Reduced-Basis approach for homogenization beyond the periodic setting

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periodic setting

Sébastien Boyaval

To cite this version:

Sébastien Boyaval. Reduced-Basis approach for homogenization beyond the periodic setting. [Research

Report] RR-6130, INRIA. 2007, pp.29. �inria-00132763v3�

inria-00132763, version 3 - 26 Feb 2007

a p p o r t

d e r e c h e r c h e

Thème NUM

Reduced-Basis approach for homogenization beyond

the periodic setting

Sébastien B

OYAVAL

N° 6130

Sébastien Boyaval

∗

ThèmeNUM Systèmes numériques ProjetMICMAC

Rapportde re her he n°6130 Février, 2007  40 pages

Abstra t: We onsiderthe omputationofaveraged oe ientsforthe homogeniza-tionofellipti partialdierential equations. Inthis problem,likeinmanymultis ale problems,a largenumber ofsimilar omputationsparametrizedbythema ros opi s aleisrequiredatthe mi ros opi s ale. Thisisaframeworkverymu hadaptedto modelorder redu tionattempts.

The purpose of this work is to show how the redu ed-basis approa h allows to speedupthe omputationofalargenumberof ellproblemswithoutanylossof pre- ision. The essential omponentsofthis redu ed-basisapproa h arethea posteriori errorestimation, whi hprovidessharp errorboundsfor theoutputsof interest, and anapproximationpro essdividedintooine andonlinestages, whi h de ouplesthe generationof the approximationspa e andits usefor Galerkin proje tions.

Key-words: Homogenization ;Redu ed-Basis Method;A posterioriEstimates

∗

CERMICS, E ole Nationale des Ponts et Chaussées, 6 & 8 avenue Blaise Pas al, Cité Des artes, 77455 Marne-la-Vallée Cedex 2, Fran e and MICMAC Proje t, INRIA, Domaine deVolu eau,BP.105-Ro quen ourt,78153LeChesnay Cedex(boyaval ermi s.enp .fr)

Résumé: Nousnousintéressonsau al ulde oe ientsmoyennéspourl'homogénéisation d'équations aux dérivées partielles elliptiques. Comme de nombreux problèmes multié helles, e problème né essite, à l'é helle mi ros opique, une grande quantité de al uls similaires entre eux, quisont paramétrés parl'é helle ma ros opique. Un tel adre seprêtetrès bienaux tentativesde rédu tiond'ordre.

Le but de e travail est de montrer omment une appro he par bases réduites permet d'a élérer le al ul d'un grand nombre de problèmes de ellules sansperte de pré ision. Les omposants essentiels de ette appro he par bases réduites sont: uneestimationd'erreuraposteriori,quifournitdesbornesd'erreurpré isespourles quantités nales intéressantes, et une pro édure d'approximation divisée entre des étapesoine etonline, equidé ouplela onstru tiondel'espa ed'approximation deson utilisationpour les proje tionsde Galerkin.

AMS subje t lassi ations. 74Q05,74S99,65N15,35J20.

Contents

1 Introdu tion 3

2 Setting of theproblem,elementsofhomogenization theoryand RB

approa h 4

2.1 Formulation oftheproblem . . . 4

2.2 General ontext for homogenization . . . 6

2.2.1 Abstra thomogenization results . . . 6

2.2.2 Theexpli it two-s ale homogenization . . . 8

2.2.3 Numeri alhomogenization strategies . . . 9

2.3 Theredu ed-basis method . . . 12

2.3.1 Theparametrized ell problem . . . 13

2.3.2 Prin iple oftheredu ed-basis method . . . 15

2.3.3 Pra ti eof theredu ed-basismethod . . . 16

3 Redu ed-basis approa h for the ell problem 17 3.1 Errorboundsfor the ell problem . . . 17

3.2 Theredu ed-basis onstru tion . . . 20

3.3 Convergen e of theredu ed-basismethodfor the ell problem . . . . 21

3.4 Errorestimate forthe asymptoti

H

1

homogenized solution . . . 23

3.5 Pra ti al inuen eof theparametrization . . . 28

4 Numeri al results 32 4.1 Denitionof the problem . . . 32

4.2 Oine omputations . . . 33

4.3 Online omputations . . . 35

5 Con lusion and perspe tives 37

1 Introdu tion

Inthis work,we studythe numeri al homogenizationof linear s alarellipti partial dierential equations (PDEs) su hasthose en ountered intheproblems of thermal diusionandele tri al ondu tion. Os illatingtestfun tions,alsotermed orre tors,

are omputedthrougharedu ed-basis (RB)approa hfor parametrized ell problems supplied withperiodi boundary onditions. Numeri al results have been obtained withsomeprototypi alparametrizationsoftheos illating oe ientsandareshown ina two-dimensional ase with one single varyingre tangular in lusion inside re t-angular ells. The method applies to all numeri al homogenization strategies that require tosolvea large numberofparametrized ell problems.

In periodi homogenization, only one ell problem has to be solved inorder to ompletely determine thehomogenized oe ient(s) to be usedinthehomogenized (ma ros opi )equation. Insharp ontrast,non-periodi homogenizationrequiresthe solution of several ell problems (infa t, theoreti ally, an innite numberof them, and in pra ti e, a large number). A homogenized oe ient is then approximated bysome average overalarge numberofmi ros opi ells. Consequently,asopposed totheperiodi asewhere the omputation islight andexa t, thenon-periodi ase asksfora omputationallydemanding andapproximate-in-nature task. Thisiswhy thedesignofa fastanda urate numeri al homogenizationmethodis onsidered as an important issuefor the treatment of non-periodi heterogeneous stru tures. The RBapproa h seemsvery well adaptedto this framework.

The arti le is organizedas follows. In se tion 2,we give a detailed presentation ofthesettingofthe problem. For thesakeof onsisten yand the onvenien eofthe reader, we also briey outline the main relevant issues in homogenization and RB theories. Inse tion3,theRBapproa hforaparametrized ellproblemisintrodu ed and we notably derive a posteriori error bounds related to the onvergen e of the RB method in thehomogenization ontext. Numeri al results for the prototypi al example of re tangular ells with one single re tanguler in lusion are presented in se tion4. Possible extensions ofour work aredis ussed inthenalse tion.

2 Setting of the problem, elements of homogenization theory and RB approa h

2.1 Formulation of the problem

Themathemati al problemunder onsideration throughoutthis arti le readsas fol-lows. We are interested in the behaviour of a sequen e of s alar fun tions

u

ǫ

that satisfy

in a bounded open set

Ω

⊂ R

, for a sequen e of s alars

ǫ > 0

. Of interest is the asymptoti limit of the sequen e

u

ǫ

when

ǫ

→ 0

, along with approximations for

u

ǫ

when

ǫ

is small.

For the sake of simpli ity, the s alar sour e term

f

is hosen in

L

2

_(Ω)

and we supplyequation(1)withthefollowing boundary onditionsonthesmooth (say

C

∞

-Lips hitz) boundary

∂Ω = Γ

D

S Γ

N

of

Ω

,

(BC)

 u

ǫ

_|

Γ

D

= 0

¯

¯

A

ǫ

_∇u

ǫ

_{· ¯n |}

Γ

N

= 1 .

(2)

But as a matter of fa t, it is well known that the homogenization results are lo al in nature and do not depend on the boundary onditions, ex ept for what regards error estimations lose to the boundary. Nordo thehomogenization results depend onthesour eterm

f

. Hen ethegeneralityof theassumptions(BC)and

f

∈ L

2

_(Ω)

, hosenhereto givea pre isemathemati al frameto thenumeri al experiments.

