partial-differential upper Omega Subscript eta Baseline right-parenthesis equals eta . We then have

Theorem 1.4

There exists a constant upper C such that

StartLayout 1st Row with Label left-parenthesis 1.15 right-parenthesis EndLabel left-parenthesis integral minus Subscript upper Omega Baseline StartAbsoluteValue nabla left-parenthesis u Superscript epsilon Baseline minus u overTilde Superscript epsilon Baseline right-parenthesis EndAbsoluteValue squared d x right-parenthesis Superscript 1 slash 2 Baseline less-than-or-equal-to StartFraction upper C Over eta EndFraction left-parenthesis double-vertical-bar upper U 0 minus upper U Subscript normal upper H normal upper M normal upper M Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper Omega Sub Subscript eta Subscript right-parenthesis Baseline plus double-vertical-bar u Superscript epsilon Baseline minus upper U 0 double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper Omega Sub Subscript eta Subscript right-parenthesis Baseline right-parenthesis period EndLayout

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 k equals 1 and choose upper I Subscript delta Baseline equals x Subscript upper K Baseline plus epsilon upper I , where x Subscript upper K is the barycenter of upper K . Here we have assumed that the quadrature point is at x Subscript upper K .

Let u overTilde Superscript epsilon be defined piecewise as follows:

(1)

u overTilde Superscript epsilon Baseline vertical-bar Subscript upper I Sub Subscript delta Subscript Baseline equals v Subscript upper K Superscript epsilon , where v Subscript upper K Superscript epsilon is the solution of Equation1.6 with the boundary condition that v Subscript upper K Superscript epsilon Baseline minus upper U Subscript normal upper H normal upper M normal upper M is periodic with period epsilon upper I and integral Underscript upper I Subscript delta Baseline Endscripts left-parenthesis u overTilde Superscript epsilon Baseline minus upper U Subscript normal upper H normal upper M normal upper M Baseline right-parenthesis left-parenthesis x right-parenthesis d x equals 0 .

(2)

left-parenthesis u overTilde Superscript epsilon Baseline minus upper U Subscript normal upper H normal upper M normal upper M Baseline right-parenthesis vertical-bar Subscript upper K Baseline is periodic with period epsilon upper I .

For this case, we can prove

Theorem 1.5

Let u overTilde Superscript epsilon be defined as above. Then

StartLayout 1st Row with Label left-parenthesis 1.16 right-parenthesis EndLabel left-parenthesis sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts double-vertical-bar nabla left-parenthesis u Superscript epsilon Baseline minus u overTilde Superscript epsilon Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper K Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline less-than-or-equal-to upper C left-parenthesis StartRoot epsilon EndRoot plus upper H right-parenthesis period EndLayout

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 script upper T Subscript upper H , it is called regular if there is a constant sigma such that

StartFraction upper H Subscript upper K Baseline Over rho Subscript upper K Baseline EndFraction less-than-or-equal-to sigma for all upper K element-of script upper T Subscript upper H Baseline

and if the quantity

upper H equals max Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts upper H Subscript upper K

approaches zero, where upper H Subscript upper K is the diameter of upper K and rho Subscript upper K is the diameter of the largest ball inscribed in upper K . script upper T Subscript upper H satisfies an inverse assumption if there exists a constant nu such that

StartFraction upper H Over upper H Subscript upper K Baseline EndFraction less-than-or-equal-to nu for all upper K element-of script upper T Subscript upper H Baseline period

A regular family of triangulation of script upper T Subscript upper H 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 k greater-than-or-equal-to 2 comma nabla Superscript k Baseline v is understood in a piecewise manner.

Theorem 1.6 (Reference10).

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

StartLayout 1st Row with Label left-parenthesis 1.17 right-parenthesis EndLabel upper W Superscript k plus 1 comma p Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis right-arrow with hook script upper C Superscript 0 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis and upper W Superscript k plus 1 comma p Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis right-arrow with hook upper W Superscript m comma q Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis period EndLayout

Then

StartLayout 1st Row with Label left-parenthesis 1.18 right-parenthesis EndLabel StartAbsoluteValue v minus upper Pi v EndAbsoluteValue Subscript m comma q comma upper K Baseline less-than-or-equal-to upper C StartAbsoluteValue upper K EndAbsoluteValue Superscript 1 slash q minus 1 slash p Baseline StartFraction upper H Subscript upper K Superscript k plus 1 Baseline Over rho Subscript upper K Superscript m Baseline EndFraction StartAbsoluteValue v EndAbsoluteValue Subscript k plus 1 comma p comma upper K Baseline period EndLayout

If script upper T Subscript upper H is regular, we have the global estimate

StartLayout 1st Row with Label left-parenthesis 1.19 right-parenthesis EndLabel StartAbsoluteValue v minus upper Pi v EndAbsoluteValue Subscript m comma q comma upper D Baseline less-than-or-equal-to upper C upper H Superscript k plus 1 minus m plus min left-brace 0 comma d left-parenthesis 1 slash q minus 1 slash p right-parenthesis right-brace Baseline StartAbsoluteValue v EndAbsoluteValue Subscript k plus 1 comma p comma upper D Baseline period EndLayout

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 p equals q equals 2 and m equals 2 comma k equals 1 , we have double-vertical-bar v minus upper Pi v double-vertical-bar Subscript 2 comma upper D Baseline less-than-or-equal-to upper C double-vertical-bar v double-vertical-bar Subscript 2 comma upper D . Hence

StartLayout 1st Row with Label left-parenthesis 1.20 right-parenthesis EndLabel double-vertical-bar upper Pi v double-vertical-bar Subscript 2 comma upper D Baseline less-than-or-equal-to double-vertical-bar v minus upper Pi v double-vertical-bar Subscript 2 comma upper D Baseline plus double-vertical-bar v double-vertical-bar Subscript 2 comma upper D Baseline less-than-or-equal-to upper C double-vertical-bar v double-vertical-bar Subscript 2 comma upper D Baseline period EndLayout

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

Theorem 1.7 (Reference10, Theorem 3.2.6).

Assume that script upper T Subscript upper H is regular, and assume also that the two pairs left-parenthesis l comma r right-parenthesis and left-parenthesis m comma q right-parenthesis with l comma m greater-than-or-equal-to 0 and r comma q element-of left-bracket 0 comma normal infinity right-bracket satisfy

l less-than-or-equal-to m and script upper P Subscript k Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis subset-of upper W Superscript l comma r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis intersection upper W Superscript m comma q Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis period

Then there exists a constant upper C equals upper C left-parenthesis sigma comma nu comma l comma r comma m comma q right-parenthesis such that

StartLayout 1st Row with Label left-parenthesis 1.21 right-parenthesis EndLabel StartAbsoluteValue v EndAbsoluteValue Subscript m comma q comma upper K Baseline less-than-or-equal-to upper C upper H Subscript upper K Superscript l minus m plus d left-parenthesis 1 slash q minus 1 slash r right-parenthesis Baseline StartAbsoluteValue v EndAbsoluteValue Subscript l comma r comma upper K EndLayout

for any v element-of script upper P Subscript k Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis .

If in addition script upper T Subscript upper H satisfies the inverse assumption, then there exists a constant upper C equals upper C left-parenthesis sigma comma nu comma l comma r comma m comma q right-parenthesis such that

StartLayout 1st Row with Label left-parenthesis 1.22 right-parenthesis EndLabel left-parenthesis sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts StartAbsoluteValue v EndAbsoluteValue Subscript m comma q comma upper K Superscript q Baseline right-parenthesis Superscript 1 slash q Baseline less-than-or-equal-to upper C upper H Superscript l minus m plus min left-brace 0 comma d left-parenthesis 1 slash q minus 1 slash r right-parenthesis right-brace Baseline left-parenthesis sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts StartAbsoluteValue v EndAbsoluteValue Subscript l comma r comma upper K Superscript r Baseline right-parenthesis Superscript 1 slash r EndLayout

for any v element-of upper X Subscript upper H and r comma q less-than normal infinity , with

StartLayout 1st Row 1st Column max Underscript upper K element-of script upper T Subscript upper H Endscripts StartAbsoluteValue v EndAbsoluteValue Subscript m comma normal infinity comma upper K 2nd Column replacing left-parenthesis sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts StartAbsoluteValue v EndAbsoluteValue Subscript m comma q comma upper K Superscript q Baseline right-parenthesis Superscript 1 slash q Baseline comma if q equals normal infinity comma 2nd Row 1st Column max Underscript upper K element-of script upper T Subscript upper H Endscripts StartAbsoluteValue v EndAbsoluteValue Subscript l comma normal infinity comma upper K 2nd Column replacing left-parenthesis sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts StartAbsoluteValue v EndAbsoluteValue Subscript l comma r comma upper K Superscript r Baseline right-parenthesis Superscript 1 slash r Baseline comma if r equals normal infinity period EndLayout

The following simple result will be used repeatedly.

