an iterative method for elliptic problems with rapidly ......and jean-christophe mourrat4,*...
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ESAIM: M2AN 55 (2021) 37โ55 ESAIM: Mathematical Modelling and Numerical Analysishttps://doi.org/10.1051/m2an/2020080 www.esaim-m2an.org
AN ITERATIVE METHOD FOR ELLIPTIC PROBLEMS WITH RAPIDLYOSCILLATING COEFFICIENTS
Scott Armstrong1, Antti Hannukainen2, Tuomo Kuusi3
and Jean-Christophe Mourrat4,*
Abstract. We introduce a new iterative method for computing solutions of elliptic equations with ran-dom rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involvesdifferent computations meant to address different length scales. However, we use here the homogenizedequation on all scales larger than a fixed multiple of the scale of oscillation of the coefficients. While theperformance of standard multigrid methods degrades rapidly under the regime of large scale separationthat we consider here, we show an explicit estimate on the contraction factor of our method whichis independent of the size of the domain. We also present numerical experiments which confirm theeffectiveness of the method, with openly available source code.
Mathematics Subject Classification. 65N55, 35B27.
Received March 27, 2020. Accepted November 19, 2020.
1. Introduction
1.1. Informal summary of results
In this paper, we introduce a new iterative method for the numerical approximation of solutions of ellipticproblems with rapidly oscillating coefficients. For definiteness, we consider the Dirichlet problem
๐๐๐ โโ ยท (a(๐ฅ)โ๐ข) = ๐ in ๐๐,
๐ข = ๐ on ๐๐๐,(1.1)
where ๐ > 0 is the length scale of the problem, which is typically very large (๐ โซ 1), and we write ๐๐ := ๐๐where ๐ โ R๐ is a bounded ๐ถ1,1 domain, in dimension ๐ > 2. The boundary condition ๐ belongs to ๐ป1(๐๐), andthe right-hand side ๐ belongs to ๐ปโ1(๐๐). The coefficients a(๐ฅ) are symmetric, uniformly elliptic and Holdercontinuous. Moreover, in order to ensure that quantitative homogenization holds on large scales, we assume thatthe coefficients are sampled by a probability measure which is Z๐-stationary and has a unit range of dependence
Keywords and phrases. Multiscale method, multigrid method, homogenization.
1 Courant Institute of Mathematical Sciences, New York University, New York, USA.2 Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland.3 Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland.4 DMA, Ecole normale superieure, CNRS, PSL Research University, Paris, France.*Corresponding author: [email protected]
c The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
38 S. ARMSTRONG ET AL.
(see below for the precise formulation of these assumptions). Our goal is to build a numerical method for thecomputation of ๐ข which remains efficient in the regime of fast oscillations of the coefficient field (which inour setting corresponds to the case in which the length scale is very large, ๐ โซ 1) and does not rely on scaleseparation for convergence (the method computes the true solution for fixed ๐ and not only in the limit ๐ โโ).
In the absence of fast oscillations of the coefficient field, contemporary technology allows to access numericalapproximations of elliptic problems involving billions of degrees of freedom. One of the most successful methodsallowing to achieve such results is the multigrid method (see [16] for benchmarks). However, the performance ofthis method degrades as the coefficient field becomes more rapidly oscillating (see for instance [35], Tab. IV).
We seek to remedy this problem by leveraging on homogenization. While standard multigrid methods usea decomposition of the elliptic problem into a series of scales, the difficulty in our context is that the sloweigenmodes of the heterogeneous operator still have fast oscillations, and are thus not easily captured through acoarse representation. We overcome this by introducing a suitable variant of the multigrid method that succeedsin replacing the heterogeneous operator by the homogenized one on length scales larger than a large but finitemultiple of the correlation length scale. The result is a new iterative method that converges exponentially fastin the number of iterations, each of which is relatively inexpensive to compute โ the memory and number ofcomputations required scale linearly in the volume, and the computation is very amenable to parallelization.We give a rigorous proof of convergence and present numerical experiments which establish the efficiency of themethod from a practical point of view.
1.2. Statement of the main result
We introduce some notation in order to state our main result. We begin with the precise assumptions on thecoefficient field. We fix parameters ฮ > 1 and ๐ผ โ (0, 1] and require our coefficient fields a(๐ฅ) to satisfy
โ๐ฅ, ๐ฆ โ R๐, |a(๐ฆ)โ a(๐ฅ)| 6 ฮ|๐ฅโ ๐ฆ|๐ผ (1.2)
andโ๐ฅ โ R๐, โ๐ โ R๐, ฮโ1|๐|2 6 ๐ ยท a(๐ฅ)๐ 6 ฮ|๐|2. (1.3)
We denote by R๐ร๐sym the set of ๐-by-๐ real symmetric matrices and define
ฮฉ :=a : R๐ โ R๐ร๐
sym satisfying (1.2) and (1.3)
.
For each Borel set ๐ โ R๐, we denote by โฑ๐ the Borel ๐-algebra on ฮฉ generated by the family of mappings
a โฆโโซ
R๐
๐a๐๐ , ๐, ๐ โ 1, . . . , ๐, ๐ โ ๐ถโ๐ (๐ ).
We also set โฑ := โฑR๐ . For each ๐ฆ โ R๐, we denote by ๐๐ฆ : ฮฉ โ ฮฉ the action of translation by ๐ฆ:
โ๐ฅ โ R๐, ๐๐ฆa(๐ฅ) := a(๐ฅ + ๐ฆ).
We assume that P is a probability measure on (ฮฉ,โฑ) satisfying:
โ stationarity with respect to Z๐-translations: for every ๐ฆ โ Z๐ and ๐ด โ โฑ ,
P [๐๐ฆ๐ด] = P [๐ด] ;
โ unit range of dependence: for every Borel sets ๐,๐ โ R๐,
dist(๐,๐ ) > 1 =โ โฑ๐ and โฑ๐ are P-independent.
AN ITERATIVE METHOD FOR RAPIDLY OSCILLATING COEFFICIENTS 39
The expectation associated with the probability measure P is denoted by E. We recall that, by classical homog-enization theory (see [3,33]), the heterogeneous operator โโยทa(๐ฅ)โ homogenizes to the homogeneous operatorโโ ยท aโ, where a โ R๐ร๐ is a deterministic, constant, positive definite matrix. For every ๐ , ๐ > 0 and randomvariable ๐, we write
๐ 6 ๐ช๐ (๐) if and only if E[exp
((๐โ1 max(๐, 0)
)๐ )]6 2. (1.4)
We also set, for every ๐ โ (0, 1],
โ(๐) :=
(log(1 + ๐โ1)
) 12 if ๐ = 2,
1 if ๐ > 3.(1.5)
For notational convenience, from now on we will suppress the explicit dependence on the spatial variable in theoperator โโ ยท a(๐ฅ)โ and simply write โโ ยท aโ.
We now state the main result of the paper. We recall that P is a probability measure on (ฮฉ,โฑ) which specifiesthe law of the coefficient field a(๐ฅ) and satisfies the assumptions stated above, that a is the homogenized matrixassociated to P, and that ๐ โ R๐ is a bounded domain with ๐ถ1,1 boundary.
Theorem 1.1 (๐ป1 contraction). For each ๐ โ (0, 2), there exists a constant ๐ถ(๐ , ๐, ฮ, ๐ผ, ๐) < โ such thatthe following statement holds. Fix ๐ > 1, ๐ โ
[๐โ1, 1
], ๐ โ ๐ปโ1(๐๐), ๐ โ ๐ป1(๐๐) and let ๐ข โ ๐ + ๐ป1
0 (๐๐) bethe solution of (1.1). Also fix a function ๐ฃ โ ๐ + ๐ป1
0 (๐๐) and define the functions ๐ข0, ๐ข, ๐ข โ ๐ป10 (๐๐) to be the
solutions of the following equations (with null Dirichlet boundary condition on ๐๐๐):(๐2 โโ ยท aโ
)๐ข0 = ๐ +โ ยท aโ๐ฃ in ๐๐,
โโ ยท aโ๐ข = ๐2๐ข0 in ๐๐,(๐2 โโ ยท aโ
) ๐ข =(๐2 โโ ยท aโ
)๐ข in ๐๐.