To x ideas, the unknown

u

ǫ

ould be thought of, either as a temperature or asan ele tri eld in a ma ros opi domain

Ω

. The tensorial oe ients for

¯

¯

A

ǫ

_(x)

would respe tively be thought of, either as temperature diusivities or as ele tri ondu tivities.

Next, let us dene, for any

ǫ > 0

,the family

¯

¯

A

ǫ

_{∈ L}

∞

_(Ω,

M

α

A

,γ

A

)

of fun tions from

Ω

totheset

M

α

A

,γ

A

ofuniformlypositivedenite

n

× n

matri es(se ondorder tensors)withuniformlypositivedenite inverses,that is,matri es

¯

¯

A

ǫ

satisfying,for all

x

∈ Ω

,

0 < α

A

| u |

2

≤

A

¯

¯

ǫ

(x)u

· u, ∀u ∈ R

n

(3)

0 < γ

A

| u |

2

≤

A

¯

¯

ǫ

(x)

−1

u

· u, ∀u ∈ R

n

.

(4)

Undersu h onditions,equations(1) -(2)arewellposedinthesenseofHadamard. Forevery

ǫ > 0

,thereexistsauniquesolution

u

ǫ

in

H

1

Γ

D

(Ω) =

u ∈ H

1

_{(Ω), u}

_|

Γ

D

= 0

that ontinuously depends on

f

,

ku

ǫ

k

H

1

(Ω)

≤ C(Ω)kfk

L

2

(Ω)

,

(5)

with some onstant

C(Ω)

that is only fun tion of

Ω

. Moreover, the sequen e of solutions

u

ǫ

isboundedin

H

1

Γ

D

(Ω)

,sothatsomesubsequen e

ǫ

′

weakly onverges to a limit

u

⋆

_{∈ H}

1

Γ

D

(Ω)

when

ǫ

′

→ 0

. We arespe i ally interested inestimating the behaviourof thisweakly- onvergent subsequen e.

In a typi al frame for the homogenization theory, the oe ients

¯

¯

A

ǫ

are as-sumed to os illatevery rapidly on a ount ofnumeroussmall heterogeneitiesinthe

domain

Ω

. For example,

ǫ

typi ally denotes the ratio of the mean period for mi- ros opi fast os illations of

¯

¯

A

ǫ

divided by the mean period for ma ros opi slow os illations of

¯

¯

A

ǫ

in

Ω

. Moreover, it is usually assumed that ma ros opi (ma ro) and mi ros opi (mi ro) s ales separate when

ǫ

is su iently small, whi h allows for theos illating oe ientsto be lo ally homogenized inthelimit

ǫ

→ 0

.

2.2 General ontext for homogenization

Asannoun edabove,thisse tion2.2in ludessomebasi sofhomogenizationtheory for linears alarellipti PDEs. Thepurposeof this summaryis onlyto olle t some elementaryresultsfor onvenien e. Readersfamiliarwiththehomogenizationtheory may thenlike to skip this se tion and pro eed to se tion 2.3, whi h introdu es the RBtheory.

2.2.1 Abstra t homogenization results

Thefollowing abstra thomogenizationresult isthebasisformanystudies thataim at omputing a numeri al approximation for

u

ǫ

when

ǫ

is small [13, 17 , 2, 11, 9 ℄. It shows that, inthe limit

ǫ

→ 0

, the small os illating s ale disappears" from the ma ros opi point of view ; that is, the mi ros opi and ma ros opi behaviours asymptoti allyseparate". Thisimpliesthatthelimitproblemiseasiertosolvethan equation(1)forsomesmall

ǫ

,sin etheformerdoesnotrequiretoresolvemi ros opi details. Moreover, a tra table approximation of

u

ǫ

when

ǫ

is small enough an be omputed fromtheasymptoti limit when

ǫ

→ 0

.

Morepre isely,

u

⋆

anbeobtained asthesolutiontotheH-limitequationfor(1) (seeequation(8)below). Itisthenan

L

2

-approximationfor

u

ǫ

when

ǫ

issmall,asthe asymptoti

L

2

-limit of

u

ǫ

when

ǫ

→ 0

. Moreover, an improved

H

1

-approximation for

u

ǫ

when

ǫ

issmall analsobe omputedwith

u

⋆

after orre tion"ofthegradient

∇u

⋆

.

Thehomogenizationofthesequen eofequations(1)isthemathemati al pro ess whi h allows to dene the H-limit equationand the

H

1

approximation for

u

ǫ

. It is performed usingthe following abstra tobje ts [14 ℄:

•

asequen eof

n

os illating testfun tions

z

ǫ

i

∈ H

1

(Ω)

su hthat,for everydire tion

(e

i

)

1≤i≤n

of theambient physi al spa e

R

n

,wehave

z

ǫ

i

⇀x

i

in

H

1

_(Ω)

and

−div( ¯¯

A

ǫ

_∇z

ǫ

_i

) =

_{−div( ¯¯}

A

⋆

e

i

)

in

H

−1

_{(Ω) ,}

•

a homogenized tensor

A

¯

dened by

¯

¯

A

ǫ

_∇z

_i

ǫ

⇀ ¯

A

¯

⋆

e

i

in

[L

2

_(Ω)]

n

_,

(6)

•

a subsequen e

u

ǫ

′

of solutions for (1)thatsatises

(

u

ǫ

′

⇀ u

⋆

in

H

1

Γ

D

(Ω)

¯

¯

A

ǫ

′

∇u

ǫ

′

⇀ ¯

A

¯

⋆

_∇u

⋆

in

[L

2

_(Ω)]

n

(7) where

u

⋆

issolution forthe H-limitor homogenized equation

−div( ¯¯

A

⋆

(x)

∇u

⋆

(x)) = f (x),

∀x ∈ Ω ,

(8)

suppliedwiththe boundary onditions (BC),

•

and anasymptoti approximation fora subsequen e

ǫ

′

of

ǫ

thatsatises

u

ǫ

′

− u

⋆

L

2

(Ω)

ǫ

′

→0

−→ 0

(9)

∇u

ǫ

′

−

n

X

i=1

z

_i

ǫ

′

∂

i

u

⋆

[L

1

loc

(Ω)]

n

ǫ

′

→0

−→ 0 ,

(10) where

∂

i

u

⋆

are the omponentsof

∇u

⋆

inea h dire tion

e

i

. Note that the latter onvergen e result (10) for

∇u

ǫ

′

also holds in

[L

2

loc

(Ω)]

n

if

u

⋆

_{∈ W}

1,∞

_(Ω)

. So, if

u

⋆

_{∈ H}

2

_(Ω)

, the orre tor result states that

u

ǫ

an be approximated withthefollowing formula,

u

ǫ

= u

⋆

+

n

X

i=1

(z

ǫ

_i

− x

i

)∂

i

u

⋆

+ r

ǫ

,

(11)

where the remainder term

r

ǫ

onverges strongly to zeroin

W

1,1

loc

(Ω)

.