Lemma 1.8

Let upper A 1 left-parenthesis x right-parenthesis and upper A 2 left-parenthesis x right-parenthesis be symmetric matrices satisfying Equation1.9. Let phi 1 be the solution of

StartLayout 1st Row with Label left-parenthesis 1.23 right-parenthesis EndLabel minus d i v left-parenthesis upper A 1 left-parenthesis x right-parenthesis nabla phi 1 left-parenthesis x right-parenthesis right-parenthesis equals d i v left-parenthesis ModifyingAbove upper A With tilde Subscript 1 Baseline left-parenthesis x right-parenthesis nabla upper F 1 left-parenthesis x right-parenthesis right-parenthesis x element-of upper Omega comma EndLayout

with either the Dirichlet or periodic boundary condition on partial-differential upper Omega . Let phi 2 be a solution of Equation1.23 with upper A 1 comma upper A overTilde Subscript 1 Baseline and upper F 1 replaced by upper A 2 comma upper A overTilde Subscript 2 Baseline and upper F 2 , respectively, and let phi 2 satisfy the same boundary condition as phi 1 . Then

StartLayout 1st Row 1st Column lamda double-vertical-bar nabla left-parenthesis phi 1 minus phi 2 right-parenthesis double-vertical-bar Subscript 0 comma upper Omega 2nd Column max Underscript x element-of upper Omega Endscripts StartAbsoluteValue left-parenthesis upper A overTilde Subscript 1 Baseline minus upper A overTilde Subscript 2 Baseline right-parenthesis left-parenthesis x right-parenthesis EndAbsoluteValue double-vertical-bar nabla upper F 1 double-vertical-bar Subscript 0 comma upper Omega Baseline plus max Underscript x element-of upper Omega Endscripts StartAbsoluteValue left-parenthesis upper A 1 minus upper A 2 right-parenthesis left-parenthesis x right-parenthesis EndAbsoluteValue double-vertical-bar nabla phi 2 double-vertical-bar Subscript 0 comma upper Omega Baseline 2nd Row with Label left-parenthesis 1.24 right-parenthesis EndLabel 1st Column Blank 2nd Column plus max Underscript x element-of upper Omega Endscripts StartAbsoluteValue ModifyingAbove upper A With tilde Subscript 2 Baseline left-parenthesis x right-parenthesis EndAbsoluteValue double-vertical-bar nabla left-parenthesis upper F 1 minus upper F 2 right-parenthesis double-vertical-bar Subscript 0 comma upper Omega Baseline period EndLayout

Proof.

Inequality Equation1.24 is a direct consequence of

lamda double-vertical-bar nabla left-parenthesis phi 1 minus phi 2 right-parenthesis double-vertical-bar Subscript 0 comma upper Omega Superscript 2 Baseline less-than-or-equal-to integral Underscript upper Omega Endscripts nabla left-parenthesis phi 1 minus phi 2 right-parenthesis dot left-parenthesis left-parenthesis upper A overTilde Subscript 2 Baseline minus upper A overTilde Subscript 1 Baseline right-parenthesis nabla upper F 1 plus left-parenthesis upper A 2 minus upper A 1 right-parenthesis nabla phi 2 plus upper A overTilde Subscript 2 Baseline nabla left-parenthesis upper F 2 minus upper F 1 right-parenthesis right-parenthesis period

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

Lemma 1.9

Let phi be the solution of

StartLayout 1st Row with Label left-parenthesis 1.25 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column minus d i v left-parenthesis a nabla phi right-parenthesis 2nd Column equals 0 3rd Column Blank 4th Column in upper Omega subset-of double-struck upper R Superscript d Baseline comma 2nd Row 1st Column phi 2nd Column equals upper V Subscript script l Baseline 3rd Column Blank 4th Column on element-of partial-differential upper Omega comma EndLayout EndLayout

where upper V Subscript script l is a linear function and a equals left-parenthesis a Subscript i j Baseline right-parenthesis satisfies

lamda upper I less-than-or-equal-to a less-than-or-equal-to upper Lamda upper I period

Then we have

StartLayout 1st Row with Label left-parenthesis 1.26 right-parenthesis EndLabel double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper Omega Baseline less-than-or-equal-to double-vertical-bar nabla phi double-vertical-bar Subscript 0 comma upper Omega Baseline and left-parenthesis integral Underscript upper Omega Endscripts nabla phi dot a nabla phi right-parenthesis Superscript 1 slash 2 Baseline less-than-or-equal-to left-parenthesis integral Underscript upper Omega Endscripts nabla upper V Subscript script l Baseline dot a nabla upper V Subscript script l Baseline right-parenthesis Superscript 1 slash 2 Baseline period EndLayout

Proof.

Notice that phi equals upper V Subscript script l on the edges of upper Omega , using the fact that nabla upper V Subscript script l is a constant in upper Omega , and integration by parts leads to

integral Underscript upper Omega Endscripts nabla left-parenthesis phi minus upper V Subscript script l Baseline right-parenthesis left-parenthesis x right-parenthesis nabla upper V Subscript script l Baseline left-parenthesis x right-parenthesis d x equals 0 comma

which implies

integral Underscript upper Omega Endscripts StartAbsoluteValue nabla phi left-parenthesis x right-parenthesis EndAbsoluteValue squared d x equals integral Underscript upper Omega Endscripts StartAbsoluteValue nabla upper V Subscript script l Baseline left-parenthesis x right-parenthesis EndAbsoluteValue squared d x plus integral Underscript upper Omega Endscripts StartAbsoluteValue nabla left-parenthesis phi minus upper V Subscript script l Baseline right-parenthesis left-parenthesis x right-parenthesis EndAbsoluteValue squared d x period

This gives the first result in Equation1.26. Multiplying Equation1.25 by phi left-parenthesis x right-parenthesis minus upper V Subscript script l Baseline left-parenthesis x right-parenthesis and integrating by parts, we obtain

StartLayout 1st Row 1st Column integral Underscript upper Omega Endscripts nabla phi left-parenthesis x right-parenthesis dot a nabla phi left-parenthesis x right-parenthesis d x 2nd Column plus integral Underscript upper Omega Endscripts nabla left-parenthesis phi minus upper V Subscript script l Baseline right-parenthesis left-parenthesis x right-parenthesis dot a nabla left-parenthesis phi minus upper V Subscript script l Baseline right-parenthesis left-parenthesis x right-parenthesis d x 2nd Row 1st Column Blank 2nd Column equals integral Underscript upper Omega Endscripts nabla upper V Subscript script l Baseline left-parenthesis x right-parenthesis dot a nabla upper V Subscript script l Baseline left-parenthesis x right-parenthesis d x period EndLayout

This gives the second part of Equation1.26.

Remark 1.10

For this result, the coefficient a equals left-parenthesis a Subscript i j Baseline right-parenthesis 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: phi minus upper V Subscript script l is periodic with period upper Omega .

2. Generalities

Here we prove Theorem 1.1. We will let upper U Subscript upper H Baseline equals upper U Subscript normal upper H normal upper M normal upper M for convenience.

Since upper U Subscript upper H is the numerical solution associated with the quadratic form upper A Subscript upper H , upper U 0 is the exact solution associated with the quadratic form upper A , defined for any upper V element-of upper H 0 Superscript 1 Baseline left-parenthesis upper D right-parenthesis as

upper A left-parenthesis upper V comma upper V right-parenthesis equals integral Underscript upper D Endscripts nabla upper V left-parenthesis x right-parenthesis dot script upper A left-parenthesis x right-parenthesis nabla upper V left-parenthesis x right-parenthesis d x period

To estimate upper U 0 minus upper U Subscript upper H , we view upper A Subscript upper H as an approximation to upper A , and we use Strang’s first lemma Reference10.