For ๐ฃ โ ๐ + ๐ป10 (๐๐) defined by ๐ฃ := ๐ฃ + ๐ข0 + ๐ข, (1.6)
we have the estimateโโ(๐ฃ โ ๐ข)โ๐ฟ2(๐๐) 6 ๐ช๐
(๐ถโ(๐)
12 ๐
12 โโ(๐ฃ โ ๐ข)โ๐ฟ2(๐๐)
). (1.7)
The function ๐ข โ ๐ป1(๐๐) appearing in Theorem 1.1 is the unknown we wish to approximate, and ๐ฃ โ ๐ป1(๐๐)should be thought of as the current approximation to ๐ข. The function ๐ฃ is then the new, updated approximationto ๐ข and the estimate (1.7) says that, if ๐ is chosen small enough, then the error in our approximation willbe reduced by a multiplicative factor of 1/2. As explained more precisely around (1.10) below, we can theniterate this procedure and obtain rapid convergence to the solution. The only assumption we make on ๐ฃ is thatit satisfies the correct boundary condition, that is, ๐ฃ โ ๐ + ๐ป1
0 (๐๐). In particular, we may begin the iterationwith ๐ฃ = ๐ as the initial guess (or any other function with the correct boundary condition). The computationof ๐ฃ reduces to solving the problems for ๐ข0, ๐ข, and ๐ข listed in the statement, and the point is that each of theseproblems is relatively inexpensive to compute, provided that ๐ is not too small. A fundamental aspect of theresult is therefore that the required smallness of the parameter ๐ (so that (1.7) gives us a strict contractionin ๐ป1) does not depend on the length scale ๐ of the problem. In other words, we may need to take ๐ to besmall, but it will still be of order one, no matter how large ๐ is.
Similarly to standard multigrid methods, the equation for ๐ข0 is meant to resolve the small-scale discrepanciesbetween ๐ข and ๐ฃ. Note that the equation for ๐ข0 can be rewritten as
(๐2 โโ ยท aโ)๐ข0 = โโ ยท aโ(๐ขโ ๐ฃ) in ๐๐.
40 S. ARMSTRONG ET AL.
The parameter ๐โ1 is the characteristic length scale of this problem, and in practice we will take it to be somefixed multiple of the scale of oscillations of the coefficients. The computation of ๐ข0 can thus be decomposedinto a large number of essentially unrelated elliptic problems posed on subdomains of side length of the order of๐โ1. In analogy with multigrid methods, we may also think of ๐โ2 as the number of elementary pre-smoothingsteps performed during one global iteration.
As announced, we then use the homogenized operator on scales larger than ๐โ1. This is what the problemfor ๐ข is meant to capture. Since the elliptic problem for ๐ข involves the homogenized operator โโ ยท aโ, it canbe solved efficiently using the standard multigrid method. We note that the equation for ๐ข can be rewritten, ifdesired, in the form
โโ ยท aโ๐ข = โโ ยท aโ(๐ขโ ๐ฃ โ ๐ข0) in ๐๐. (1.8)
The final step of the iteration, involving the definition of ๐ข, is meant to add back some small-scale detailsto the function ๐ข. It is analogous to the post-smoothing step in the standard ๐ -cycle implementation of themultrigrid method, and the parameter ๐โ2 represents the number of post-smoothing steps.
We next discuss the more probabilistic aspects involved in the statement of Theorem 1.1. Since the coefficientfield is random, the statement of this theorem can only be valid with high probability, but not almost surely.Indeed, with non-zero probability, the coefficient field can be essentially arbitrary, and on such small-probabilityevents, the idea of leveraging on homogenization can only perform badly (recall that we aim for a convergenceresult for large but fixed ๐, as opposed to asymptotic convergence). It may help the intuition to observe that,by Chebyshevโs inequality, the assumption of (1.4) implies that
โ๐ฅ > 0, P [๐ > ๐๐ฅ] 6 2 exp(โ๐ฅ๐ ), (1.9)
and that conversely, the assumption of (1.9) implies that ๐ 6 ๐ช๐ (๐ถ๐) for some constant ๐ถ(๐ ) < โ (see [3],Lem. A.1).
We remark that Theorem 1.1 is new even when restricted to the subclass of periodic coefficient fields. Inthis case, both the probabilistic part of the estimate as well as the logarithmic factor of โ(๐) are not present,and (1.7) can be replaced with the simpler form
โโ(๐ฃ โ ๐ข)โ๐ฟ2(๐๐) 6 ๐ถ๐12 โโ(๐ฃ โ ๐ข)โ๐ฟ2(๐๐).
We stress that the probabilistic statement in (1.7) is valid for each fixed choice of ๐ข, ๐ฃ โ ๐ป1(๐๐). In fact,further work inspired by the first version of the present paper allowed to obtain the following stronger, uniformestimate [20]. For each ๐ โ (0, 2), there exist a constant ๐ถ(๐ , ๐, ๐, ฮ, ๐) < โ and, for each ๐ > 1 and ๐ โ [๐โ1, 1],a random variable ๐ณ๐ ,๐,๐ : ฮฉ โ [0, +โ] satisfying
๐ณ๐ ,๐,๐ 6 ๐ช๐ (๐ถ)
such that, for every ๐ข, ๐ฃ โ ๐ป1(๐๐) and ๐ฃ as in the statement of Theorem 1.1,
โโ(๐ฃ โ ๐ข)โ๐ฟ2(๐๐) 6 ๐ณ๐ ,๐,๐ โ(๐)12 ๐
12 (log ๐)
1๐ โโ(๐ฃ โ ๐ข)โ๐ฟ2(๐๐) . (1.10)
Moreover, the proof given in [20] does not require that the coefficient field be Holder continuous. As is apparentin (1.10), the price one has to pay for the uniformity of this estimate in the functions ๐ข and ๐ฃ is a slightdegradation of the contraction factor, by a slowly diverging logarithmic factor of the domain size. Due torandomness, uniform estimates such as (1.10) must necessarily contain some logarithmic divergence in thedomain size. Indeed, consider for instance the case of a coefficient field given by a random checkerboard inwhich we toss a fair coin, independently for each ๐ง โ Z๐, the coefficient field in ๐ง + [0, 1)๐ to be either ๐ผ๐ or 2๐ผ๐.Then, with probability tending to one as ๐ tends to infinity, there will be in the domain ๐๐ a region of spaceof side length of the order of (log ๐)
1๐ where the coefficient field is constant equal to ๐ผ๐. If the support of the
solution we seek is concentrated in this region, then the iteration described in Theorem 1.1 will perform badlyunless ๐โ1 is chosen larger than (log ๐)
1๐ .
AN ITERATIVE METHOD FOR RAPIDLY OSCILLATING COEFFICIENTS 41
The iteration proposed in Theorem 1.1 requires the user to make a judicious choice of the length scale ๐โ1.Ideally, it would be preferable to devise an adaptive method which discovers a good choice for ๐โ1 automatically.The contraction of the iteration would then be guaranteed with probability one, and more subtle probabilisticquantifiers would instead enter into the complexity analysis of the method. A suitably designed adaptive algo-rithm would likely also work on more general coefficient fields than those considered here, allowing for instanceto drop the assumption of stationarity. An assumption of approximate local stationarity would then also enterinto the complexity analysis of the method. We leave the development of such adaptive methods to future work.
The method proposed here also requires that the user computes a beforehand. An efficient method for doingso was presented in [23,29]; see also [11,17] and references therein for previous work on this problem. Moreover,one can check that in order to guarantee the contraction property of the iteration described in Theorem 1.1, sayby a factor of 1/2, a coarse approximation of a, which may be off by a small but fixed positive amount, suffices.