In a nutshell, the homogenization of the sequen e of equations (1) has allowed toderiveanabstra t homogenizedproblem, (6) -(8),thesolution

u

⋆

ofwhi h anbe orre tedwith(10) into an

H

1

approximationof

u

ǫ

in thelimit

ǫ

→ 0

. Butwela kanexpli itexpressionforthehomogenizedtensor

¯

¯

A

⋆

togetanexpli it asymptoti limit

u

⋆

. Thatiswhy,thoughit isnotrequired bythepreviousabstra t theory,thes aleseparationinthebehaviouroftheos illating oe ients

¯

¯

A

ǫ

isoften assumedto be expli itly en oded, using some spe i postulated form for

¯

¯

A

ǫ

. This allowstoderiveanexpli itexpressionofthehomogenizedproblem,andevenanerror estimate in terms of

ǫ

for the orre tion error

r

ǫ

in (11) , whi h allows to quantify thehomogenizationapproximation error.

2.2.2 The expli it two-s ale homogenization

To get expli it expressions for the homogenized problem, some parti ular depen-den e of the family

¯

¯

A

ǫ

on the spa e variable

x

is often assumed, like in two-s ale homogenizationforinstan e. Namely,ona ountofthes aleseparationassumption and the lo al dependen e of the homogenization pro ess, one of the most ommon assumption isthe lo al periodi ity for

¯

¯

A

ǫ

,whi h an bemade pre iseasfollows. It isassumedthat tensors

¯

¯

A

ǫ

aretra esoffun tions oftwo oupled variables on thesetlo allydened bya fastmi ros opi variable

ǫ

−1

_x

linearly oupled withthe slow ma ros opi variable

x

in

Ω

:

¯

¯

A

ǫ

(x) = ¯

A

¯



x,

x

ǫ



,

(12)

where,for any

x

∈ Ω

,thefun tion

¯

¯

A(x,

_·)

¯

¯

A(x,

·) : y ∈ R

n

→ ¯¯

A(x, y)

∈ R

n×n

is 1-periodi in ea h of the

n

dire tions

(e

i

)

1≤i≤n

, whi h makes the lo al os illa-tions ompletely determined when

ǫ

→ 0

. The domain

Y = [0, 1]

n

of the periodi pattern is alled the ell and is identied with the

n

-dimensional torus.

¯

¯

A(x,

·)

is said to be Y-periodi . Note that the properties of the tensors

¯

¯

A

ǫ

imply

¯

¯

A

_∈

L

∞

(Ω, L

∞

(Y,

_M

α

A

,γ

A

))

.

Now, undertheassumption (12) oflo al periodi ity,one possible mannerto get expli it expressions for the homogenized problem is to perform a formal two-s ale analysiswiththe following Ansatz

u

ǫ

(x) = u

0



x,

x

ǫ



+ ǫu

1



x,

x

ǫ



+ ǫ

2

u

2



x,

x

ǫ



. . .

(13)

where, for any

x

∈ Ω

, thefun tions

u

i

(x,

·)

are Y-periodi . The rst two terms of theAnsatz(13) areshownto oïn ide withthe

H

1

approximation(11)for

u

ǫ

[4,1 ℄. InsertingtheAnsatz(13)intoequation(1)givesthefollowingexpli itexpressions for theobje tspreviously dened bytheabstra thomogenization result:

•

the fun tion

u

0

= u

⋆

_(x)

does not depend on the fastvariable

ǫ

−1

_x

and is the

L

2

approximationfor

u

ǫ

givenbythe onvergen e result(9),

•

thegradient

∇

y

u

1

(x,

·)

linearlydependson

∇

x

u

⋆

_(x)

,

u

1



x,

x

ǫ



=

n

X

i=1

∂

i

u

⋆

(x)w

i



x,

x

ǫ



+ ˜

u

1

(x),

•

the

n

ell fun tions

w

i

(x,

·)

, parametrized by their ma ros opi position

x

∈ Ω

, aresolutions to thefollowing

n

ell problems,

−div

y

( ¯

A(x, y)

¯

· [e

i

+

∇

y

w

i

(x, y)]) = 0,

∀y ∈ R

n

,

(14)

andthe orre tors

z

ǫ

i

nowread

z

ǫ

i

= x

i

+ ǫw

i

(x, x/ǫ)

,

•

theentries

 ¯¯

A

⋆

_(x)

i,j



1≤i,j≤n

ofthehomogenizedmatrix

¯

¯

A

⋆

anbeexpli itly

om-putedwiththe ell fun tions

w

i

(x,

·)

,

¯

¯

A

⋆

(x)

i,j

=

Z

Y

¯

¯

A(x, y)[e

i

+

∇

y

w

i

(x, y)]

· e

j

dy,

(15)

•

the

H

1

approximationfor

u

ǫ

isnowtra table andwrites

u

ǫ

= u

⋆

+ ǫ

n

X

i=1

w

i

∂

i

u

⋆

+ r

ǫ

,

(16) where, provided

u

⋆

_{∈ W}

2,∞

_(Ω)

, the orre tion error

r

ǫ

an be estimated to lo ally s ale as

ǫ

(far enough from the boundary layer), and to globally s ale as

√

ǫ

,

kr

ǫ

k

H

1

ΓD

(ω)

≤ C

1

ǫ

ku

⋆

_k

W

2

,∞

_(ω)

,

∀ω ⋐ Ω,

(17)

kr

ǫ

k

H

1

ΓD

(Ω)

≤ C

2

√

ǫ

_ku

⋆

_k

_W

2,∞

_(Ω)

,

(18)

with onstants

C

i

dependingonly on

Ω

.

To sumup,the lo alperiodi ityassumption(12)allowsto ompletely determine thehomogenized problem through expli it two-s ale expressions. The derivation of thehomogenizedequationinthe aseoflo allyperiodi oe ientsservesasabasis for manynumeri al homogenization strategies.

2.2.3 Numeri al homogenization strategies

Underlo alperiodi ityassumption(12),atwo-s aleexpli ithomogenizationstrategy for a sequen eof linears alarellipti PDEslike (1) readsasfollows intheframe of Finite-Element approximations for the s alarellipti problems(14) and(8).

Algorithm 1 (Two-s ale homogenization strategy) Tohomogenizethesequen e of PDEs (1):

1. solve the parametrized ell problems (14) at ea h point

x

∈ Ω

where the value of

¯

¯

A

⋆

(x)

is ne essary to ompute the FE matrix of the homogenized problem (8),

2. store the fun tions

w

i

for the future omputation of the

H

1

approximation of

u

ǫ

,

3. assemblethe FEmatrix asso iated withthehomogenizedoperator

−div( ¯¯

A

⋆

_∇·)

,

4. solve the ma ros opi homogenized problem (8),

5. build the

H

1

approximation (16)for

u

ǫ

with

u

⋆

and

w

i

.

On the one hand, in many pra ti al situations, it is very ommon to assume thatthe tensors

¯

¯

A

ǫ

satisfyassumption (12). Indeed, inpra ti e,

¯

¯

A

ǫ

isoften known for some given

ǫ = ǫ

0

only. So, the asymptoti stru ture

¯

¯

A

ǫ

of the problem with os illating oe ients in

Ω

has to be onstru ted from theonly member

¯

¯

A

ǫ

0

. Now, totakeadvantageoftheexa t expli itexpressions given bythetwo-s aleanalysis,it ispreferable to builda family

¯

¯

A

ǫ

that satisesassumption (12), whenpossible. As-sumption(12)thenseemsfullyjustiedformanyappli ationsfromthepra titioner's point of view. Then, themain numeri al di ulty ofthe two-s ale homogenization istherststep,thatisthea urate omputationofalargenumberof ellfun tions. Thisisthe main issueaddressedinthis arti le.