Using Equation1.26 with upper Omega equals upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis and Equation1.9, for any upper V element-of upper X Subscript upper H , we have

StartLayout 1st Row 1st Column upper A Subscript upper H Baseline left-parenthesis upper V comma upper V right-parenthesis 2nd Column lamda sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts StartAbsoluteValue upper K EndAbsoluteValue sigma-summation Underscript x Subscript script l Baseline element-of upper K Endscripts omega Subscript script l Baseline integral minus Subscript upper I Sub Subscript delta Subscript left-parenthesis x Sub Subscript script l Subscript right-parenthesis Baseline StartAbsoluteValue nabla upper V Subscript script l Baseline left-parenthesis x right-parenthesis EndAbsoluteValue squared d x 2nd Row 1st Column Blank 2nd Column equals lamda sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts StartAbsoluteValue upper K EndAbsoluteValue sigma-summation Underscript x Subscript script l Baseline element-of upper K Endscripts omega Subscript script l Baseline StartAbsoluteValue nabla upper V left-parenthesis x Subscript script l Baseline right-parenthesis EndAbsoluteValue squared 3rd Row with Label left-parenthesis 2.1 right-parenthesis EndLabel 1st Column Blank 2nd Column equals lamda double-vertical-bar nabla upper V double-vertical-bar Subscript 0 Superscript 2 Baseline period EndLayout

Similarly, for any upper V comma upper W element-of upper X Subscript upper H Baseline , we obtain

StartLayout 1st Row 1st Column StartAbsoluteValue upper A Subscript upper H Baseline left-parenthesis upper V comma upper W right-parenthesis EndAbsoluteValue 2nd Column sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts StartAbsoluteValue upper K EndAbsoluteValue sigma-summation Underscript x Subscript script l Baseline element-of upper K Endscripts omega Subscript script l Baseline left-parenthesis integral minus Subscript upper I Sub Subscript delta Subscript left-parenthesis x Sub Subscript script l Subscript right-parenthesis Baseline nabla upper V Subscript script l Baseline dot a Superscript epsilon Baseline nabla upper V Subscript script l Baseline right-parenthesis Superscript one-half Baseline left-parenthesis integral minus Subscript upper I Sub Subscript delta Subscript left-parenthesis x Sub Subscript script l Subscript right-parenthesis Baseline nabla upper W Subscript script l Baseline dot a Superscript epsilon Baseline nabla upper W Subscript script l Baseline right-parenthesis Superscript one-half Baseline 2nd Row 1st Column Blank 2nd Column upper Lamda sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts StartAbsoluteValue upper K EndAbsoluteValue sigma-summation Underscript x Subscript script l Baseline element-of upper K Endscripts omega Subscript script l Baseline StartAbsoluteValue nabla upper V left-parenthesis x Subscript script l Baseline right-parenthesis EndAbsoluteValue StartAbsoluteValue nabla upper W left-parenthesis x Subscript script l Baseline right-parenthesis EndAbsoluteValue 3rd Row 1st Column Blank 2nd Column equals upper Lamda sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts integral Underscript upper K Endscripts StartAbsoluteValue nabla upper V left-parenthesis x right-parenthesis EndAbsoluteValue StartAbsoluteValue nabla upper W left-parenthesis x right-parenthesis EndAbsoluteValue d x 4th Row with Label left-parenthesis 2.2 right-parenthesis EndLabel 1st Column Blank 2nd Column upper Lamda double-vertical-bar nabla upper V double-vertical-bar Subscript 0 Baseline double-vertical-bar nabla upper W double-vertical-bar Subscript 0 Baseline period EndLayout

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 ModifyingAbove upper A With caret Subscript upper H .

ModifyingAbove upper A With caret Subscript upper H Baseline left-parenthesis upper V comma upper W right-parenthesis equals sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts ModifyingAbove upper A With caret Subscript upper K Baseline left-parenthesis upper V comma upper W right-parenthesis with ModifyingAbove upper A With caret Subscript upper K Baseline left-parenthesis upper V comma upper W right-parenthesis equals StartAbsoluteValue upper K EndAbsoluteValue sigma-summation Underscript x Subscript script l Baseline element-of upper K Endscripts omega Subscript script l Baseline left-parenthesis nabla upper W dot script upper A nabla upper V right-parenthesis left-parenthesis x Subscript script l Baseline right-parenthesis period

Classical results on numerical integration Reference11, Theorem 6 give for any upper V comma upper W element-of upper X Subscript upper H Baseline ,

StartLayout 1st Row with Label left-parenthesis 2.3 right-parenthesis EndLabel StartAbsoluteValue ModifyingAbove upper A With caret Subscript upper K Baseline left-parenthesis upper V comma upper W right-parenthesis minus integral Underscript upper K Endscripts nabla upper W dot script upper A nabla upper V d x EndAbsoluteValue less-than-or-equal-to upper C upper H Superscript m Baseline double-vertical-bar upper V double-vertical-bar Subscript m comma upper K Baseline double-vertical-bar nabla upper W double-vertical-bar Subscript 0 comma upper K Baseline 1 less-than-or-equal-to m less-than-or-equal-to k period EndLayout

Moreover, for any upper V comma upper W element-of upper X Subscript upper H Baseline , if double-vertical-bar upper V double-vertical-bar Subscript k plus 1 and double-vertical-bar upper W double-vertical-bar Subscript 2 are bounded, we have Reference11, Theorem 8,

StartLayout 1st Row with Label left-parenthesis 2.4 right-parenthesis EndLabel StartAbsoluteValue ModifyingAbove upper A With caret Subscript upper H Baseline left-parenthesis upper V comma upper W right-parenthesis minus upper A left-parenthesis upper V comma upper W right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C upper H Superscript k plus 1 Baseline double-vertical-bar upper V double-vertical-bar Subscript k plus 1 Baseline double-vertical-bar upper W double-vertical-bar Subscript 2 Baseline period EndLayout

Proof of Theorem 1.1.

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

double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript 1 Baseline less-than-or-equal-to upper C inf Underscript upper V element-of upper X Subscript upper H Baseline Endscripts left-parenthesis double-vertical-bar upper U 0 minus upper V double-vertical-bar Subscript 1 Baseline plus sup Underscript upper W element-of upper X Subscript upper H Baseline Endscripts StartFraction StartAbsoluteValue upper A Subscript upper H Baseline left-parenthesis upper V comma upper W right-parenthesis minus upper A left-parenthesis upper V comma upper W right-parenthesis EndAbsoluteValue Over double-vertical-bar upper W double-vertical-bar Subscript 1 Baseline EndFraction right-parenthesis period

Let upper V equals upper Pi upper U 0 and using Equation1.19 with m equals 1 comma p equals q equals 2 , we have

StartLayout 1st Row with Label left-parenthesis 2.5 right-parenthesis EndLabel inf Underscript upper V element-of upper X Subscript upper H Baseline Endscripts double-vertical-bar upper U 0 minus upper V double-vertical-bar Subscript 1 Baseline less-than-or-equal-to double-vertical-bar upper U 0 minus upper Pi upper U 0 double-vertical-bar Subscript 1 Baseline less-than-or-equal-to upper C upper H Superscript k Baseline period EndLayout

It remains to estimate StartAbsoluteValue upper A Subscript upper H Baseline left-parenthesis upper V comma upper W right-parenthesis minus upper A left-parenthesis upper V comma upper W right-parenthesis EndAbsoluteValue for upper V equals upper Pi upper U 0 and upper W element-of upper X Subscript upper H . Using Equation2.3, we get

StartLayout 1st Row 1st Column StartAbsoluteValue upper A Subscript upper H Baseline left-parenthesis upper V comma upper W right-parenthesis minus upper A left-parenthesis upper V comma upper W right-parenthesis EndAbsoluteValue 2nd Column StartAbsoluteValue upper A Subscript upper H Baseline left-parenthesis upper V comma upper W right-parenthesis minus ModifyingAbove upper A With caret Subscript upper H Baseline left-parenthesis upper V comma upper W right-parenthesis EndAbsoluteValue plus StartAbsoluteValue ModifyingAbove upper A With caret Subscript upper H Baseline left-parenthesis upper V comma upper W right-parenthesis minus upper A left-parenthesis upper V comma upper W right-parenthesis EndAbsoluteValue 2nd Row with Label left-parenthesis 2.6 right-parenthesis EndLabel 1st Column Blank 2nd Column left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis double-vertical-bar nabla upper V double-vertical-bar Subscript 0 Baseline plus upper C upper H Superscript k Baseline double-vertical-bar upper V double-vertical-bar Subscript k Baseline right-parenthesis double-vertical-bar nabla upper W double-vertical-bar Subscript 0 Baseline period EndLayout

This gives Equation1.11

To get the L squared estimate, we use the Aubin-Nitsche dual argument Reference10. To this end, consider the following auxiliary problem: Find w element-of upper H 0 Superscript 1 Baseline left-parenthesis upper D right-parenthesis such that

StartLayout 1st Row with Label left-parenthesis 2.7 right-parenthesis EndLabel upper A left-parenthesis v comma w right-parenthesis equals left-parenthesis upper U 0 minus upper U Subscript upper H Baseline comma v right-parenthesis for all v element-of upper H 0 Superscript 1 Baseline left-parenthesis upper D right-parenthesis period EndLayout

The standard regularity result reads Reference24

StartLayout 1st Row with Label left-parenthesis 2.8 right-parenthesis EndLabel double-vertical-bar w double-vertical-bar Subscript 2 Baseline less-than-or-equal-to upper C double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript 0 Baseline period EndLayout

Putting v equals upper U 0 minus upper U Subscript upper H into the right-hand side of Equation2.7, we obtain