The proof of Theorem 1.1 can be modified so that the ๐ฟ2 norms in (1.7) are replaced by ๐ฟ๐ norms, forany exponent ๐ < โ. Up to some additional logarithmic factors in ๐, the contraction factor in the estimatewould then be of order ๐
1๐ rather than ๐
12 . This modification requires the application of large-scale Calderonโ
Zygmund-type ๐ฟ๐ estimates which can be found in Chapter 7 of [3]. The main required modification to theproof of Theorem 1.1 is simply to upgrade the two-scale expansion result of Theorem 3.1 from ๐ = 2 to largerexponents by adapting the argument of Theorem 7.10 from [3].
1.3. Previous works
There has been a lot of work on numerical algorithms that become sharp only in the limit of infinite scaleseparation (see for instance [1,6,8,9,24,28,30] and the references therein). That is, the error between the truesolution ๐ข and its numerical approximation becomes small only as ๐ โ โ. Such algorithms typically have acomputational complexity scaling sublinearly with the volume of the domain. An example of such a method inthe context of the homogenization problem considered here is to compute an approximation of the solution tothe homogenized equation. In addition to relying on scale separation, we note that such a sublinear method canonly give an accurate global approximation in a weaker space such as ๐ฟ2, but not in stronger norms such as ๐ป1
which are sensitive to small scale oscillations.We now turn our attention to numerical algorithms that, like ours, converge to the true solution for each finite
value of ๐. As pointed out in [19, 25], direct applications of standard multigrid methods result in coarse-scalesystems that do not capture the relevant large-scale properties of the problem. Indeed, standard coarseningprocedures produce effective coefficients that are simple arithmetic averages of the original coefficient field,instead of the homogenized coefficients. To remedy this problem, Griebel and Knapek [19, 25] propose moresubtle, matrix-dependent choices for the restriction and prolongation operators. The idea is to try to approxi-mate a Schur complement calculation, while preserving some calculability constraints such as matrix sparsity.The method proposed there is shown numerically to perform better than simple averaging, but no theoreticalguarantee is provided.
In [12, 13], the authors propose, in the periodic setting, to solve local problems for the correctors, deducelocally homogenized coefficients, and build coarsened operators from these. For the special two-dimensional casewith a(๐ฅ) = a(๐ฅ1 โ ๐ฅ2) for some 1-periodic a โ ๐ถ([0, 1]; R2ร2
sym), they show (in our notation) that ๐(๐53 log ๐)
smoothing steps suffice to guarantee the contractivity of the two-step multigrid method (assuming that thechosen coarsening scale is a bounded multiple of the oscillation scale). For comparison, this roughly correspondsto the choice of ๐ โ ๐โ
56 in our method. They also report better numerical performance than predicted by their
theoretical arguments.Beyond our current assumption of stationarity of the coefficient field, one can look for numerical methods for
the resolution of general elliptic problems with rapidly oscillating coefficients. Possibly the simplest such methodis to rely on the uniform ellipticity assumption (1.3) and appeal to a preconditioned conjugate gradient method,using the standard Laplacian as a preconditioner. However, the norm that is contracted at each iteration of
42 S. ARMSTRONG ET AL.
this algorithm is the ๐ฟ2 norm, as opposed to a contraction of the ๐ป1 norm as obtained in the present paper1.Moreover, the performance of this method degrades quickly if the ellipticity ratio ฮ becomes large. In contrast,using some of the results and techniques of [2,7], generalizations of Theorem 1.1 have now been obtained in thehighly degenerate case of perforated media of percolation type, for which ฮ = โ; see [21].
Algebraic multigrid methods are intended to solve completely arbitrary linear systems of equations, by auto-matically discovering a hierarchy of coarsened problems [34]. In practice, it is however necessary to make somejudicious choices of coarsening operators. In a sense, the present contribution as well as those of [12, 13, 19, 25]are descriptions of specific coarsening procedures which, under stronger assumptions such as stationarity, areshown to have fast convergence properties.
Many alternative approaches to the computation of elliptic problems with arbitrary coefficient fields have beendeveloped. We mention in particular, without going into details, hierarchical matrices [5], generalized multiscalefinite element methods [4, 10, 18], polyharmonic splines [32], local orthogonal decompositions [27], subspacecorrection methods [26] and gamblets [31]. While methods such as gamblets have been shown theoreticallyto have essentially linear complexity under weaker assumptions than those explored in the present paper, theconstruction and storage of the hierarchy of gamblets may actually be quite expensive in practice (we are notaware of large-scale computations that use gamblets; the main numerical example in [31] has 224 โ 1.7 ร 107
degrees of freedom). Methods such as local orthogonal decompositions introduce an intermediate scale, oftendenoted by ๐ป, inbetween the microscopic and the macroscopic scales, and an adapted basis of local functions iscomputed at this level. In this framework, the numerical error is bounded from below by a multiple of ๐ป. Themethod presented in Theorem 1.1 shares some aspects of this idea in that it also introduces an intermediatescale ๐โ1; however, the final numerical error is not constrained by this choice, and can be made arbitrarily lowirrespectively of the value of ๐.
The very recent work [15] probably comes closest to the goals of the present work. Under assumptions similarto ours, they analyze the performance of the method of local orthogonal decompositions (LOD). We believethat the method presented here has fundamental advantages over LOD. One aspect is that, as explained above,the error in LOD is bounded from below by a multiple of the intermediate scale (often denoted by ๐ป), whilethis is not so in our approach (in which the intermediate scale is ๐โ1). Moreover, our result guarantees anapproximation of the true solution in ๐ป1, while the results of [15] only provide with ๐ฟ2 estimates. Finally,our method is very easy to implement, as can be verified by the curious reader since the source code of ournumerical tests is openly available, see (4.1); on the other hand, the finite but possibly large overlaps of thelocal orthogonal frame might in practice cause significant computational overheads.
1.4. Organization of the paper
We introduce some more notation in Section 2. Section 3 is devoted to the proof of Theorem 1.1. We reporton our numerical results in Section 4. Finally, an appendix recalls some classical Sobolev and elliptic estimatesfor the readerโs convenience.
2. Notation
In this section, we collect some notation used throughout the paper. Recall that the notation ๐ช๐ (ยท) wasdefined in (1.4). We will need the following fact, which says that ๐ช๐ is behaving like a norm: for each ๐ โ (0,โ),there exists ๐ถ๐ < โ (with ๐ถ๐ = 1 for ๐ > 1) such that the following triangle inequality for ๐ช๐ (ยท) holds: forany measure space (๐ธ,๐ฎ, ๐), measurable function ๐ : ๐ธ โ (0,โ) and jointly measurable family ๐(๐ง)๐งโ๐ธ ofrandom variables, we have (see [3], Lem. A.4)
โ๐ง โ ๐ธ, ๐(๐ง) 6 ๐ช๐ (๐(๐ง)) =โโซ
๐ธ
๐ d๐ 6 ๐ช๐
(๐ถ๐
โซ๐ธ
๐ d๐
). (2.1)
1Naturally, the ๐ฟ2 and ๐ป1 norms become equivalent after discretization; but a theoretical guarantee of contraction in ๐ป1 ensuresthat the high frequencies can be resolved efficiently after only a few steps of the iteration, irrespectively of the size of the meshrefinement.
AN ITERATIVE METHOD FOR RAPIDLY OSCILLATING COEFFICIENTS 43
We denote by (๐1, . . . , ๐๐) the canonical basis of R๐, and write ๐ต(๐ฅ, ๐) โ R๐ for the Euclidean ball centered at๐ฅ โ R๐ and of radius ๐ > 0. For a Borel set ๐ โ R๐, we denote its Lebesgue measure by |๐ |. If |๐ | < โ, thenfor every ๐ โ [1,โ) and ๐ โ ๐ฟ๐(๐ ) we write the scaled ๐ฟ๐ norm of ๐ by
โ๐โ๐ฟ๐(๐ ) :=(|๐ |โ1
โซ๐
๐๐
) 1๐
= |๐ |โ1๐ โ๐โ๐ฟ๐(๐ ).