On the other hand, for some appli ations where the heterogeneities are highly non-periodi , one may want to build the sequen e

¯

¯

A

ǫ

dierently, or even skip the expli it onstru tionofthesequen e

¯

¯

A

ǫ

. Forexample,thea tual onstru tionpro ess of heterogeneities may suggest another sequen e

¯

¯

A

ǫ

for whi h the error estimation of the

H

1

approximation would then be more pre ise and meaningful. Or it may seem too di ult to expli itly build su h a sequen e

¯

¯

A

ǫ

that satises (12) from the knowledge of some

¯

¯

A

ǫ

0

only. In su h ases, many numeri al homogenization strategieshave beendeveloppedtotreatthenumeri al homogenizationofos illating oe ientsthatare not lo allyperiodi .

Toour knowledge, mostoftheexistingnumeri alhomogenizationstrategiesmay be lassiedinoneofthetwofollowing ategories. Theyeitherrelyondierentspa e assumptions than lo al periodi ity for the os illating oe ients (e.g. reiterated homogenization [15 ℄, sto hasti homogenization [6℄, deformed periodi oe ients [8℄, sto hasti ally deformed periodi oe ients [5 ℄), and still allow to derive exa t (but not always fully expli it) expressions for the homogenized equation and the error estimateoftheapproximation.

Or,thenumeri alhomogenizationstrategiesaremu h oarserandonlyrelyonthe assumption that expli it s ale separation allow for the behaviour of the os illating oe ients at some small

ǫ

to be numeri ally homogenized. Those strategies are then approximate-in-nature. They manage to approximate quite a large lass of heterogeneous problems, but may be omputationally very demanding. They may also la k sharp error estimates. Example are theMultis ale nite-element method (MsFEM) [13, 2℄, the Heterogeneous multis ale method (HMM) [9 ℄, or the re ent variationalapproa h for non-linear monotoneellipti operatorsproposedin[11℄...

Now, in any of thetwo previously des ribed situations where the numeri al ho-mogenization strategiesrequire the omputation of a large number of parametrized ell problems, the RB approa h proposed here-after is likely to bring some addi-tional omputational e ien y. As a matter of fa t, most numeri al approximate homogenization strategies are only slight modi ations of the exa t two-s ale ho-mogenization strategy proposed above in the frame of lo al periodi ityassumption (12), and they do require the omputation of a large number of parametrized ell fun tions. Formanyme hani alappli ations,thisowestotheassumedexisten eofa Representativevolume elements(RVE),whi hleadsto general ellproblems atea h point

x

of the ma ro domain [10 ℄. One simple example of a possibly approximate numeri al homogenizationstrategy thatrequiresthe omputation ofalargenumber ofparametrized ell fun tionsisbasedon thefollowingtheorem, proved byJikovet al. in[14 ℄. Theorem 1 Let

¯

¯

A

ǫ

be a sequen e of matri es in

L

∞

_(Ω,

_M

α,η

)

that denes a se-quen eoflinears alarellipti problemslike(1). TheH-limitof

¯

¯

A

ǫ

isthehomogenized tensor

¯

¯

A

⋆

.

For any

x

∈ Ω

and

ǫ > 0

, and any su iently small

h > 0

, let us dene a sequen e of lo ally periodi matri es

¯

¯

A

ǫ

h

(inthe sense of (12) ),

¯

¯

A

ǫ

_h

(x) = ¯

A

¯

ǫ

(x + h[ǫ

−1

x]) ,

(19)

where

[ǫ

−1

_x]

denotes the integer part of

ǫ

−1

_x

.

Then, for every

1

≤ i ≤ n

, there exists a unique sequen e of periodi solutions

w

ǫ,h

_i

(x,

_·)

inthequotiented Sobolevspa e

H

1

#

(Y )/R

of

Y

-periodi fun tionsin

H

1

_{(Y )}

that satisfy the

n

ell problems

−div( ¯¯

A

ǫ

(x + hy)

· [e

i

+

∇

y

w

ǫ,h

i

(y)]) = 0,

∀y ∈ Y

(20)

in the

n

-torus

Y = [0, 1]

n

Forea h point

x

∈ Ω

and

ǫ > 0

, we dene a matrix

¯

¯

A

⋆

ǫ,h

made of elements

¯

¯

A

⋆

_ǫ,h

(x)e

i

· e

j

=

Z

Y

¯

¯

A

ǫ

(x + hy)

_{· [e}

i

+

∇

y

w

i

ǫ,h

(x, y)]

· [e

j

+

∇

y

w

ǫ,h

j

(x, y)] dy

for any

(i, j)

in

{1, 2, . . . , n}

2

. Then,there exists a subsequen e

h

′

_{→ 0}

su h that

lim

h

′

→0

ǫ→0

lim

¯

¯

A

⋆

_ǫ,h

′

(x) = ¯

A

¯

⋆

(x) .

Thistheoremshowsthat, foranyfamily

¯

¯

A

ǫ

,itisalwayspossibleto approximate the exa t homogenized problem with the same expli it expressions than those ob-tained under the lo al periodi ity assumption (12), after lo al periodization of

¯

¯

A

ǫ

like in(19) .

So, onsidering the lands apefor thehomogenization theoryasdes ribedabove, among alternatives way of improving the numeri al homogenization strategies, we hoosehereto on entrateonspeedingupthenumeri altreatmentofalargenumber ofparametrized ellfun tions,ratherthan,forexample,reningtheapproximations leading to expli it ell problems for larger lassof os illating oe ients.

In the sequel, for the sake of simpli ity, we assume thatthe sequen eof tensors

¯

¯

A

ǫ

satises assumption (12)and apply theRBapproa h to thetwo-s ale numeri al homogenization strategy (Algorithm 1). Yet, the RB approa h may apply as well with any numeri al homogenization strategy that onsists of rst approximating

¯

¯

A

ǫ

by some sequen e of tensors

¯

¯

A

ǫ

_h

like in (19) , the latter leading to an expli it approximation for the homogenized problemafter solving parametrized generalized ell problems like (20). Let us now on entrate on de reasing as mu h as possible the omputational ost ofsolving (14) for manyparameter values

x

∈ Ω

.

2.3 The redu ed-basis method

Two riti al observations allow to think that an output-oriented model order re-du tion te hnique like the RB method is likely to improve the repeated numeri al treatmentofparametrized ellproblems(14). First,onlyoutputs ofthe ellfun tions arerequiredto solvethehomogenized problem,and ana posterioriestimation gives sharperror bounds forthose outputs.

Se ond, as extensively dis ussed above, the numerous parametrized ell prob-lemsarising from numeri al homogenization strategies an be solved independently for ea h value of the parameter. Thus, a omputational pro edure based on an of-ine/onlineapproa h shouldnaturallyallowforaredu tionofthe omputationtime

inthelimit of manyqueries. Inparti ular, a large numberof (and theoreti ally, an innityof) parametrized ell problems o urs inthelimit

ǫ

→ 0

ofthe homogeniza-tionstrategies, inorderto omputethe homogenized problem(8)withnon-periodi oe ients. And the number of homogenized problems to ompute and solve an alsobevery largeinpra ti e, for instan eintheframe ofparameterestimation and optimization problems.

Thesetwo observations motivateanRBapproa hfortheparametrized ell prob-lems(14), whi h should signi antly de reasetheexpense of omputationsinterms of CPUtime for the homogenization problems where theoine stage is short om-pared to the online stage, or where the oine stage is not even an issue (like in real-timeengineeringproblemsforinstan e)[18℄. Wearenowgoingtointrodu ethe basi softheredu ed-basismethod,wellknowntoexperts,whomaywanttodire tly pro eedto the Se tion 3.