StartLayout 1st Row 1st Column double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript 0 Superscript 2 2nd Column equals upper A left-parenthesis upper U 0 minus upper U Subscript upper H Baseline comma w minus upper Pi w right-parenthesis plus left-parenthesis upper A Subscript upper H Baseline left-parenthesis upper U Subscript upper H Baseline comma upper Pi w right-parenthesis minus upper A left-parenthesis upper U Subscript upper H Baseline comma upper Pi w right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column equals upper A left-parenthesis upper U 0 minus upper U Subscript upper H Baseline comma w minus upper Pi w right-parenthesis 3rd Row 1st Column Blank 2nd Column plus left-parenthesis upper A Subscript upper H Baseline left-parenthesis upper U Subscript upper H Baseline minus upper Pi upper U 0 comma upper Pi w right-parenthesis minus upper A left-parenthesis upper U Subscript upper H Baseline minus upper Pi upper U 0 comma upper Pi w right-parenthesis right-parenthesis 4th Row with Label left-parenthesis 2.9 right-parenthesis EndLabel 1st Column Blank 2nd Column plus left-parenthesis upper A Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis minus upper A left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis right-parenthesis period EndLayout

Using Equation2.6 with k equals 1 , we bound the first two terms in the right-hand side of the above identity as

StartAbsoluteValue upper A left-parenthesis upper U 0 minus upper U Subscript upper H Baseline comma w minus upper Pi w right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript 1 Baseline double-vertical-bar w minus upper Pi w double-vertical-bar Subscript 1 Baseline less-than-or-equal-to upper C upper H double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript 1 Baseline double-vertical-bar w double-vertical-bar Subscript 2

and

StartAbsoluteValue upper A Subscript upper H Baseline left-parenthesis upper U Subscript upper H Baseline minus upper Pi upper U 0 comma upper Pi w right-parenthesis minus upper A left-parenthesis upper U Subscript upper H Baseline minus upper Pi upper U 0 comma upper Pi w right-parenthesis EndAbsoluteValue less-than-or-equal-to left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper C upper H right-parenthesis double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript 1 Baseline double-vertical-bar upper Pi w double-vertical-bar Subscript 1 Baseline period

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

StartLayout 1st Row 1st Column upper A Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis minus upper A left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis 2nd Column equals left-parenthesis upper A Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis minus ModifyingAbove upper A With caret Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column plus left-parenthesis ModifyingAbove upper A With caret Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis minus upper A left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis right-parenthesis period EndLayout

It follows from Equation2.4 that

StartAbsoluteValue ModifyingAbove upper A With caret Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis minus upper A left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C upper H Superscript k plus 1 Baseline double-vertical-bar upper U 0 double-vertical-bar Subscript k plus 1 Baseline double-vertical-bar w double-vertical-bar Subscript 2 Baseline period

By definition of e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis and using Equation1.20, we get

StartAbsoluteValue upper A Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis minus ModifyingAbove upper A With caret Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma upper Pi w right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis double-vertical-bar nabla upper Pi upper U 0 double-vertical-bar Subscript 0 Baseline double-vertical-bar w double-vertical-bar Subscript 2 Baseline period

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

It remains to prove Equation1.13. As in Reference37, for any point z element-of upper D , we define the regularized Green’s function upper G Superscript z Baseline element-of upper H 0 Superscript 1 Baseline left-parenthesis upper D right-parenthesis and the discrete Green’s function upper G Subscript upper H Superscript z Baseline element-of upper X Subscript upper H as

StartLayout 1st Row with Label left-parenthesis 2.10 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper A left-parenthesis upper G Superscript z Baseline comma upper V right-parenthesis 2nd Column equals left-parenthesis delta Subscript z Baseline comma partial-differential upper V right-parenthesis for all upper V element-of upper H 0 Superscript 1 Baseline left-parenthesis upper D right-parenthesis comma 2nd Row 1st Column upper A left-parenthesis upper G Subscript upper H Superscript z Baseline comma upper V right-parenthesis 2nd Column equals left-parenthesis delta Subscript z Baseline comma partial-differential upper V right-parenthesis for all upper V element-of upper X Subscript upper H Baseline comma EndLayout EndLayout

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

StartLayout 1st Row with Label left-parenthesis 2.11 right-parenthesis EndLabel double-vertical-bar upper G Superscript z Baseline minus upper G Subscript upper H Superscript z Baseline double-vertical-bar Subscript 1 comma 1 Baseline less-than-or-equal-to upper C and double-vertical-bar upper G Subscript upper H Superscript z Baseline double-vertical-bar Subscript 1 comma 1 Baseline less-than-or-equal-to upper C StartAbsoluteValue ln upper H EndAbsoluteValue period EndLayout

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 upper G Superscript z and upper G Subscript upper H Superscript z , a simple manipulation gives

StartLayout 1st Row 1st Column partial-differential left-parenthesis upper U 0 minus upper U Subscript upper H Baseline right-parenthesis left-parenthesis z right-parenthesis 2nd Column equals upper A left-parenthesis upper G Superscript z Baseline comma upper U 0 minus upper Pi upper U 0 right-parenthesis plus upper A left-parenthesis upper G Superscript z Baseline comma upper Pi upper U 0 minus upper U Subscript upper H Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column equals upper A left-parenthesis upper G Superscript z Baseline minus upper G Subscript upper H Superscript z Baseline comma upper U 0 minus upper Pi upper U 0 right-parenthesis plus upper A left-parenthesis upper G Subscript upper H Superscript z Baseline comma upper U 0 minus upper U Subscript upper H Baseline right-parenthesis 3rd Row 1st Column Blank 2nd Column equals upper A left-parenthesis upper G Superscript z Baseline minus upper G Subscript upper H Superscript z Baseline comma upper U 0 minus upper Pi upper U 0 right-parenthesis plus upper A Subscript upper H Baseline left-parenthesis upper U Subscript upper H Baseline comma upper G Subscript upper H Superscript z Baseline right-parenthesis minus upper A left-parenthesis upper U Subscript upper H Baseline comma upper G Subscript upper H Superscript z Baseline right-parenthesis 4th Row 1st Column Blank 2nd Column equals upper A left-parenthesis upper G Superscript z Baseline minus upper G Subscript upper H Superscript z Baseline comma upper U 0 minus upper Pi upper U 0 right-parenthesis plus left-parenthesis upper A Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis minus upper A left-parenthesis upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis right-parenthesis 5th Row 1st Column Blank 2nd Column plus left-parenthesis upper A Subscript upper H Baseline left-parenthesis upper U Subscript upper H Baseline minus upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis minus upper A left-parenthesis upper U Subscript upper H Baseline minus upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis right-parenthesis period EndLayout

Using Equation2.11, we obtain

StartLayout 1st Row 1st Column double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript 1 comma normal infinity 2nd Column upper C double-vertical-bar upper U 0 minus upper Pi upper U 0 double-vertical-bar Subscript 1 comma normal infinity Baseline plus StartAbsoluteValue upper A left-parenthesis upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis minus upper A Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis EndAbsoluteValue 2nd Row 1st Column Blank 2nd Column plus StartAbsoluteValue upper A left-parenthesis upper U Subscript upper H Baseline minus upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis minus upper A Subscript upper H Baseline left-parenthesis upper U Subscript upper H Baseline minus upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis EndAbsoluteValue period EndLayout

Using Equation2.6, we get

StartLayout 1st Row 1st Column vertical-bar upper A left-parenthesis upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis minus upper A Subscript upper H Baseline left-parenthesis upper Pi upper U 0 comma 2nd Column upper G Subscript upper H Superscript z Baseline right-parenthesis vertical-bar left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper C upper H Superscript k Baseline right-parenthesis sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts double-vertical-bar upper Pi upper U 0 double-vertical-bar Subscript k comma upper K Baseline double-vertical-bar nabla upper G Subscript upper H Superscript z Baseline double-vertical-bar Subscript 0 comma upper K Baseline 2nd Row 1st Column Blank 2nd Column upper C left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper H Superscript k Baseline right-parenthesis sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts double-vertical-bar upper Pi upper U 0 double-vertical-bar Subscript k comma normal infinity comma upper K Baseline double-vertical-bar nabla upper G Subscript upper H Superscript z Baseline double-vertical-bar Subscript upper L Sub Superscript 1 Subscript left-parenthesis upper K right-parenthesis Baseline 3rd Row 1st Column Blank 2nd Column upper C left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper H Superscript k Baseline right-parenthesis StartAbsoluteValue ln upper H EndAbsoluteValue double-vertical-bar upper U 0 double-vertical-bar Subscript k plus 1 comma normal infinity Baseline comma EndLayout

where we have used the inverse inequality Equation1.21.