For each ๐ โ N, we denote by ๐ป๐(๐ ) the classical Sobolev space on ๐ , whose norm is given by
โ๐โ๐ป๐(๐ ) :=โ
06|๐ฝ|6๐
โ๐๐ฝ๐โ๐ฟ2(๐ ).
In the expression above, the parameter ๐ฝ = (๐ฝ1, . . . , ๐ฝ๐) is a multi-index in N๐, and we used the notation
|๐ฝ| :=๐โ
๐=1
๐ฝ๐ and ๐๐ฝ๐ = ๐๐ฝ1๐ฅ1ยท ยท ยท ๐๐ฝ๐
๐ฅ๐๐.
Whenever |๐ | < โ, we define the scaled Sobolev norm by
โ๐โ๐ป๐(๐ ) :=โ
06|๐ฝ|6๐
|๐ ||๐ฝ|โ๐
๐ โ๐๐ฝ๐โ๐ฟ2(๐ ).
We denote by ๐ป10 (๐ ) the completion in ๐ป1(๐ ) of the space ๐ถโ๐ (๐ ) of smooth functions with compact support
in ๐ . We write ๐ปโ1(๐ ) for the dual space to ๐ป10 (๐ ), which we endow with the (scaled) norm
โ๐โ๐ปโ1(๐ ) := sup|๐ |โ1
โซ๐
๐ ๐, ๐ โ ๐ป10 (๐ ), โ๐โ๐ป1(๐ ) 6 1
.
The integral sign above is an abuse of notation and should be understood as the duality pairing between ๐ปโ1(๐ )and ๐ป1
0 (๐ ). The spaces ๐ปโ1(๐ ) and ๐ป10 (๐ ) can be continuously embedded into the space of distributions, and
we make sure that the duality pairing is consistent with the integral expression above whenever ๐ and ๐ aresmooth functions. For every ๐ > 0 and ๐ฅ โ R๐, we denote the time-slice of the heat kernel which has lengthscale ๐ by
ฮฆ๐(๐ฅ) := (4๐๐2)โ๐2 exp
(โ ๐ฅ2
4๐2
)ยท (2.2)
We denote by ๐ โ ๐ถโ๐ (R๐) the standard mollifier
๐(๐ฅ) :=
๐๐ exp
(โ(1โ |๐ฅ|2)โ1
)if |๐ฅ| < 1,
0 if |๐ฅ| > 1,(2.3)
where the constant ๐๐ is chosen so thatโซ
R๐ ๐ = 1. For ๐ โ ๐ฟ๐(R๐) and ๐ โ ๐ฟ๐โฒ(R๐) with 1๐ + 1
๐โฒ = 1, we denotethe convolution of ๐ and ๐ by
๐ * ๐(๐ฅ) :=โซ
R๐
๐(๐ฆ)๐(๐ฅโ ๐ฆ) d๐ฆ.
3. Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. We begin by introducing the notion of (first-order)corrector : for each ๐ โ R๐, the corrector in the direction of ๐ is the function ๐๐ โ ๐ป1
loc(R๐) solving
โโ ยท a (๐ +โ๐๐) = 0 in R๐,
44 S. ARMSTRONG ET AL.
and such that the mapping ๐ฅ โฆโ โ๐๐(๐ฅ) is Z๐-stationary and satisfies
E
[โซ[0,1]๐
โ๐๐
]= 0.
The corrector ๐๐ is unique up to an additive constant (see [3], Def. 4.2 for instance). We also recall that onecan define the homogenized matrix a โ R๐ร๐
sym via the formula
โ๐ โ R๐, a๐ = E
[โซ[0,1]๐
a(๐ +โ๐๐)
],
or equivalently,
โ๐ โ R๐, ๐ ยท a๐ = E
[โซ[0,1]๐
(๐ +โ๐๐) ยท a(๐ +โ๐๐)
],
and in particular, as a consequence of (1.3), we have
โ๐ โ R๐, ฮโ1|๐|2 6 ๐ ยท a๐ 6 ฮ|๐|2. (3.1)
For each ๐ โ 1, . . . , ๐ and ๐ > 0, we denote
๐(๐)๐๐
:= ๐๐๐โ ๐๐๐
* ฮฆ๐โ1 .
A key ingredient in the proof of Theorem 1.1 is the following quantitative two-scale expansion for the oper-ator
(๐2 โโ ยท aโ
). It is the only input from the quantitative theory of stochastic homogenization used in this
paper and it follows from some estimates which can be found in [3].
Theorem 3.1 (Two-scale expansion and error estimate). For each ๐ โ (0, 2), there exists a constant๐ถ(๐ , ๐, ฮ, ๐ผ, ๐) < โ such that, for every ๐ > 1, ๐ โ [๐โ1, 1], and ๐ฃ โ ๐ป1
0 (๐๐) โฉ๐ป2(๐๐), defining
๐ค := ๐ฃ +๐โ
๐=1
๐(๐)๐๐
๐๐ฅ๐๐ฃ, (3.2)
we have the estimate
โโ ยท (aโ๐ค โ aโ๐ฃ)โ๐ปโ1(๐๐) 6 ๐ช๐
(๐ถโ(๐)โ๐ฃโ๐ป2(๐๐) + ๐ถ๐
๐2 โ๐ฃโ๐ป1(๐๐)
). (3.3)
Moreover, for every ๐ โ [0, ๐] and ๐ฃ โ ๐ป10 (๐๐) such that(
๐2 โโ ยท aโ)๐ฃ =
(๐2 โโ ยท aโ
)๐ฃ, (3.4)
we have the estimate
โ๐ฃ โ ๐คโ๐ป1(๐๐) +(๐ + ๐โ1
)โ๐ฃ โ ๐ฃโ๐ฟ2(๐๐) +
(๐ + ๐โ1
)2 โ๐ฃ โ ๐ฃโ๐ปโ1(๐๐)
6 ๐ช๐
(๐ถ(๐โ(๐) + ๐
๐2
)โ๐ฃโ๐ป1(๐๐) + ๐ถโ(๐)
12 โ๐ฃโ
12๐ป1(๐๐)
โ๐ฃโ12๐ป2(๐๐)
+ ๐ถโ(๐)โ๐ฃโ๐ป2(๐๐)
). (3.5)
The proof of Theorem 3.1 follows that of a similar result from Chapter 6 of [3]. The main difference here isthe presence of the zeroth order term with the factor of ๐2, which presents no additional difficulty. We beginby recalling the concept of a flux corrector and stating some estimates on the correctors proved in [3].
AN ITERATIVE METHOD FOR RAPIDLY OSCILLATING COEFFICIENTS 45
For each ๐ โ R๐, we denote the (centered) flux of the corrector ๐๐ by
(g๐,๐)16๐6๐ = g๐ := a(๐ +โ๐๐)โ a๐.
Since โ ยท g๐ = 0, the flux of the corrector admits a representation as the โcurlโ of some vector potential, byHelmholtzโs theorem. This vector potential, the flux corrector, will be useful for the proof of Theorem 3.1.For each ๐ โ R๐, the vector potential (S๐,๐๐)16๐,๐6๐ is a matrix-valued random field with entries in ๐ป1
loc(R๐)satisfying, for each ๐, ๐ โ 1, . . . , ๐,
S๐,๐๐ = โS๐,๐๐,
โ ยท S๐ = g๐, (3.6)
and such that ๐ฅ โฆโ โS๐,๐๐(๐ฅ) is a stationary random field with mean zero. In (3.6), we used the shorthandnotation
(โ ยท S๐)๐ :=๐โ
๐=1
๐๐ฅ๐S๐,๐๐ .
The conditions above do not specify the flux corrector uniquely. One way to โfix the gaugeโ is to enforce that,for each ๐, ๐ โ 1, . . . , ๐,
โS๐,๐๐ = ๐๐ฅ๐g๐,๐ โ ๐๐ฅ๐g๐,๐ .