2.3.1 The parametrized ell problem

Let

X

be the quotientedspa e

H

1

#

(Y )/R

of

Y

-periodi fun tionsthatbelongtothe Sobolev spa e

H

1

_{(Y )}

. The Hilbertspa e

X

isimbued withthe

˜

H

1

_{(Y )}

-norm

kuk

X

=

Z

Y

∇u · ∇u



1/2

indu ed by the innerprodu t

u, v

X

=

R

Y

∇u · ∇v

for any

(u, v)

∈ X × X

. Inthe dualspa e

X

′

of

X

,thedualnormis dened for any

g

∈ X

′

by

kgk

X

′

= sup

v∈X

g(v)

kvk

X

.

For any

x

∈ Ω

,we dene:

•

a ontinuousand oer ive bilinear formin

X × X

parametrizedby

x

∈ Ω

,

a(u, v; x) =

Z

Y

¯

¯

A(x, y)

_{∇u(y) · ∇v(y)dy, ∀(u, v) ∈ X × X ,}

for whi h

α

A

and

γ

−1

A

arerespe tively oer ivityand ontinuity onstants,

•

and

n

ontinuous linearforms in

X

also parametrizedby

x

∈ Ω

,

f

i

(v; x) =

−

Z

Y

¯

¯

Now,for anyinteger

i

,

1

≤ i ≤ n

,thei-th ellproblem(14)forthe ellfun tions

w

i

(x,

·)

rewrites inthe following weak form: Find

w

i

(x,

·) ∈ X

solutionfor

a(w

i

(x,

·), v; x) = f

i

(v; x) ,

∀v ∈ X ,

(21) where

x

∈ Ω

playstherole ofa parameter.

We set

M

i

=

{w

i

(x, y), x

∈ Ω}

the solution subspa e of the i-th ell problem (21)indu ed bythevariations of

x

in

Ω

,and

M = {w

i

(x, y), x

∈ Ω, 1 ≤ i ≤ N} =

N

[

i=1

M

i

theglobal solutionsubspa e, thatisthereunion ofallsolution subspa esfor all ell problems.

Remark 1 Note at this point that

M

i

and

M

should be seen as spa es indu ed by the familyof oe ients

 ¯¯

A

⋆

_(x,

_·)



x∈Ω

,and not

x

. Itis indeed always possible (and oftenuseful)touseotherexpli itquantitiesthan

x

asparameters tomap

M

i

and

M

, provided that the variations of the parameters inside a given range of values indu e thesamefamilyof oe ientsandthesame orresponding ellfun tionsthan

x

∈ Ω

. Besides,forthesakeofsimpli ityinthepresentationoftheRBmethod,thetensor

¯

¯

A(x,

_·)

willbeassumedsymmetri inthefollowing. Hen e,inthe omputationofthe homogenizedtensor

¯

¯

A

⋆

_(x)

,theonly interesting outputforthe

n

solutions

w

i

(x,

·)

is given bya symmetri matrix

s

of size

n

× n

(somewhat similar to a omplian e in theterminology ofme hani s). Theentries

(s

ij

)

1≤i,j≤n

of thematrix

s

aregiven by

s

ij

(x) =

−f

j

(w

i

(x,

·); x) =

Z

Y

¯

¯

A(x, y)

_∇w

i

(x, y)

· e

j

dy .

(22)

But note that the RB approa h still applies with non-symmetri tensors

¯

¯

A(x,

_·)

moduloslight modi ations 1

.

ThepurposeoftheRBmethodistospeedupthe omputationofalargenumber of solutions

w

i

(x,

·) ∈ X

of (21)for many parameter values

x

∈ Ω

while ontrolling theapproximationerror for theoutput

s

.

1

When the tensor

¯

¯

A

(x, ·)

is not symmetri , a dual problem, adjoint to the problem (21) , is introdu ed. The dualproblem anbe solvedsimilarly to theprimal" problem (21) withanRB method,inadualRBproje tionspa e. Last,theoutputshouldberewrittenlike

s

plusanadditional termthata ountsfortheresidualerrorduetotheRBproje tionoftheequation(21) . This primal-dualapproa hismoreextensivelydes ribedin[18 ℄forinstan e.

2.3.2 Prin iple of the redu ed-basis method

The purpose of most orderredu tion te hniques like theRBmethod isto solvethe weakformofaPDElike(21)throughaGalerkinproje tionmethodwithaHilbertian basisthatisadapted" to the solution subspa e

M

.

Forinstan e, aHilbertianbasisthatisadaptedtotheequations(21)when

x

∈ Ω

and

1

≤ i ≤ n

is an orthonormal family

(ξ

j

)

j∈N

(orthonormal with respe t to the ambient inner produ t

(

·, ·)

X

) su h that

ˆ theambient solution spa e

X ⊂ span{ξ

j

, j

∈ N}

is separable,

ˆ and,for anite

N

-dimensional subspa e

X

N

= span

{ξ

j

, 1

≤ j ≤ N}

of

X

,the Galerkin approximations

w

iN

(x,

·) ∈ X

N

for

w

i

(x,

·)

that satisfy,for any

x

in

Ω

and

1

≤ i ≤ N

,

a(w

iN

(x,

·), v; x) = f

i

(v; x) ,

∀v ∈ X

N

,

(23) aresu iently" lose to

w

i

(x,

·)

for agiven tolerable pre ision.

But the previous denition is only vaguely stated until the tolerable pre ision" is mathemati ally dened.

One possible way ofdening su ient pre ision is to ontrol theapproximation errorfor

w

i

(x,

·) ∈ X

withthenatural norm

k · k

X

ofthe ellproblem. The redu ed-basis method rather proposes to ontrol the approximation error for some linear outputlike

s

,whi hisnotverydierentinthepresent asewhere

¯

¯

A(x,

_·)

issymmetri 2

,asitwill be made learer inse tion3.1.

TheRBmethodisbasedonthe omputation ofa basisforaGalerkinproje tion spa e in

X

thatis adapted" to the ell problem (14) inthesense of the minimiza-tionofthe outputapproximationerror. Theapproximationerrorsaremadeexpli it through rigorous a posteriori estimates, whi h allow to a posteriori ertify the e- ien yofthemodelorderredu tion,thatis, the onvergen e oftheRBmethodwhen thesize

N

of theGalerkinproje tion spa ein reases.

2

Ingeneral non-symmetri ases,the approximationerror for linear outputs like

s

anbe ex-pressedasaprodu toftwoapproximationerrors, onefor theparametrizedsolutions

w

i

(x, ·)

and anotheroneforsomedualquantitythatissolutionfortheproblemdualto(21) . Buthere,be ause ofthesymmetryandof thespe i natureof theoutput,theso- alled omplian einreferen e to me hani s, the dual problem is unne essary and the approximation error for the output an be dire tlyexpressedasthesquareoftheapproximationerrorforthe ellfun tion.

2.3.3 Pra ti e of the redu ed-basis method

TheRBmethod omputes abasisfor

X

N

from anapproximation

M

p

of

M

,

M

p

=

{w

i

(x,

·), x

k

∈ D, 1 ≤ i ≤ n} ,

indu ed bya dis rete sample

D = {x

k

, 1

≤ k ≤ p}

,

p

× n > N

,of parameter values distributedoverthe parameterspa e

Ω

(

D ⊂ Ω

). Thisistermedastheoine stage. Su h a model order redu tion is e ient if the tolerable pre ision for the output approximation error is rea hed with a

N

-dimensional adapted basis" when

N

is m uhsmallerthan

N

,where

N

isthenumberofdegrees offreedom ne essaryfor a generi numeri al method,liketheFEmethod,to rea hthesamepre ision. We all redu edbasissu h an adaptedbasis"

(ξ

j

)

1≤j≤n×N

.