Similarly, we have

StartLayout 1st Row 1st Column Blank 2nd Column StartAbsoluteValue upper A left-parenthesis upper U Subscript upper H Baseline minus upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis minus upper A Subscript upper H Baseline left-parenthesis upper U Subscript upper H Baseline minus upper Pi upper U 0 comma upper G Subscript upper H Superscript z Baseline right-parenthesis EndAbsoluteValue 2nd Row 1st Column Blank 2nd Column left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper C upper H right-parenthesis sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts double-vertical-bar upper U Subscript upper H Baseline minus upper Pi upper U 0 double-vertical-bar Subscript 1 comma upper K Baseline double-vertical-bar nabla upper G Subscript upper H Superscript z Baseline double-vertical-bar Subscript 0 comma upper K Baseline 3rd Row 1st Column Blank 2nd Column upper C left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper H right-parenthesis sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts double-vertical-bar upper U Subscript upper H Baseline minus upper Pi upper U 0 double-vertical-bar Subscript 1 comma normal infinity comma upper K Baseline double-vertical-bar nabla upper G Subscript upper H Superscript z Baseline double-vertical-bar Subscript 0 comma 1 comma upper K Baseline 4th Row 1st Column Blank 2nd Column upper C left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper H right-parenthesis StartAbsoluteValue ln upper H EndAbsoluteValue double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript 1 comma normal infinity Baseline 5th Row 1st Column Blank 2nd Column plus upper C left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper H right-parenthesis StartAbsoluteValue ln upper H EndAbsoluteValue upper H Superscript k Baseline double-vertical-bar upper U 0 double-vertical-bar Subscript k plus 1 comma normal infinity Baseline period EndLayout

A combination of the above three estimates yields

StartLayout 1st Row 1st Column double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript 1 comma normal infinity 2nd Column upper C upper H Superscript k Baseline plus upper C left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper H right-parenthesis StartAbsoluteValue ln upper H EndAbsoluteValue double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript 1 comma normal infinity Baseline 2nd Row 1st Column Blank 2nd Column plus upper C left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper H Superscript k Baseline right-parenthesis StartAbsoluteValue ln upper H EndAbsoluteValue double-vertical-bar upper U 0 double-vertical-bar Subscript k plus 1 comma normal infinity Baseline period EndLayout

If e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis StartAbsoluteValue ln upper H EndAbsoluteValue less-than upper C 0 colon equals 1 slash left-parenthesis 2 upper C right-parenthesis , then there exits a constant upper H 0 such that for all upper H less-than-or-equal-to upper H 0 ,

upper C left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper H right-parenthesis StartAbsoluteValue ln upper H EndAbsoluteValue less-than-or-equal-to 1 slash 2 plus upper C upper H StartAbsoluteValue ln upper H EndAbsoluteValue less-than 1 period

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

Combining the foregoing proof for the L squared and W Superscript 1 comma normal infinity estimates, using the Green’s function defined in Reference39, we obtain

Remark 2.1

Under the same condition for the W Superscript 1 comma normal infinity estimate in Theorem Equation1.1, we have

double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript Baseline less-than-or-equal-to upper C left-parenthesis e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis plus upper H Superscript k plus 1 Baseline right-parenthesis StartAbsoluteValue ln upper H EndAbsoluteValue squared period

3. Estimating e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis

In this section, we estimate e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis for problems with locally periodic coefficients. The estimate of e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis for problems with random coefficients can be found in the Appendix.

We assume that a Superscript epsilon Baseline left-parenthesis x right-parenthesis equals a left-parenthesis x comma x slash epsilon right-parenthesis , where a Superscript epsilon is smooth in x and periodic in y with period upper I . Define kappa equals left floor delta slash epsilon right floor , and we introduce ModifyingAbove upper V With caret Subscript script l as

StartLayout 1st Row with Label left-parenthesis 3.1 right-parenthesis EndLabel ModifyingAbove upper V With caret Subscript script l Baseline left-parenthesis x right-parenthesis equals upper V Subscript script l Baseline left-parenthesis x right-parenthesis plus epsilon chi Superscript k Baseline left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis StartFraction partial-differential upper V Subscript script l Baseline Over partial-differential x Subscript k Baseline EndFraction left-parenthesis x right-parenthesis comma EndLayout

where StartSet chi Superscript j Baseline EndSet Subscript j equals 1 Superscript d is defined as: For j equals 1 comma ellipsis comma d , chi Superscript j Baseline left-parenthesis x comma y right-parenthesis is periodic in y with period upper I and satisfies

StartLayout 1st Row with Label left-parenthesis 3.2 right-parenthesis EndLabel minus StartFraction partial-differential Over partial-differential y Subscript i Baseline EndFraction left-parenthesis a Subscript i k Baseline StartFraction partial-differential chi Superscript j Baseline Over partial-differential y Subscript k Baseline EndFraction right-parenthesis left-parenthesis x comma y right-parenthesis equals StartFraction partial-differential Over partial-differential y Subscript i Baseline EndFraction a Subscript i j Baseline left-parenthesis x comma y right-parenthesis in upper I comma integral Underscript upper I Endscripts chi Superscript j Baseline left-parenthesis x comma y right-parenthesis d y equals 0 period EndLayout

Given StartSet chi Superscript j Baseline EndSet Subscript j equals 1 Superscript d , the homogenized coefficient script upper A equals left-parenthesis script upper A Subscript i j Baseline left-parenthesis x right-parenthesis right-parenthesis is given by

script upper A Subscript i j Baseline left-parenthesis x right-parenthesis equals integral minus Subscript upper I Baseline left-parenthesis a Subscript i j Baseline plus a Subscript i k Baseline StartFraction partial-differential chi Superscript j Baseline Over partial-differential y Subscript k Baseline EndFraction right-parenthesis left-parenthesis x comma y right-parenthesis d y period

Note that StartSet chi Superscript j Baseline EndSet Subscript j equals 1 Superscript d is smooth and bounded in all norms.

First let us consider the case when upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis equals x Subscript script l Baseline plus epsilon upper I , and Equation1.6 is solved with the periodic boundary condition. Denote by ModifyingAbove v With caret Subscript script l Superscript epsilon the solution of Equation1.6 with the coefficients a Superscript epsilon Baseline left-parenthesis x right-parenthesis replaced by a left-parenthesis x Subscript script l Baseline comma x slash epsilon right-parenthesis . ModifyingAbove v With caret Subscript script l Superscript epsilon may be viewed as a perturbation of v Subscript script l Superscript epsilon . Using Lemma Equation1.8, we get

StartLayout 1st Row with Label left-parenthesis 3.3 right-parenthesis EndLabel double-vertical-bar nabla left-parenthesis v Subscript script l Superscript epsilon Baseline minus ModifyingAbove v With caret Subscript script l Superscript epsilon Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper I Sub Subscript epsilon Subscript Baseline less-than-or-equal-to upper C epsilon double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript epsilon Subscript Baseline period EndLayout

Observe that ModifyingAbove v With caret Subscript script l Superscript epsilon Baseline equals ModifyingAbove upper V With caret Subscript script l . A direct calculation yields

StartLayout 1st Row 1st Column left-parenthesis nabla upper W dot left-parenthesis script upper A Subscript upper H Baseline minus script upper A right-parenthesis nabla upper V right-parenthesis left-parenthesis x Subscript script l Baseline right-parenthesis 2nd Column equals integral minus Subscript upper I Sub Subscript epsilon Baseline nabla w Subscript script l Superscript epsilon Baseline dot left-bracket a left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis minus a left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis right-bracket nabla v Subscript script l Superscript epsilon Baseline d x 2nd Row 1st Column Blank 2nd Column plus integral minus Subscript upper I Sub Subscript epsilon Subscript Baseline nabla w Subscript script l Superscript epsilon Baseline dot a left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis nabla left-parenthesis v Subscript script l Superscript epsilon Baseline minus ModifyingAbove v With caret Subscript script l Superscript epsilon Baseline right-parenthesis d x period EndLayout

Using Equation3.3, we get

StartLayout 1st Row with Label left-parenthesis 3.4 right-parenthesis EndLabel e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis less-than-or-equal-to upper C epsilon period EndLayout

Next we consider the more general case when upper I Subscript delta is a cube of size delta not necessarily equal to epsilon . The following analysis applies equally well to the case when the period of a left-parenthesis x comma dot right-parenthesis is of general and even nonpolygonal shape. This situation arises in some examples of composite materials Reference30. We will show that if delta is much larger than epsilon , 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:

StartLayout 1st Row 1st Column left-parenthesis nabla upper W dot script upper A nabla upper V right-parenthesis left-parenthesis x Subscript script l Baseline right-parenthesis 2nd Column equals nabla upper W Subscript script l Baseline left-parenthesis x right-parenthesis dot script upper A left-parenthesis x Subscript script l Baseline right-parenthesis nabla upper V Subscript script l Baseline left-parenthesis x right-parenthesis 2nd Row with Label left-parenthesis 3.5 right-parenthesis EndLabel 1st Column Blank 2nd Column equals integral minus Subscript upper I Sub Subscript kappa epsilon Subscript left-parenthesis x Sub Subscript script l Subscript right-parenthesis Baseline nabla upper W Subscript script l Baseline dot a left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis nabla ModifyingAbove upper V With caret Subscript script l Baseline d x period EndLayout

We first establish some estimates on the solution of the cell problem Equation1.6. We will write upper I Subscript delta instead of upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis if there is no risk of confusion.