This latter choice then defines S๐,๐๐ uniquely, up to the addition of a constant. We refer to Section 6.1 of [3] formore precision on this construction. We set
S(๐)๐ := S๐ โ S๐ * ฮฆ๐โ1 . (3.7)
The fundamental ingredient for the proof of Theorem 3.1 is the following proposition, which quantifies theconvergence to zero of the spatial averages of the gradients of the correctors.
Proposition 3.2 (Corrector estimates). For each ๐ โ (0, 2), there exists a constant ๐ถ(๐ , ๐, ฮ, ๐ผ, ๐) < โ suchthat for every ๐ โ (0, 1), ๐ฅ โ R๐ and ๐, ๐, ๐ โ 1, . . . , ๐,
|โ๐๐๐(๐ฅ)| 6 ๐ช๐ (๐ถ) , (3.8)
|(โ๐๐๐* ฮฆ๐โ1) (๐ฅ)|+ |(โS๐๐,๐๐ * ฮฆ๐โ1) (๐ฅ)| 6 ๐ช๐
(๐ถ๐
๐2
), (3.9)
๐(๐)๐๐
(๐ฅ)+S(๐)
๐๐,๐๐(๐ฅ)
= ๐ช๐ (๐ถโ(๐)) . (3.10)
Proof. By Lemma 4.4 of [3], we haveโโ๐๐๐
โ๐ฟ2(๐ต(0,1)) 6 ๐ช๐ (๐ถ) .
By the assumption of (1.2), we can apply standard Schauder estimates, see e.g. Theorems 3.1 and 3.8 of [22],to deduce (3.8). The estimates in (3.9) are proved in Theorem 4.9 and Proposition 6.2 of [3]. The estimates in(3.10) also follow from Theorem 4.9 and Proposition 6.2 of [3], combined with the assumption of (1.2) and theSchauder estimate in Corollary 3.2 and Theorem 3.8 of [22].
In the next lemma, we provide a convenient representation of โ ยท aโ๐ค in terms of the correctors.
Lemma 3.3. Let ๐ > 0, ๐ฃ โ ๐ป1(๐๐), and let ๐ค โ ๐ป1(๐๐) be defined by (3.2). Then
โ ยท (aโ๐ค โ aโ๐ฃ) = โ ยท F,
where the ๐-th component of the vector field F is given by
F๐ :=๐โ
๐,๐=1
(a๐๐๐
(๐)๐๐โ S(๐)
๐๐,๐๐
)๐๐ฅ๐ ๐๐ฅ๐
๐ฃ +๐โ
๐,๐=1
(a๐๐
(๐๐ฅ๐ ๐๐๐
* ฮฆ๐โ1
)+ ๐๐ฅ๐S๐,๐๐ * ฮฆ๐โ1
)๐๐ฅ๐
๐ฃ. (3.11)
46 S. ARMSTRONG ET AL.
Proof. The argument is very similar to that for Lemma 6.6 of [3], the main difference being that the definitionof ๐
(๐)๐๐ is slightly different from that of ๐๐
๐๐there. We recall the argument here for the readerโs convenience.
Observe that, for each ๐ โ 1, . . . , ๐,
๐๐ฅ๐๐ค =
๐โ๐=1
((๐ฟjk + ๐๐ฅ๐
๐๐๐
)๐๐ฅ๐
๐ฃ โ(๐๐ฅ๐
๐๐๐* ฮฆ๐โ1
)๐๐ฅ๐
๐ฃ + ๐(๐)๐๐
๐๐ฅ๐๐๐ฅ๐
๐ฃ)
. (3.12)
We start by studying the contribution of the first summand. By (3.6) and (3.7), we have, for every ๐,๐ โ 1, . . . , ๐,
๐โ๐=1
๐๐ฅ๐S(๐)
๐๐,๐๐ =๐โ
๐=1
(a๐๐
(๐ฟjk + ๐๐ฅ๐
๐๐๐
)โ a๐๐๐ฟjk โ ๐๐ฅ๐
S๐๐,๐๐ * ฮฆ๐โ1
).
We deduce that, for each ๐ โ 1, . . . , ๐,
๐โ๐,๐=1
a๐๐
(๐ฟjk + ๐๐ฅ๐
๐๐๐
)๐๐ฅ๐
๐ฃ =๐โ
๐,๐=1
(a๐๐๐ฟjk + ๐๐ฅ๐
S(๐)๐๐,๐๐ + ๐๐ฅ๐
S๐๐,๐๐ * ฮฆ๐โ1
)๐๐ฅ๐
๐ฃ, (3.13)
and thus
๐โ๐,๐,๐=1
๐๐ฅ๐
(a๐๐
(๐ฟjk + ๐๐ฅ๐ ๐๐๐
)๐๐ฅ๐
๐ฃ)
= โ ยท aโ๐ฃ +๐โ
๐,๐,๐=1
๐๐ฅ๐
(๐๐ฅ๐S
(๐)๐๐,๐๐ ๐๐ฅ๐
๐ฃ)
+๐โ
๐,๐,๐=1
๐๐ฅ๐
((๐๐ฅ๐
S๐๐,๐๐ * ฮฆ๐โ1
)๐๐ฅ๐
๐ฃ).
By the skew-symmetry of S(๐)๐ , we have
0 =๐โ
๐,๐,๐=1
๐๐ฅ๐๐๐ฅ๐
(S(๐)
๐๐,๐๐ ๐๐ฅ๐๐ฃ)
=๐โ
๐,๐,๐=1
๐๐ฅ๐
(๐๐ฅ๐
S(๐)๐๐,๐๐ ๐๐ฅ๐
๐ฃ)
+๐โ
๐,๐,๐=1
๐๐ฅ๐
(S(๐)
๐๐,๐๐ ๐๐ฅ๐๐๐ฅ๐
๐ฃ)
,
and thus
๐โ๐,๐,๐=1
๐๐ฅ๐
(a๐๐
(๐ฟjk + ๐๐ฅ๐ ๐๐๐
)๐๐ฅ๐
๐ฃ)
= โ ยท aโ๐ฃ โ๐โ
๐,๐,๐=1
๐๐ฅ๐
(S(๐)
๐๐,๐๐ ๐๐ฅ๐ ๐๐ฅ๐๐ฃ)
+๐โ
๐,๐,๐=1
๐๐ฅ๐
((๐๐ฅ๐
S๐๐,๐๐ * ฮฆ๐โ1
)๐๐ฅ๐
๐ฃ).
Recalling (3.12), we obtain the announced result.
We next present the proof of Theorem 3.1, which can be compared to the one of Theorem 6.9 from [3].
Proof of Theorem 3.1. We will proceed by proving first (3.3), and then the ๐ป1, ๐ฟ2 and ๐ปโ1 estimates appearingin (3.5), in this order. We decompose the arguments into seven steps.
AN ITERATIVE METHOD FOR RAPIDLY OSCILLATING COEFFICIENTS 47
Step 1. We prove (3.3). In view of Lemma 3.3, it suffices to show that, for the vector field F defined in (3.11),
โFโ๐ฟ2(๐๐) 6 ๐ช๐
(๐ถโ(๐)โ๐ฃโ๐ป2(๐๐) + ๐ถ๐
๐2 โ๐ฃโ๐ป1(๐๐)
). (3.14)
We estimate each of the terms appearing in the definition of F. By Proposition 3.2 and (2.1), we have, forevery ๐, ๐, ๐ โ 1, . . . , ๐,(
a๐๐๐(๐)๐๐โ S(๐)
๐๐,๐๐
)๐๐ฅ๐
๐๐ฅ๐๐ฃ
๐ฟ2(๐๐)6 ๐ช๐
(๐ถโ(๐)โ๐ฃโ๐ป2(๐๐)
),
as well as (a๐๐
(๐๐ฅ๐
๐๐๐* ฮฆ๐โ1
)+ ๐๐ฅ๐
S๐,๐๐ * ฮฆ๐โ1
)๐๐ฅ๐
๐ฃ
๐ฟ2(๐๐)6 ๐ช๐
(๐ถ๐
๐2 โ๐ฃโ๐ป1(๐๐)
),
and thus (3.14) follows.