Now, the RB treatment of(21) begins withthe omputation ofa sample of ell fun tionsthatindu es

M

p

. The ell fun tionsofthenitespa e

M

p

shouldthenbe approximatedbeforethemodelorderredu tionispossible. An a urateandgeneri numeri almethod

3

,withalargenumber

N

ofdegreesoffreedom,likeanFEmethod with a ne mesh for

Y

for instan e, should be used at the beginning of the oine stageto ompute an approximation

M

N

p

for

M

p

.

Then, the sample of solutions

M

p

from whi h the basis

(ξ

j

)

1≤j≤n×N

is built should be as representative as possible of

M

. In the absen e of information,

D

should be hosen arbitrarily. So, it is not only quite often impossible to hoose a priori the right parameter sample

D

for

X

N

in order to minimize the output approximation error over every

N

-dimensional ve tor subspa e of

M

, but besides,

D

should not be too large so that the redu ed-basis onstru tion is fast ompared to theonline stage. Hen ethe ne essityfor a reliable a posteriori ontrol of theRB approximation method, whi h allows to build fast a redu ed basis

(ξ

j

)

1≤j≤n×N

, as itwill be seeninse tion 3.2.

After building a redu ed basis for the ve tor eld

X

N

, the Galerkin proje tion method is applied to the weak form for (21) at any

x

∈ Ω

. That is, inthis online stage, the previous redu ed basis is assumed to span a ve tor eld

X

N

su iently lose to the solutionmanifold

M

oftheparametrized PDE sothatwe an ompute fast a su iently a urate Galerkin approximation in

X

N

for the solution of the parametrizedPDE at anyparameter value

x

∈ Ω

.

3

Note that the time needed to ompute a (possibly large) sample of

p

a urate FE approxi-mations an also be anissue, that anbe dealt withby apre-pro essing stage a ording to the parametrization,asitwillbemade learerinse tion3.2.

3 Redu ed-basis approa h for the ell problem

TheRBapproa hforequation(21)needstoa posterioriestimatetheapproximation errorfor Galerkinsolutions ofthe ell problem. Ina se ondstage,thisallowsfor an a posteriori estimationof theapproximationerror on theoutput

s

.

3.1 Error bounds for the ell problem

The purpose of this se tion is to derive the error bound (28) for the ell funtion s, solutions of equation (21). This allows to a posteriori estimate the approximation errorfor Galerkin solutions ofthe ell problem,and their outputsthrough theerror bound (33) . To this end, let us introdu e the linear operator

T

x

: X → X

so that, for any

u

∈ X

and

x

∈ Ω

,

(T

x

u, v)

X

= a(u, v; x),

∀v ∈ X .

Theexisten eofsu h anoperatordire tlyleans ontheRiesz-Fré hetrepresentation Theoremin the Hilbert spa e

X

.

For any

1

≤ i ≤ N

and

x

∈ Ω

,the Galerkinapproximation error

kw

i

(x,

·) − w

iN

(x,

·)k

X

(24)

an be bounded starting fromthe following equality,

a(w

i

(x,

·) − w

iN

(x,

·), v; x) = f

i

(v; x)

− a(w

iN

(x, y), v; x),

∀v ∈ X ,

(25) whi his easilyobtained bysubstra tionof (23)from (21).

Letus dene theparametrized bilinearresidual forms

g

i

in

X × X

su hthat, for allparameter values

x

∈ Ω

and

1

≤ i ≤ n

,

g

i

(u, v; x) = a(u, v; x)

− f

i

(v; x),

∀(u, v) ∈ X × X .

Then, equation (25) with

v = w

i

(x,

·) − w

iN

(x,

·)

allows to immediately derive the following estimates through the dual norm of the residual linear form for

w

i

(x,

·)

dened in

X

v

_{→ g}

i

(w

iN

(x,

·), v; x) .

First,owing to the oer ivityof thebilinearform

a

,weobtain thelower bound:

for theGalerkinapproximation error.

Se ond, inview of the ontinuity of the bilinear form

a

,we obtain the superior bound:

kg

i

(w

iN

(x,

·), v; x)k

X

′

≤ γ

−1

A

kw

i

(x,

·) − w

iN

(x,

·)k

X

,

(27) for theGalerkinapproximation error.

Finally,notethatitis possible to ompute thedualnorm ofthelinearform

v

→ g

i

(w

iN

(x,

·), v; x) = −a(w

i

(x,

·) − w

iN

(x,

·), v; x)

usingtheRiesz-Fré hetrepresentant

T

x

_(w

i

(x,

·) − w

iN

(x,

·))

intheHilbertspa e

X

, andthatone an obtainnumeri al approximations for

α

A

and

γ

−1

A

,either usingthe spe tral properties of the matri es resulting from the Galerkin proje tion in large generi solutionspa esduringtheoinestage,orusingpropertiesofthe parametriza-tionlikeinse tion3.5. So,theGalerkinapproximationerror(24) anbeaposteriori bounded using estimations (26)and (27).

For

x

∈ Ω

and

1

≤ i ≤ N

,we dene a posterioriestimators

∆

N

(w

i

(x,

·))

forthe Galerkinapproximation errors(24), usingthe previoussuperior bounds,by

∆

N

(w

i

(x,

·)) =

ka(w

i

(x,

_{·) − w}

iN

(x,

·), ·; x)k

X

′

α

A

.

(28)

The ee tivities

η

N

(w

i

(x,

·))

orrespondingto theestimators

∆

N

(w

i

(x,

·))

,

η

N

(w

i

(x,

·)) =

∆

N

(w

i

(x,

·))

kw

i

(x,

·) − w

iN

(x,

·)k

X

,

(29)

satisfythe following inequalities independently of

N

,

1

_{≤ η}

N

(w

i

(x,

·)) ≤

γ

_A

−1

α

A

,

(30)

whi hshows thestability ofthe error estimator

∆

N

(w

i

(x,

·))

.

Last, Galerkinapproximationsfor thehomogenized andtheoutputmatrixwrite

¯

¯

A

⋆

_N

(x)

i,j

=

Z

Y

¯

¯

A(x, y)[e

i

+

∇

y

w

iN

(x, y)]

· e

j

dy,

(31)

s

N

_ij

(x) =

Z

Y

¯

¯

A(x, y)

_∇w

iN

(x, y)

· e

j

dy .

(32)

The a posteriori superior bound

∆

N

(w

i

(x,

·))

for the Galerkin approximation error (24) will now allow us to derive a simple superior bound for output approximation errors. Indeed, we have for any

1

≤ i, j ≤ N

and

x

∈ Ω

,

| s

ij

(x)

− s

N

ij

(x)

| = | f

j

(w

i

(x,

·) − w

iN

(x,

·); x) |

=

_{| a(w}

j

(x,

·), w

i

(x,

·) − w

iN

(x,

·); x) |

=

_{| a(w}

j

(x,

·) − w

jN

(x,

·), w

i

(x,

·) − w

iN

(x,

·); x) |

≤ α

A

∆

N

(w

i

(x,

·))∆

N

(w

j

(x,

·))

sin e

w

j

(x,

·)

and

w

i

(x,

·)

aresolutions for (21), and

w

iN

(x,

·)

issolution for (23). Wearenallyinpossessionofanaposteriorisuperiorbound

∆

s

ij,N

(x)

forGalerkin approximations oftheoutput

s

ij

(x)

,

∆

s

_ij,N

(x) =

ka(w

i

(x,

·) − w

iN

(x,

·), ·; x)k

X

′

ka(w

j

(x,

·) − w

jN

(x,

·), ·; x)k

X

′

α

A

.