Lemma 3.1

There exists a constant upper C independent of epsilon and delta such that for each script l ,

StartLayout 1st Row with Label left-parenthesis 3.6 right-parenthesis EndLabel double-vertical-bar nabla v Subscript script l Superscript epsilon Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript kappa epsilon Subscript Baseline less-than-or-equal-to upper C left-parenthesis left-parenthesis StartFraction epsilon Over delta EndFraction right-parenthesis Superscript 1 slash 2 Baseline plus delta right-parenthesis double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline period EndLayout

Proof.

We still denote by ModifyingAbove v With caret Subscript script l Superscript epsilon the solution of Equation1.6 with the coefficient a Superscript epsilon Baseline left-parenthesis x right-parenthesis replaced by a left-parenthesis x Subscript script l Baseline comma x slash epsilon right-parenthesis . Using Lemma Equation1.8, we get

StartLayout 1st Row with Label left-parenthesis 3.7 right-parenthesis EndLabel double-vertical-bar nabla left-parenthesis v Subscript script l Superscript epsilon Baseline minus ModifyingAbove v With caret Subscript script l Superscript epsilon Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline less-than-or-equal-to upper C delta double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline period EndLayout

Define theta Subscript script l Superscript epsilon Baseline equals ModifyingAbove v With caret Subscript script l Superscript epsilon Baseline minus ModifyingAbove upper V With caret Subscript script l , which obviously satisfies

StartLayout 1st Row with Label left-parenthesis 3.8 right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row 1st Column minus d i v left-parenthesis a left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis nabla theta Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis right-parenthesis 2nd Column equals 0 3rd Column Blank 4th Column x element-of upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis comma 2nd Row 1st Column theta Subscript script l Superscript epsilon Baseline left-parenthesis x right-parenthesis 2nd Column equals minus epsilon chi Superscript k Baseline left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis StartFraction partial-differential upper V Subscript script l Baseline Over partial-differential x Subscript k Baseline EndFraction left-parenthesis x right-parenthesis 3rd Column Blank 4th Column x element-of partial-differential upper I Subscript delta Baseline left-parenthesis x Subscript script l Baseline right-parenthesis period EndLayout EndLayout

Note that theta Subscript script l Superscript epsilon 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 x prime equals x slash delta over upper I Subscript delta and epsilon prime equals epsilon slash delta .

StartLayout 1st Row with Label left-parenthesis 3.9 right-parenthesis EndLabel double-vertical-bar nabla theta Subscript script l Superscript epsilon Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline less-than-or-equal-to upper C left-parenthesis StartFraction epsilon Over delta EndFraction right-parenthesis Superscript 1 slash 2 Baseline double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline period EndLayout

This together with Equation3.7 gives

StartLayout 1st Row with Label left-parenthesis 3.10 right-parenthesis EndLabel double-vertical-bar nabla left-parenthesis v Subscript script l Superscript epsilon Baseline minus ModifyingAbove upper V With caret Subscript script l Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline less-than-or-equal-to upper C left-parenthesis left-parenthesis StartFraction epsilon Over delta EndFraction right-parenthesis Superscript 1 slash 2 Baseline plus delta right-parenthesis double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline period EndLayout

A straightforward calculation gives

StartLayout 1st Row with Label left-parenthesis 3.11 right-parenthesis EndLabel double-vertical-bar nabla ModifyingAbove upper V With caret Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript kappa epsilon Subscript Baseline less-than-or-equal-to upper C left-parenthesis StartFraction epsilon Over delta EndFraction right-parenthesis Superscript 1 slash 2 Baseline double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline comma EndLayout

which together with Equation3.10 leads to

StartLayout 1st Row 1st Column double-vertical-bar nabla v Subscript script l Superscript epsilon Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript kappa epsilon Subscript 2nd Column double-vertical-bar nabla ModifyingAbove upper V With caret Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript kappa epsilon Subscript Baseline plus double-vertical-bar nabla left-parenthesis v Subscript script l Superscript epsilon Baseline minus ModifyingAbove upper V With caret Subscript script l Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript kappa epsilon Subscript Baseline 2nd Row 1st Column Blank 2nd Column double-vertical-bar nabla ModifyingAbove upper V With caret Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript kappa epsilon Subscript Baseline plus double-vertical-bar nabla left-parenthesis v Subscript script l Superscript epsilon Baseline minus ModifyingAbove upper V With caret Subscript script l Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline 3rd Row 1st Column Blank 2nd Column upper C left-parenthesis left-parenthesis StartFraction epsilon Over delta EndFraction right-parenthesis Superscript 1 slash 2 Baseline plus delta right-parenthesis double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline period EndLayout

This gives Equation3.6.

As in Equation3.6, we also have

StartLayout 1st Row with Label left-parenthesis 3.12 right-parenthesis EndLabel double-vertical-bar nabla v Subscript script l Superscript epsilon Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript left-parenthesis kappa minus 2 right-parenthesis epsilon Subscript Baseline less-than-or-equal-to upper C left-parenthesis left-parenthesis StartFraction epsilon Over delta EndFraction right-parenthesis Superscript 1 slash 2 Baseline plus delta right-parenthesis double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline period EndLayout

Theorem 3.2

StartLayout 1st Row with Label left-parenthesis 3.13 right-parenthesis EndLabel e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis less-than-or-equal-to upper C left-parenthesis StartFraction epsilon Over delta EndFraction plus delta right-parenthesis period EndLayout

Proof.

Note that v Subscript script l Superscript epsilon Baseline equals left-parenthesis v Subscript script l Superscript epsilon Baseline minus ModifyingAbove v With caret Subscript script l Superscript epsilon Baseline right-parenthesis plus theta Subscript script l Superscript epsilon Baseline plus ModifyingAbove upper V With caret Subscript script l . We have

left-parenthesis nabla upper W dot left-parenthesis script upper A Subscript upper H Baseline minus script upper A right-parenthesis nabla upper V right-parenthesis left-parenthesis x Subscript script l Baseline right-parenthesis equals colon upper I 1 plus upper I 2 plus upper I 3 comma

where

StartLayout 1st Row 1st Column upper I 1 2nd Column equals integral minus Subscript upper I Sub Subscript delta Subscript Baseline nabla w Subscript script l Superscript epsilon Baseline dot a left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis nabla left-parenthesis v Subscript script l Superscript epsilon Baseline minus ModifyingAbove v With caret Subscript script l Superscript epsilon Baseline right-parenthesis d x comma upper I 2 equals integral minus Subscript upper I Sub Subscript delta Subscript Baseline nabla w Subscript script l Superscript epsilon Baseline dot a left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis nabla theta Subscript script l Superscript epsilon Baseline d x comma 2nd Row 1st Column upper I 3 2nd Column equals integral minus Subscript upper I Sub Subscript delta Subscript Baseline nabla w Subscript script l Superscript epsilon Baseline dot a left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis nabla ModifyingAbove upper V With caret Subscript script l Baseline d x minus nabla upper W Subscript script l Baseline dot script upper A left-parenthesis x Subscript script l Baseline right-parenthesis nabla upper V Subscript script l Baseline period EndLayout

Using Equation3.7 and Equation2.2, we bound upper I 1 as

StartLayout 1st Row 1st Column StartAbsoluteValue upper I 1 EndAbsoluteValue 2nd Column upper Lamda delta Superscript negative d Baseline double-vertical-bar nabla left-parenthesis v Subscript script l Superscript epsilon Baseline minus ModifyingAbove v With caret Subscript script l Superscript epsilon Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline double-vertical-bar nabla w Subscript script l Superscript epsilon Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline 2nd Row 1st Column Blank 2nd Column upper C delta Superscript 1 minus d Baseline double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline double-vertical-bar nabla upper W Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline less-than-or-equal-to upper C delta StartAbsoluteValue nabla upper V Subscript script l Baseline EndAbsoluteValue StartAbsoluteValue nabla upper W Subscript script l Baseline EndAbsoluteValue period EndLayout

Using the symmetry of a Superscript epsilon , upper I 2 equals integral minus Subscript upper I Sub Subscript delta Baseline nabla theta Subscript script l Superscript epsilon Baseline dot a left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis nabla w Subscript script l Superscript epsilon Baseline d x and