Step 2. In order to show (3.5), we first need to evaluate the contribution of a boundary layer. For every โ > 0,we write ๐โ := โโ๐๐(โโ1 ยท ) (recall the definition of ๐ in (2.3)) and
๐๐,โ := ๐ฅ โ ๐๐ : dist(๐ฅ, ๐๐๐) > โ . (3.15)
With the definition of โ(๐) given in (1.5), we set
๐ :=(1R๐โ๐๐,2โ(๐)
* ๐โ(๐)
) ๐โ๐=1
๐(๐)๐๐
๐๐ฅ๐๐ฃ.
We will use the function ๐ as a test function for an upper bound on the size of the actual boundary layerin the next step. In this step, we show that there exists ๐ถ(๐ , ๐, ฮ, ๐ผ, ๐) < โ such that
โโ๐โ๐ฟ2(๐๐) 6 ๐ช๐
(๐ถ โ(๐)
12 โ๐ฃโ
12๐ป1(๐๐)
โ๐ฃโ12๐ป2(๐๐)
+ ๐ถโ(๐)โ๐ฃโ๐ป2(๐๐)
)(3.16)
andโ๐โ๐ฟ2(๐๐) 6 ๐ช๐
(๐ถ โ(๐)
32 โ๐ฃโ
12๐ป1(๐๐)
โ๐ฃโ12๐ป2(๐๐)
). (3.17)
By the chain rule,
โโ๐โ๐ฟ2(๐๐) 6 ๐ถ
๐โ๐=1
(|โ๐ฃ|โ(๐)
+ |โ2๐ฃ|)
๐(๐)๐๐
+ |โ๐ฃ|
โ๐(๐)
๐๐
๐ฟ2(๐๐โ๐๐,3โ(๐))
.
By Proposition 3.2 and (2.1), we have|โ2๐ฃ|
๐(๐)
๐๐
๐ฟ2(๐๐โ๐๐,3โ(๐))
6 ๐ช๐
(๐ถโ(๐)โโ2๐ฃโ๐ฟ2(๐๐)
).
Similarly, |โ๐ฃ|โ(๐)
๐(๐)
๐๐
๐ฟ2(๐๐โ๐๐,3โ(๐))
6 ๐ช๐
(๐ถโโ๐ฃโ๐ฟ2(๐๐โ๐๐,3โ(๐))
), (3.18)
and by Proposition A.1,
โโ๐ฃโ๐ฟ2(๐๐โ๐๐,3โ(๐)) 6 ๐ถ โ(๐)12 ๐
๐2 โ๐ฃโ
12๐ป1(๐๐)
โ๐ฃโ12๐ป2(๐๐)
. (3.19)
Finally, using again Proposition 3.2 and (2.1), we have|โ๐ฃ|
โ๐(๐)
๐๐
๐ฟ2(๐๐โ๐๐,3โ(๐))
6 ๐ช๐
(๐ถโโ๐ฃโ๐ฟ2(๐๐โ๐๐,3โ(๐))
),
and we can appeal once more to (3.19) to estimate the norm of โ๐ฃ on the right side above. This completesthe proof of (3.16). The estimate (3.17) follows from (3.18) and (3.19).
48 S. ARMSTRONG ET AL.
Step 3. We now evaluate the size of the boundary layer ๐ โ ๐ป1(๐๐) defined as the solution ofโงโชโชโจโชโชโฉ(๐2 โโ ยท aโ
)๐ = 0 in ๐๐,
๐ =๐โ
๐=1
๐(๐)๐๐
๐๐ฅ๐๐ฃ on ๐๐๐.
(3.20)
Since ๐ and ๐ share the same boundary condition on ๐๐๐, by the variational formulation of (3.20), we haveโซ๐๐
(๐2๐2 +โ๐ ยท aโ๐
)6โซ
๐๐
(๐2๐ 2 +โ๐ ยท aโ๐
).
By the result of the previous step, we thus obtain, for every ๐ โ [0, ๐],
๐โ๐โ๐ฟ2(๐๐) + โโ๐โ๐ฟ2(๐๐) 6 ๐ช๐
(๐ถ โ(๐)
12 โ๐ฃโ
12๐ป1(๐๐)
โ๐ฃโ12๐ป2(๐๐)
+ ๐ถโ(๐)โ๐ฃโ๐ป2(๐๐)
). (3.21)
Step 4. We are now prepared to prove that
โโ(๐ฃ โ ๐ค)โ๐ฟ2(๐๐) + ๐โ๐ฃ โ ๐คโ๐ฟ2(๐๐)
6 ๐ช๐
(๐ถ(๐โ(๐) + ๐
๐2
)โ๐ฃโ๐ป1(๐๐) + ๐ถโ(๐)
12 โ๐ฃโ
12๐ป1(๐๐)
โ๐ฃโ12๐ป2(๐๐)
+ ๐ถโ(๐)โ๐ฃโ๐ป2(๐๐)
). (3.22)
For concision, we define๐ณ1 := โ โ โ ยท (aโ๐ฃ โ aโ๐ค)โ๐ปโ1(๐๐),
and recall that, by (3.3),๐ณ1 6 ๐ช๐
(๐ถโ(๐)โ๐ฃโ๐ป2(๐๐) + ๐ถ๐
๐2 โ๐ฃโ๐ป1(๐๐)
). (3.23)
Moreover, by (3.4) and (3.20),
โโ ยท (aโ๐ฃ โ aโ๐ค) = โโ ยท aโ(๐ฃ โ ๐ค) + ๐2(๐ฃ โ ๐ฃ)= โโ ยท aโ(๐ฃ โ ๐ค + ๐) + ๐2(๐ฃ โ ๐ฃ + ๐).
Since ๐ฃ โ ๐ค + ๐ โ ๐ป10 (๐๐), we deduce that
|๐๐|โ1
โซ๐๐
(โ(๐ฃ โ ๐ค + ๐) ยท aโ(๐ฃ โ ๐ค + ๐) + ๐2(๐ฃ โ ๐ค + ๐)(๐ฃ โ ๐ฃ + ๐)
)6 ๐ณ1โโ(๐ฃ โ ๐ค + ๐)โ๐ฟ2(๐๐),
and by the uniform ellipticity of a and Holderโs inequality,
โโ(๐ฃ โ ๐ค + ๐)โ2๐ฟ2(๐๐) + ๐2โ๐ฃ โ ๐ค + ๐โ2๐ฟ2(๐๐) 6 ๐ถ๐ณ1โโ(๐ฃ โ ๐ค + ๐)โ๐ฟ2(๐๐)
+ ๐2โ๐ค โ ๐ฃโ๐ฟ2(๐๐) โ๐ฃ โ ๐ค + ๐โ๐ฟ2(๐๐).
Using Proposition 3.2 and (2.1), we verify that
โ๐ค โ ๐ฃโ๐ฟ2(๐๐) 6 ๐ช๐
(๐ถโ(๐)โ๐ฃโ๐ป1(๐๐)
). (3.24)
Combining these two estimates with (3.23) and Youngโs inequality, we obtain that
โโ(๐ฃ โ ๐ค + ๐)โ๐ฟ2(๐๐) + ๐โ๐ฃ โ ๐ค + ๐โ๐ฟ2(๐๐) 6 ๐ช๐
(๐ถ(๐โ(๐) + ๐
๐2
)โ๐ฃโ๐ป1(๐๐) + ๐ถโ(๐)โ๐ฃโ๐ป2(๐๐)
).
An application of (3.21) then yields the announced estimate (3.22).