(33)

Numeri alapproximations for

∆

s

ij,N

(x)

willallowustobuildfastaredu edbasisfor ellproblems(21). Notethat

∆

s

ij,N

(x)

s alesastheprodu t

∆

N

(w

i

(x,

·))∆

N

(w

j

(x,

·))

, hen ethe interestofmodelorder redu tionte hniques forsolutions

w

i

(x,

·)

without mu h lossof pre ision foroutput

s(x)

.

Remark 2 Note thatfortheoutputerror boundstos alelikethe square ofthe error bound for the ell fun tions,it has been essential tohave the followingorthogonality propertyfor any

x

∈ Ω

,

a(w

i

(x,

·) − w

iN

(x,

·), w

jN

(x,

·); x) = 0 , ∀1 ≤ i, j ≤ N.

That is why we have hosen to build only one RB proje tion spa e

X

N

, spanned by all the parametrized ell fun tions

w

i

(x

k

, y)

when

1

≤ i ≤ n

and

x

k

∈ D

. Yet, note that without this hoi e, the same s aling an still be obtained with

n

distin t RB proje tion spa es

(X

iN

)

1≤i≤n

for ea h of the

n

solution subspa es

M

i

, provided one slightly modies the denition of the output. Namely, another output matrix

σ

and its RB approximation

σ

N

should then be dened, starting from

s

and

s

N

, by adding a residual error. Their entries read, for

1

≤ i, j ≤ n

,

σ

ij

(x) =

−f

j

(w

i

(x,

·); x) + g

i

(w

i

(x,

·), w

j

(x,

·); x)

(34)

σ

N

_ij

(x) =

_−f

j

(w

iN

(x,

·); x) + g

i

(w

iN

(x,

·), w

jN

(x,

·); x) ,

(35)

where

σ = s

be ausethetensor

¯

¯

A(x,

·)

issymmetri ,and

σ

N

_{= s}

N

onlywhentheRB proje tion spa e isthe same for all solution subspa es

M

i

(asabove). Interestingly,

thesame additionalresidual termin the output

σ

also ariseswhenthe tensor

¯

¯

A(x,

_·)

isnotsymmetri . Itisthenevaluatedwithdual ellfun tions,solutionsforaproblem dual to the ell problems (21) [18℄.

3.2 The redu ed-basis onstru tion

Let us hoose a sample

D = {x

k

, 1

≤ k ≤ p}

of

p

values for the parameter

x

in

Ω

. Unlesssomephysi alpropertiesofthesystemguidesthe hoi efor

D

,theparameter values

x

k

should be

p

realizations of a random variable uniformly distributed over

Ω

. To a uratelysolve equation(21) forea h parameter value

x

k

,we hoose anFE methodwitha

N

-dimensionalFEve torspa e

X

N

. Typi ally,

N

isverylargeforthe FE approximations to be a urate. The

(n

× p)

FE approximations

w

iN

(x

k

,

·)

i,k

for

w

i

(x

k

,

·)

i,k

spanan

(n

× p)

-dimensional ve tor spa e

X

n×p

⊂ X

that ontains theapproximation

M

N

p

=

{w

iN

(x

k

, y), x

k

∈ D}

ofthesolution subspa e

M

.

Remark 3 Computing the

(n

× p)

FEapproximations

w

iN

(x

k

,

·)

i,k

an be ome a umbersomepreliminarytask tothe redu ed basis onstru tion whenthe orrespond-ingFE matri es are di ult toassemble. Thatis why the RB method also in ludes somepre-pro essing toassemblefastthoseFEmatri es. Su hapre-pro essingisvery simple in the ase where the parametrization of the os illating oe ients is ane (this terminology will be made learer in se tion 3.5). It might be more di ult in other more general ases. In the present work, we only treat the ane ase. Some more elaborate resultsin non-ane ases,that are based on theextrapolationmethod introdu ed in [3, 12℄ forinstan e, will appear in [7℄.

First, in the oine stage of the RB approa h, we would like to build a

N

-dimensional RB proje tion subspa e

X

N

⊂ X

that also ontains a very lose ap-proximation of

M

N

p

, thus of

M

.

X

N

will be spannedby a redu ed basis

(ξ

j

)

1≤j≤N

madeof

N

ve torsof

X

n×p

,with

N < n

×p

. Moreover,

N

≪ N

shouldbesu iently smallfor the modelorder redu tionto allow asigni antgain of omputation time. Then, in the online stage, for any

x

∈ Ω

,

w

i

(x, y)

is to be approximated, using the RB method, by some ve tor

w

iN

(x, y)

in the Galerkin proje tion spa e

X

N

of size

N

thatwrites

w

iN

(x, y) =

N

X

j=1

The redu ed basis

(ξ

j

(y))

1≤j≤N

of

X

N

is built in order to best ontrol the

ap-proximationerror for outputs through the a posteriorierror bounds derived above. Thisisperformedinthe oine stage asfollows.

Algorithm 2 (Oine algorithm) Webuildaredu ed basis

(ξ

j

(y))

1≤j≤N

from

Span

{w

i

(x

k

,

·), x

k

∈

D, 1 ≤ i ≤ n}

as follows:

1. for some ouple

k

0

_{(1), i}

0

₍₁₎

,

1

≤ k

0

₍₁₎

_{≤ p}

and

1

≤ i

0

₍₁₎

_{≤ n}

, ompute the a urate FE approximation

w

i

0

(1)N

(x

k

0

(1)

, y)

for

w

i

0

(1)

(x

k

0

(1)

, y)

, element of

M

N

p

=

{w

iN

(x

k

, y), x

k

∈ D, 1 ≤ i ≤ n}

, 2. set

j = 1

,

ξ

1

(y) =

w

iN

(x

k

, y)

kw

iN

(x

k

,

·)k

X

, 3. while

j < N

,

(a) ompute for every

x

k

∈ D

and

1

≤ i ≤ n

the

(n

× p)

RB approximations

w

ij

(x

k

, y)

∈ X

j

= span

{ξ

k

, 1

≤ k ≤ j}

for the

n

ell problems (23), (b) for

k

0

_{(j + 1), i}

0

_{(j + 1)}

=

argmax

1≤k≤p,1≤i≤n

∆

j

(w

ij

(x

k

,

·))

kw

ij

(x

k

,

·)k

X

, ompute the

a - urate FE approximation

w

i

0

(j+1)N

(x

k

0

(j+1)

, y)

for

w

i

0

(j+1)

(x

k

0

(j+1)

, y)

, element of

M

N

p

=

{w

iN

(x

k

, y), x

k

∈ D, 1 ≤ i ≤ n}

, ( ) set

ξ

j+1

(y) =

R

j+1

(y)

kR

j+1

(y)

k

X

where

R

j+1

isthe remainderof the proje tion on the

j

-dimensionalredu ed basis,

R

j+1

(y) = w

i

0

(j+1)

(x

k

0

(j+1)

, y)

−

j

X

k=1

(w

_i

0

(j+1)

(x

k

0

(j+1)

,

·), ξ

k

)

X

ξ

k

(y),

(d) do

j = j + 1

.