StartLayout 1st Row 1st Column upper I 2 2nd Column equals integral minus Subscript upper I Sub Subscript delta Baseline nabla left-parenthesis theta Subscript script l Superscript epsilon Baseline plus epsilon chi Superscript k Baseline left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis StartFraction partial-differential upper V Subscript script l Baseline Over partial-differential x Subscript k Baseline EndFraction left-parenthesis 1 minus rho Superscript epsilon Baseline right-parenthesis right-parenthesis dot a left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis nabla w Subscript script l Superscript epsilon Baseline d x 2nd Row 1st Column Blank 2nd Column minus integral minus Subscript upper I Sub Subscript delta Subscript Baseline nabla left-parenthesis epsilon chi Superscript k Baseline left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis StartFraction partial-differential upper V Subscript script l Baseline Over partial-differential x Subscript k Baseline EndFraction left-parenthesis 1 minus rho Superscript epsilon Baseline right-parenthesis right-parenthesis dot a left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis nabla w Subscript script l Superscript epsilon Baseline d x comma EndLayout

where rho Superscript epsilon Baseline left-parenthesis x right-parenthesis element-of script upper C 0 Superscript normal infinity Baseline left-parenthesis upper I Subscript delta Baseline right-parenthesis comma StartAbsoluteValue nabla rho Superscript epsilon Baseline EndAbsoluteValue less-than-or-equal-to upper C slash epsilon , and

StartLayout 1st Row with Label left-parenthesis 3.14 right-parenthesis EndLabel rho Superscript epsilon Baseline left-parenthesis x right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column if dist left-parenthesis x comma partial-differential upper I Subscript delta Baseline right-parenthesis greater-than-or-equal-to 2 epsilon comma 2nd Row 1st Column 0 2nd Column if dist left-parenthesis x comma partial-differential upper I Subscript delta Baseline right-parenthesis less-than-or-equal-to epsilon period EndLayout EndLayout

Using Equation1.6 and theta Subscript script l Superscript epsilon Baseline plus epsilon chi Superscript k Baseline left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis StartFraction partial-differential upper V Subscript script l Baseline Over partial-differential x Subscript k Baseline EndFraction left-parenthesis 1 minus rho Superscript epsilon Baseline right-parenthesis element-of upper H 0 Superscript 1 Baseline left-parenthesis upper I Subscript delta Baseline right-parenthesis , integrating by parts makes the first term in the right-hand side of upper I 2 vanish; therefore we write upper I 2 as

upper I 2 equals minus integral minus Subscript upper I Sub Subscript delta Subscript Baseline a Subscript i j Baseline left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis StartFraction partial-differential w Subscript script l Superscript epsilon Baseline Over partial-differential x Subscript i Baseline EndFraction StartFraction partial-differential chi Superscript k Baseline Over partial-differential y Subscript j Baseline EndFraction StartFraction partial-differential upper V Subscript script l Baseline Over partial-differential x Subscript k Baseline EndFraction left-parenthesis 1 minus rho Superscript epsilon Baseline right-parenthesis d x plus epsilon integral minus Subscript upper I Sub Subscript delta Subscript Baseline a Subscript i j Baseline left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis StartFraction partial-differential w Subscript script l Superscript epsilon Baseline Over partial-differential x Subscript i Baseline EndFraction chi Superscript k Baseline StartFraction partial-differential upper V Subscript script l Baseline Over partial-differential x Subscript k Baseline EndFraction StartFraction partial-differential rho Superscript epsilon Baseline Over partial-differential x Subscript j Baseline EndFraction d x period

Using Equation3.12, we bound upper I 2 as

StartAbsoluteValue upper I 2 EndAbsoluteValue less-than-or-equal-to upper C delta Superscript negative d Baseline double-vertical-bar nabla w Subscript script l Superscript epsilon Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript left-parenthesis kappa minus 2 right-parenthesis epsilon Subscript Baseline double-vertical-bar nabla upper V Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript left-parenthesis kappa minus 2 right-parenthesis epsilon Subscript Baseline less-than-or-equal-to upper C left-parenthesis StartFraction epsilon Over delta EndFraction plus delta squared right-parenthesis StartAbsoluteValue nabla upper W Subscript script l Baseline EndAbsoluteValue StartAbsoluteValue nabla upper V Subscript script l Baseline EndAbsoluteValue period

Using Equation3.2 and integrating by parts, we obtain

integral minus Subscript upper I Sub Subscript delta Subscript Baseline nabla w Subscript script l Superscript epsilon Baseline dot a left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis nabla ModifyingAbove upper V With caret Subscript script l Baseline d x equals integral minus Subscript upper I Sub Subscript delta Subscript Baseline nabla upper W Subscript script l Baseline dot a left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis nabla ModifyingAbove upper V With caret Subscript script l Baseline d x comma

which together with Equation3.5 gives

StartLayout 1st Row 1st Column upper I 3 2nd Column equals integral minus Subscript upper I Sub Subscript delta Baseline nabla w Subscript script l Superscript epsilon Baseline left-bracket a left-parenthesis x comma StartFraction x Over epsilon EndFraction right-parenthesis minus a left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis right-bracket nabla ModifyingAbove upper V With caret Subscript script l Baseline d x 2nd Row 1st Column Blank 2nd Column plus StartFraction 1 Over delta Superscript d Baseline EndFraction integral Underscript upper I Subscript delta Baseline minus upper I Subscript kappa epsilon Baseline Endscripts nabla upper W Subscript script l Baseline dot a left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis nabla ModifyingAbove upper V With caret Subscript script l Baseline d x plus left-parenthesis StartAbsoluteValue kappa epsilon slash delta EndAbsoluteValue Superscript d Baseline minus 1 right-parenthesis nabla upper W Subscript script l Baseline dot script upper A left-parenthesis x Subscript script l Baseline right-parenthesis nabla upper V Subscript script l Baseline period EndLayout

The last term of upper I 3 is bounded by

StartAbsoluteValue StartAbsoluteValue kappa epsilon slash delta EndAbsoluteValue Superscript d Baseline minus 1 EndAbsoluteValue StartAbsoluteValue nabla upper W Subscript script l Baseline dot script upper A left-parenthesis x Subscript script l Baseline right-parenthesis nabla upper V Subscript script l Baseline EndAbsoluteValue less-than-or-equal-to upper C StartFraction epsilon Over delta EndFraction StartAbsoluteValue nabla upper V Subscript script l Baseline EndAbsoluteValue StartAbsoluteValue nabla upper W Subscript script l Baseline EndAbsoluteValue comma

where we have used StartAbsoluteValue StartAbsoluteValue kappa epsilon slash delta EndAbsoluteValue Superscript d Baseline minus 1 EndAbsoluteValue less-than-or-equal-to upper C epsilon slash delta . Using Equation3.11, we get

StartLayout 1st Row 1st Column delta Superscript negative d Baseline StartAbsoluteValue integral Underscript upper I Subscript delta Baseline minus upper I Subscript kappa epsilon Baseline Endscripts nabla upper W Subscript script l Baseline dot a left-parenthesis x Subscript script l Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis nabla ModifyingAbove upper V With caret Subscript script l Baseline d x EndAbsoluteValue 2nd Column upper C StartFraction 1 Over delta Superscript d Baseline EndFraction double-vertical-bar nabla ModifyingAbove upper V With caret Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript kappa epsilon Subscript Baseline double-vertical-bar nabla upper W Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript minus upper I Sub Subscript kappa epsilon Subscript Baseline 2nd Row 1st Column Blank 2nd Column upper C left-parenthesis StartFraction epsilon Over delta EndFraction plus delta squared right-parenthesis StartAbsoluteValue nabla upper V Subscript script l Baseline EndAbsoluteValue StartAbsoluteValue nabla upper W Subscript script l Baseline EndAbsoluteValue period EndLayout

Consequently, we obtain

StartLayout 1st Row 1st Column StartAbsoluteValue upper I 3 EndAbsoluteValue 2nd Column upper C delta Superscript 1 minus d Baseline double-vertical-bar nabla w Subscript script l Superscript epsilon Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline double-vertical-bar nabla ModifyingAbove upper V With caret Subscript script l Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript delta Subscript Baseline plus upper C left-parenthesis StartFraction epsilon Over delta EndFraction plus delta squared right-parenthesis StartAbsoluteValue nabla upper V Subscript script l Baseline EndAbsoluteValue StartAbsoluteValue nabla upper W Subscript script l Baseline EndAbsoluteValue 2nd Row 1st Column Blank 2nd Column upper C left-parenthesis StartFraction epsilon Over delta EndFraction plus delta right-parenthesis StartAbsoluteValue nabla upper V Subscript script l Baseline EndAbsoluteValue StartAbsoluteValue nabla upper W Subscript script l Baseline EndAbsoluteValue period EndLayout

Combining the estimates for upper I 1 comma upper I 2 and upper I 3 gives the desired result Equation3.13.

Remark 3.3

An explicit expression for v Subscript script l Superscript epsilon is available in one dimension, from which we may show that the bound for e left-parenthesis normal upper H normal upper M normal upper M right-parenthesis is sharp.

4. Reconstruction and compression

4.1. Reconstruction procedure

Next we consider how to construct better approximations to u Superscript epsilon from upper U Subscript upper H . We will restrict ourselves to the case when k equals 1 .

Proof of Theorem 1.4.