AN ITERATIVE METHOD FOR RAPIDLY OSCILLATING COEFFICIENTS 49
Step 5. In this step, we complete the proof of the fact that โ๐ฃโ๐คโ๐ป1(๐๐) is bounded by the right side of (3.5).In view of (3.22), it suffices to show that
๐โ1โ๐ฃ โ ๐คโ๐ฟ2(๐๐) 6 ๐ช๐
(๐ถ(๐โ(๐) + ๐
๐2
)โ๐ฃโ๐ป1(๐๐) + ๐ถโ(๐)
12 โ๐ฃโ
12๐ป1(๐๐)
โ๐ฃโ12๐ป2(๐๐)
+ ๐ถโ(๐)โ๐ฃโ๐ป2(๐๐)
).
(3.25)
By (3.16) and (3.22), we have
โโ(๐ฃ โ ๐ค + ๐ )โ๐ฟ2(๐๐) 6 ๐ช๐
(๐ถ(๐โ(๐) + ๐
๐2
)โ๐ฃโ๐ป1(๐๐) + ๐ถโ(๐)
12 โ๐ฃโ
12๐ป1(๐๐)
โ๐ฃโ12๐ป2(๐๐)
+ ๐ถโ(๐)โ๐ฃโ๐ป2(๐๐)
).
The estimate (3.25) then follows by the Poincare inequality and (3.17).
Step 6. We now complete the proof that(๐ + ๐โ1
)โ๐ฃ โ ๐ฃโ๐ฟ2(๐๐) is bounded by the right side of (3.5). For
๐ > ๐โ1, the result follows from (3.22) and (3.24), while ๐ 6 ๐โ1, it follows from (3.5) and (3.24).
Step 7. We finally complete the proof of (3.5) by showing the estimate for the ๐ปโ1 norm of ๐ฃโ ๐ฃ. If ๐ 6 ๐โ1,then the conclusion is immediate from the estimate on the ๐ฟ2 norm of ๐ฃ โ ๐ฃ, by scaling. Otherwise, by theequations for ๐ฃ and ๐ฃ, we have
๐2(๐ฃ โ ๐ฃ) = โ ยท (aโ๐ฃ โ aโ๐ฃ) ,
and moreover,
โโ ยท (aโ๐ฃ โ aโ๐ฃ)โ๐ปโ1(๐๐) 6 โโ ยท (aโ๐ค โ aโ๐ฃ)โ๐ปโ1(๐๐) + โโ ยท (aโ๐ฃ โ aโ๐ค)โ๐ปโ1(๐๐)
6 โโ ยท (aโ๐ค โ aโ๐ฃ)โ๐ปโ1(๐๐) + ๐ถ โโ๐ฃ โโ๐คโ๐ฟ2(๐๐) .
The terms on the right side above have been estimated in (3.3) and (3.22) respectively, so the proof iscomplete.
We next give the proof of the main result.
Proof of Theorem 1.1. Let ๐ข, ๐ฃ, ๐ข0, ๐ข, ๐ข โ ๐ป1(๐๐) be as in the statement of Theorem 1.1. We first show the apriori estimates
๐โ๐ข0โ๐ฟ2(๐๐) + โโ๐ข0โ๐ฟ2(๐๐) 6 ๐ถโ๐ขโ ๐ฃโ๐ป1(๐๐), (3.26)
andโ๐ขโ๐ป1(๐๐) + ๐โ1โ๐ขโ๐ป2(๐๐) 6 ๐ถโ๐ขโ ๐ฃโ๐ป1(๐๐). (3.27)
By the variational formulation of the equation for ๐ข0 โ ๐ป10 (๐๐), we haveโซ
๐๐
(๐2๐ข2
0 +โ๐ข0 ยท aโ๐ข0
)=โซ
๐๐
โ๐ข0 ยท aโ(๐ขโ ๐ฃ).
By Holderโs and Youngโs inequalities and the uniform ellipticity of a, we get (3.26). Using the equation (1.8)satisfied by ๐ข โ ๐ป1
0 (๐๐) and the estimate (3.26), we deduce
โโ๐ขโ๐ฟ2(๐๐) 6 ๐ถโโ(๐ขโ ๐ฃ โ ๐ข0)โ๐ฟ2(๐๐)
6 ๐ถโโ(๐ขโ ๐ฃ)โ๐ฟ2(๐๐).
By Proposition A.2 and the ๐ฟ2 estimate in (3.26), we also have
โ๐ขโ๐ป2(๐๐) 6 ๐ถ๐2โ๐ข0โ๐ฟ2(๐๐) 6 ๐ถ๐โ๐ขโ ๐ฃโ๐ป1(๐๐),
50 S. ARMSTRONG ET AL.
as announced in (3.27).We now introduce the two-scale expansion
๐ค := ๐ข +๐โ
๐=1
๐(๐)๐๐
๐๐ฅ๐๐ข.
Using the equation for ๐ข in (1.8) and Theorem 3.1 with ๐ = 0, we obtain
โ๐ฃ + ๐ข0 + ๐ค โ ๐ขโ๐ป1(๐๐) 6 ๐ช๐
(๐ถ๐
๐2 โ๐ขโ๐ป1(๐๐) + ๐ถโ(๐)
12 โ๐ขโ
12๐ป1(๐๐)
โ๐ขโ12๐ป2(๐๐)
+ ๐ถโ(๐)โ๐ขโ๐ป2(๐๐)
),
and thus, by (3.27),โ๐ฃ + ๐ข0 + ๐ค โ ๐ขโ๐ป1(๐๐) 6 ๐ช๐
(๐ถโ(๐)
12 ๐
12 โ๐ขโ ๐ฃโ๐ฟ2(๐๐)
). (3.28)
In order to complete the proof of Theorem 1.1, there remains to estimate the ๐ป1 norm of ๐คโ๐ข. By the equationfor ๐ข, Theorem 3.1 and (3.27), we have
โ๐ค โ ๐ขโ๐ป1(๐๐) 6 ๐ช๐
(๐ถ(๐โ(๐) + ๐
๐2
)โ๐ขโ๐ป1(๐๐)
+ ๐ถโ(๐)12 โ๐ขโ
12๐ป1(๐๐)
โ๐ขโ12๐ป2(๐๐)
+ ๐ถโ(๐)โ๐ขโ๐ป2(๐๐)
)6 ๐ช๐
(๐ถโ(๐)
12 ๐
12 โ๐ขโ ๐ฃโ๐ป1(๐๐)
),
as desired.
4. Numerical results
In this section, we report on numerical tests demonstrating the performance of the iterative method describedin Theorem 1.1. The code used in the tests can be consulted at
https://github.com/ahannuka/homo_mg. (4.1)
Throughout this section, we consider a two-dimensional random checkerboard coefficient field ๐ฅ โฆโ a(๐ฅ), whichis defined as follows: we give ourselves a family (๐(๐ง))๐งโZ2 of independent random variables such that for every๐ง โ Z2,
P [๐(๐ง) = 1] = P [๐(๐ง) = 9] =12ยท
We then set, for every ๐ฅ โ ๐ง + [0, 1)2,a(๐ฅ) := ๐(๐ง) ๐ผ2,
where ๐ผ2 denotes the 2-by-2 identity matrix. For this particular coefficient field, the homogenized matrix can becomputed analytically as a = 3๐ผ2 (see [3], Exercise 2.3). When such an analytical expression does not exist, thehomogenized coefficient can be approximated numerically, for example, by using the method presented in [29].
For each ๐ > 0, we write ๐๐ := (0, ๐)2. We aim to compute the solution to the continuous partial differentialequation in (1.1) with ๐ = 0 (null Dirichlet boundary condition) and load function ๐ = 1. We discretize thisproblem using a first-order finite element method. Let ๐ฏ be a triangular mesh of the domain ๐๐ constructedby first dividing each cell ๐ง + [0, 1)2 (๐ง โ Z2) into two triangles, and then using three levels of uniform meshrefinement. This results into a sufficiently fine mesh to capture the oscillations present in the exact solution ๐ข.The first order finite element space
๐โ :=๐ข โ ๐ป1
0 (๐๐) | ๐ข|๐พ โ ๐ 1(๐พ) โ๐พ โ ๐ฏ
AN ITERATIVE METHOD FOR RAPIDLY OSCILLATING COEFFICIENTS 51
0010500
50
100
Figure 1. Left panel: a typical realization of the coefficient field a(๐ฅ), with ๐ = 100 (yellowcoresponds to the value 1 and blue to the value 9). Right panel: corresponding solution.
x
0 20 40 60 80 1000
50
100
150
200
250
x
0 20 40 60 80 1000
50
100
150
200
250
Figure 2. Left panel: FE-solution to the heterogeneous problem, and right panel: FE-solutionto the corresponding homogenized problem. Both solutions are plotted along the line ๐ฆ = 55.The fast oscillation in the left figure is clearly visible.
with standard nodal basis is used in all computations. The finite element solution ๐ขโ โ ๐โ satisfies
โ๐ฃโ โ ๐โ, (aโ๐ขโ,โ๐ฃโ) = (๐, ๐ฃโ). (4.2)
A typical realization of the coefficient field a(๐ฅ) and of the corresponding exact solution ๐ข are visualizedin Figure 1, with the choice of ๐ = 100. The high-frequency oscillations in the solution are clearly visible inFigure 2, where the solution is visualized along the line ๐ฆ = 55.
Our interest lies in the contraction factor of the iterative procedure. The contraction factor is studied byfirst solving the finite dimensional problem (4.2) exactly using a direct solver. Then a sequence of approximatesolutions ๐ข(๐)
โ ๐๐=1 is generated by starting from ๐ข
(1)โ = 0 and applying the iterative procedure described in
52 S. ARMSTRONG ET AL.
-4.8 -4.6 -4.4 -4.2 -4 -3.8 -3.60
2
4
6
8
10
12
14
16
18
20N
0 2 4 6 8 10 12
Iteration step
10 -10
10 -8
10 -6
10 -4
10 -2
10 0
10 2
10 4
Err
or
in H
1-s
em
inorm
Figure 3. Left panel: empirical distribution of the factor ๐ for ๐ = 0.1 and ๐ = 100, based on100 runs. Right panel: error in the ๐ป1 seminorm for ๐ = 100 and ๐ = 0.1, after each iteration.The method converges after 8 iterations.
Theorem 1.1. The logarithm of the error โโ(๐ขโ ๐ข(๐)โ )โ๐ฟ2(๐๐) is computed for each ๐ โ 1, . . . , 10, a regression
line is fitted, and the slope of this line is denoted by ๐. It is our numerical estimate of the logarithm of thecontraction factor; roughly speaking,
๐ โ log
(โโ(๐ขโ โ ๐ข
(๐+1)โ )โ๐ฟ2(๐๐)
โโ(๐ขโ โ ๐ข(๐)โ )โ๐ฟ2(๐๐)
)
(โlogโ denotes the natural logarithm.) The iteration is said to converge when the relative error is smaller than10โ9. Past this threshold, the error between the exact and the iterative solutions is smaller than the accuracyof the discretization itself, and thus cannot be measured.
Since the coefficient field is random, the contraction factor will vary for different realizations of a. For thechoice of ๐ = 0.1 and ๐ = 100, the empirical distribution of the contraction factor is given in Figure 3, basedon one hundered samples of the coefficient field. Apart from the purposes of displaying this histogram, each ofour estimates for ๐ is an average over ten realizations of the coefficient field.
In our first test, the parameter ๐ is fixed to ๐ = 0.1, 0.2, and then 0.4. The size of the domain ๐ is variedbetween 10 and 200. The averaged contraction factor is visualized on the left side of Figure 4. The resultsare in excellent agreement with Theorem 1.1. After a pre-asymptotic region, the contraction factor becomesindependent of the size of the domain ๐. The pre-asymptotic region is due to the fact that for small valuesof ๐, the pre- and post-smoothing steps are essentially sufficient to solve the equation. We emphasize that thecontraction factor remains very good, of the order of 0.1, even for the relatively large value of ๐ = 0.4.
In the second test, the size of the domain ๐ takes values ๐ = 100, 200, and 300, while ๐ is variedbetween 0.01 and 0.5. For each ๐, the exponent of the averaged contraction factor is computed based onten simulation runs. The results are presented on the right side of Figure 4. After a pre-asymptotic region, theexponential of the contraction factor behaves like ๐1/2, as predicted by Theorem 1.1. The pre-asymptotic regionis roughly characterized by the scaling ๐ . 10๐โ1.
AN ITERATIVE METHOD FOR RAPIDLY OSCILLATING COEFFICIENTS 53
0 50 100 150 200
r
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
=0.1
=0.2
=0.4
10-2
10-1
100
10-6
10-5
10-4
10-3
10-2
10-1
100
r=100
r=200
r=3001/2
Figure 4. Left panel: averaged factor ๐ as a function of ๐, for ๐ = 0.1, 0.2, and 0.4. Rightpanel: exponential of the averaged factor ๐ as a function of ๐ for ๐ = 100, 200, and 300. In allcases, the average is computed from ten simulation runs.
Appendix A. Sobolev estimates
In this appendix, we prove an estimate for the norm of a function restricted to a layer close to the boundaryof a domain. The estimate is an integrated version of a trace theorem. For convenience, we will also recall astandard ๐ป2 estimate for homogeneous elliptic equations. As in (3.15), for every โ > 0, we write
๐๐,โ := ๐ฅ โ ๐๐ : dist(๐ฅ, ๐๐๐) > โ .
Proposition A.1 (Trace estimate). There exists ๐ถ(๐, ๐) < โ such that for every ๐ > 1, โ โ (0, ๐] and๐ โ ๐ป1(๐๐),
๐โ๐ โ๐โ2๐ฟ2(๐๐โ๐๐,โ)6 ๐ถ โ โ๐โ๐ฟ2(๐๐) โ๐โ๐ป1(๐๐).
Proof. Denote by n๐,๐ก the unit normal vector to ๐๐๐,โ, which we extend to ๐๐,โ harmonic continuation. Since๐ is ๐ถ1,1, there exists ๐ถ(๐, ๐) < โ such that for every ๐ก โ (0, ๐/๐ถ], we have โโn๐,๐กโ๐ฟโ(๐๐,๐ก) 6 ๐ถ๐โ1. It thusfollows that โซ
๐๐๐,๐ก
๐2 =โซ
๐๐,๐ก
โ ยท (๐2 n๐,๐ก)
6 ๐ถ๐โ1โ๐โ2๐ฟ2(๐๐) + ๐ถโ๐โ๐ฟ2(๐๐)โโ๐โ๐ฟ2(๐๐)
6 ๐ถ๐๐โ๐โ๐ฟ2(๐๐) โ๐โ๐ป1(๐๐).
By the coarea formula, for every โ 6 ๐/๐ถ, we have
โ๐โ2๐ฟ2(๐๐โ๐๐,โ)=โซ โ
0
โ๐โ2๐ฟ2(๐๐๐,๐ก)d๐ก.
Combining the previous two displays yields
โ๐โ2๐ฟ2(๐๐โ๐๐,โ)6 ๐ถโ ๐๐ โ๐โ๐ฟ2(๐๐) โ๐โ๐ป1(๐๐),
which is the announced result. The case โ > ๐/๐ถ is immediate.
54 S. ARMSTRONG ET AL.
Proposition A.2 (๐ป2 estimate). Let a โ R๐ร๐sym satisfy (3.1). There exists a constant ๐ถ(ฮ, ๐, ๐) < โ such that
for every ๐ข โ ๐ป10 (๐๐) and ๐ โ ๐ฟ2(๐๐), if
โโ ยท aโ๐ข = ๐,
then ๐ข โ ๐ป2(๐๐) andโ๐ขโ๐ป2(๐๐) 6 ๐ถโ๐โ๐ฟ2(๐๐).
Proof. See Theorem 6.3.2.4 of [14].
Acknowledgements. SA was partially supported by the NSF Grant DMS-1700329. AH was partially supported by theStenback stiftelse. TK was supported by the Academy of Finland. JCM was partially supported by the ANR Grant LSD(ANR-15-CE40-0020-03).
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