Subtracting Equation1.1 from Equation1.14, we obtain

StartLayout Enlarged left-brace 1st Row 1st Column minus d i v left-parenthesis a Superscript epsilon Baseline left-parenthesis x right-parenthesis nabla left-parenthesis u overTilde Superscript epsilon Baseline minus u Superscript epsilon Baseline right-parenthesis left-parenthesis x right-parenthesis right-parenthesis 2nd Column equals 0 3rd Column Blank 4th Column x element-of upper Omega Subscript eta Baseline comma 2nd Row 1st Column left-parenthesis u overTilde Superscript epsilon Baseline minus u Superscript epsilon Baseline right-parenthesis left-parenthesis x right-parenthesis 2nd Column equals upper U Subscript upper H Baseline left-parenthesis x right-parenthesis minus u Superscript epsilon Baseline left-parenthesis x right-parenthesis 3rd Column Blank 4th Column x element-of partial-differential upper Omega Subscript eta Baseline period EndLayout

Using classical interior estimates for elliptic equation Reference24, we have

double-vertical-bar nabla left-parenthesis u overTilde Superscript epsilon Baseline minus u Superscript epsilon Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper Omega Baseline less-than-or-equal-to StartFraction upper C Over eta EndFraction double-vertical-bar u overTilde Superscript epsilon Baseline minus u Superscript epsilon Baseline double-vertical-bar Subscript 0 comma upper Omega Sub Subscript eta Subscript Baseline period

Using the Hopf maximum principle, we get

StartLayout 1st Row 1st Column StartFraction 1 Over eta squared EndFraction integral minus Subscript upper Omega Sub Subscript eta Baseline StartAbsoluteValue left-parenthesis u overTilde Superscript epsilon Baseline minus u Superscript epsilon Baseline right-parenthesis left-parenthesis x right-parenthesis EndAbsoluteValue squared d x 2nd Column StartFraction upper C Over eta squared EndFraction double-vertical-bar u overTilde Superscript epsilon Baseline minus u Superscript epsilon Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper Omega Sub Subscript eta Subscript right-parenthesis Superscript 2 Baseline less-than-or-equal-to StartFraction upper C Over eta squared EndFraction double-vertical-bar u Superscript epsilon Baseline minus upper U Subscript upper H Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis partial-differential upper Omega Sub Subscript eta Subscript right-parenthesis Superscript 2 Baseline 2nd Row 1st Column Blank 2nd Column StartFraction upper C Over eta squared EndFraction left-parenthesis double-vertical-bar upper U 0 minus upper U Subscript upper H Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper Omega Sub Subscript eta Subscript right-parenthesis Superscript 2 Baseline plus double-vertical-bar u Superscript epsilon Baseline minus upper U 0 double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper Omega Sub Subscript eta Subscript right-parenthesis Superscript 2 Baseline right-parenthesis period EndLayout

A combination of the above two results implies Theorem 1.4.

Proof of Theorem 1.5.

Denote upper I Subscript epsilon Baseline left-parenthesis x Subscript upper K Baseline right-parenthesis equals x Subscript upper K Baseline plus epsilon upper I and define ModifyingAbove u With caret Superscript epsilon as the solution of

StartLayout 1st Row with Label left-parenthesis 4.1 right-parenthesis EndLabel minus d i v left-parenthesis a left-parenthesis x Subscript upper K Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis nabla ModifyingAbove u With caret Superscript epsilon Baseline left-parenthesis x right-parenthesis right-parenthesis equals 0 in upper I Subscript epsilon Baseline left-parenthesis x Subscript upper K Baseline right-parenthesis comma EndLayout

with the boundary condition that ModifyingAbove u With caret Superscript epsilon Baseline minus upper U Subscript upper H is periodic on partial-differential upper I Subscript epsilon Baseline left-parenthesis x Subscript upper K Baseline right-parenthesis and

integral Underscript upper I Subscript epsilon Baseline left-parenthesis x Subscript upper K Baseline right-parenthesis Endscripts left-parenthesis ModifyingAbove u With caret Superscript epsilon Baseline minus upper U Subscript upper H Baseline right-parenthesis d x equals 0 comma

where x Subscript upper K is the barycenter of upper K .

It is easy to verify that ModifyingAbove u With caret Superscript epsilon takes the explicit form

StartLayout 1st Row with Label left-parenthesis 4.2 right-parenthesis EndLabel ModifyingAbove u With caret Superscript epsilon Baseline left-parenthesis x right-parenthesis equals upper U Subscript upper H Baseline left-parenthesis x right-parenthesis plus epsilon chi Superscript k Baseline left-parenthesis x Subscript upper K Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis StartFraction partial-differential upper U Subscript upper H Baseline Over partial-differential x Subscript k Baseline EndFraction left-parenthesis x right-parenthesis period EndLayout

Note that the periodic extension of ModifyingAbove u With caret Superscript epsilon Baseline minus upper U Subscript upper H is still epsilon chi Superscript k Baseline left-parenthesis x Subscript upper K Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis StartFraction partial-differential upper U Subscript upper H Baseline Over partial-differential x Subscript k Baseline EndFraction left-parenthesis x right-parenthesis . This means that ModifyingAbove u With caret Superscript epsilon is also well defined for the whole of upper K and takes the same explicit form as Equation4.2.

Using integral Underscript upper I Endscripts chi Superscript k Baseline left-parenthesis x Subscript upper K Baseline comma y right-parenthesis d y equals 0 for k equals 1 comma ellipsis comma d and that nabla upper U Subscript upper H is a piecewise constant on upper K , we obtain

StartLayout 1st Row with Label left-parenthesis 4.3 right-parenthesis EndLabel integral Underscript upper I Subscript epsilon Baseline left-parenthesis x Subscript upper K Baseline right-parenthesis Endscripts left-parenthesis ModifyingAbove u With caret Superscript epsilon Baseline minus upper U Subscript upper H Baseline right-parenthesis left-parenthesis x right-parenthesis d x equals integral Underscript upper I Subscript epsilon Baseline left-parenthesis x Subscript upper K Baseline right-parenthesis Endscripts epsilon chi Superscript k Baseline left-parenthesis x Subscript upper K Baseline comma StartFraction x Over epsilon EndFraction right-parenthesis StartFraction partial-differential upper U Subscript upper H Baseline Over partial-differential x Subscript k Baseline EndFraction left-parenthesis x right-parenthesis d x equals 0 period EndLayout

As in Equation3.7, we have

double-vertical-bar nabla left-parenthesis u overTilde Superscript epsilon Baseline minus ModifyingAbove u With caret Superscript epsilon Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper I Sub Subscript epsilon Subscript left-parenthesis x Sub Subscript upper K Subscript right-parenthesis Baseline less-than-or-equal-to upper C epsilon double-vertical-bar nabla upper U Subscript upper H Baseline double-vertical-bar Subscript 0 comma upper I Sub Subscript epsilon Subscript left-parenthesis x Sub Subscript upper K Subscript right-parenthesis Baseline period

From the construction of u overTilde Superscript epsilon , we have for any x 1 element-of upper K ,

double-vertical-bar nabla left-parenthesis u overTilde Superscript epsilon Baseline minus ModifyingAbove u With caret Superscript epsilon Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper I Sub Subscript epsilon Subscript left-parenthesis x 1 right-parenthesis Baseline equals double-vertical-bar nabla left-parenthesis u overTilde Superscript epsilon Baseline minus ModifyingAbove u With caret Superscript epsilon Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper I Sub Subscript epsilon Subscript left-parenthesis x Sub Subscript upper K Subscript right-parenthesis Baseline period

Since nabla upper U Subscript upper H is constant over upper K , we get

StartLayout 1st Row with Label left-parenthesis 4.4 right-parenthesis EndLabel double-vertical-bar nabla left-parenthesis u overTilde Superscript epsilon Baseline minus ModifyingAbove u With caret Superscript epsilon Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper K Baseline less-than-or-equal-to upper C epsilon double-vertical-bar nabla upper U Subscript upper H Baseline double-vertical-bar Subscript 0 comma upper K Baseline period EndLayout

Adding up for all upper K element-of script upper T Subscript upper H and using the a priori estimate double-vertical-bar nabla upper U Subscript upper H Baseline double-vertical-bar Subscript 0 Baseline less-than-or-equal-to upper C double-vertical-bar f double-vertical-bar Subscript 0 , we obtain

StartLayout 1st Row with Label left-parenthesis 4.5 right-parenthesis EndLabel left-parenthesis sigma-summation Underscript upper K element-of script upper T Subscript upper H Baseline Endscripts double-vertical-bar nabla left-parenthesis u overTilde Superscript epsilon Baseline minus ModifyingAbove u With caret Superscript epsilon Baseline right-parenthesis double-vertical-bar Subscript 0 comma upper K Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline less-than-or-equal-to upper C epsilon double-vertical-bar nabla upper U Subscript upper H Baseline double-vertical-bar Subscript 0 Baseline less-than-or-equal-to upper C epsilon period EndLayout