3.3 Convergen e of the redu ed-basis method for the ell problem

Theapriori onvergen e ofGalerkinapproximations forsolutionsof ontinuousand oer ive ellipti equations like (21) is lassi al. It usually relies on the following lemma(see e.g. [19℄ for aproof).

Lemma 1 (Céa Lemma) For any

1

≤ i ≤ n

, let

w

i

be the solution of (21) and

w

iN

its approximation in some

N

-dimensional Galerkin proje tion spa e

X

N

⊂ X

. Thenwe have, for any

x

∈ Ω

,

kw

i

(x, y)

− w

iN

(x, y)

k

X

≤

s

γ

_A

−1

α

A

inf

w(y)∈X

N

kw

i

(x, y)

− w(y)k

X

.

To on ludethatRBapproximationslike

w

iN

∈ X

N

apriori onvergeto

w

i

∈ X

when

N

→ ∞

, it is then usual to use Lemma 1 in order to a priori prove the onvergen e oftheapproximation method.

Lemma 2 If there exists a dense separable subspa e

V

of

M

and an appli ation

r

N

:

V → X

N

su h that

lim

N →∞

kv − r

N

(v)

k = 0, ∀v ∈ V ,

(36)

then, by Céa Lemma, for any

x

∈ Ω

and

1

≤ i ≤ n

, RB approximations

w

iN

(x,

·)

onverge to

w

i

(x,

·)

inthe followingsense

lim

N →∞

kw

i

(x, y)

− w

iN

(x, y)

k

X

.

(37)

That is, for all

ǫ > 0

, there exists a positive integer

N (ǫ)

su h that,

∀x ∈ Ω

and

1

_{≤ i ≤ N}

,

kw

i

(x,

·) − w

iN

(x,

·)k

X

≤ ǫ, ∀N ≥ N(ǫ) .

(38)

Let us then naturally hoose

V = M

, and

r

N

as the proje tion operator from

M

to

X

N

for the inner produ t in

X

. Unfortunately, the onvergen e assumed in (36) an only be shown insofaras we have information about

D

, whi h amounts to knowing how the parameter values are sele ted to build

X

N

as

N

in reases. Su h an assumption is unrealisti sin e, to hoose the right parameter values

x

k

for

D

, one shouldalready know

M

or some spe tralrepresentation of it[16℄. So thes ope of Lemma 2 seems strongly limited, as any a prioi analysis of the RB method in general.

As a matter fa t, the RBmethod is a pra ti al method of order redu tion and anonlybea posteriorishownto onvergeusingreliable and omputationally unex-pensiveerror boundsthat an be evaluated alongtheRBapproximations.

Note last that, by denition, the Galerkin proje tion spa e

X

N

is built to on-vergeto the manifold

M

N

=

{w

iN

(x, y), x

∈ Ω}

indu edbytheFEapproximations

w

iN

(x, y)

in thesense that, for some given parameter

x

∈ Ω

and

1

≤ i ≤ n

, there existsfor all

ǫ > 0

a positive integer

N

i

(ǫ, x)

su hthat

kw

i

(x, y)

− w

iN

(x, y)

k

X

≤ ǫ, ∀N ≥ N

i

(ǫ, x) .

So,letusassumethatwehaveanerrorestimatefor

inf

w(y)∈X

n×N

kw

i

(x, y)

− w(y)k

X

that is global in parameter spa e

Ω

like in Lemma 2, in the limit

N

→ ∞

. Even then, on a ount of the pointwise onvergen e of FE approximations in parameter spa e, the RBmethod an only onverge inthefollowing sense

lim

N →∞

N →∞

lim

kw

i

(x, y)

− w

iN

(x, y)

k

X

= 0 ,

(39)

where

w

iN

(x,

·)

impli itly dependson

N

and where thelimitsfor

N

and

N

are not reversible. Yet,iftheerrorestimateisalsoglobalinparameterspa e

Ω

withrespe t to thelimit

N → ∞

,thenthelimit for

N

and

N

beinverted.

3.4 Error estimate for the asymptoti

H

1

homogenized solution

Inthe frame of thetwo-s ale homogenization strategy,the asymptoti

H

1

homoge-nizedapproximation for

u

ǫ

_(x)

inthelimit

ǫ

→ 0

is

u

0

(x) + ǫ u

1



x,

x

ǫ



= u

⋆

(x) + ǫ

X

1≤i≤n

w

i



x,

x

ǫ



∂

i

u

⋆

(x) ,

whi hstrongly onverges to

u

ǫ

_(x)

in

H

1

Γ

D

(Ω)

when

ǫ

→ 0

if

u

⋆

_{∈ W}

2,∞

_(Ω)

. In this approximation, thehomogenized solution

u

⋆

is thesolution to the varia-tionalformulation (40)of thehomogenizedequation (8)

Z

Ω

¯

¯

A

⋆

_∇u

⋆

_{· ∇v =}

Z

Ω

f v +

Z

Γ

N

v,

_{∀v ∈ H}

_Γ

1

_D

(Ω) .

(40)

Butinpra ti e, one an only ompute anRBapproximation

¯

¯

A

⋆

_N

for

¯

¯

A

⋆

,namely withentries

 ¯¯

_A

⋆

N

(x)



i,j

=

Z

Y

¯

¯

A(x, y)dy



i,j

− s

N

ij

(x) ,

whi h should be taken into a ount to estimate the approximation error for the asymptoti

H

1

homogenizedapproximation. Thefollowinglemma(3)willshowhow theaposteriori ontroloftheRBapproximationallowsto ontroltheapproximation error forthe asymptoti

H

1

Letusthendene anapproximationfortheasymptoti

H

1

homogenized approx-imation,

u

⋆

_N

(x) + ǫ

X

1≤i≤n

w

iN



x,

x

ǫ



∂

i

u

⋆

N

(x) ,

where

w

iN

istheRBapproximationfor

w

i

denedinpreviousse tions,and

u

⋆

N

isan

approximationfor

u

⋆

thatis solutionin

W

h

hom

for thedis retevariational problem

Z

Ω

¯

¯

A

⋆

_N

∇u

⋆

N

· ∇v =

Z

Ω

f v +

Z

Γ

N

v,

∀v ∈ W

h

hom

,

(41) with

W

h

hom

⊂ H

1

Γ

D

(Ω)

a dis rete FE Galerkin proje tion spa e asso iated with a meshof size

h

hom

for

Ω

. We have the following result.

Lemma 3 Assume that

Γ

D

is a measurable subset of

∂Ω

with a positive

(n

− 1)−

dimensional measure (when

n > 1

) so that a Poin aré inequality holds for elements of the Sobolev spa e

H

1

Γ

D

(Ω) =

{v ∈ H

1

_{(Ω), v}

_|

Γ

D

= 0

}

.

If the approximations

w

iN

(x,

·)

onverge to

w

i

(x,

·)

for all parametervalues

x

in

Ω

inthe sense

lim

N →∞

1≤i≤n

max



esssup

x∈Ω

kw

i

(x, y)

_{− w}

iN

(x, y)

k

X



= 0 ,

(42)

then the asymptoti

L

2

homogenized approximation

u

⋆

N

onverges to

u

⋆

,and sodoes theapproximation forthe asymptoti

H

1

homogenized approximationof

u

ǫ

. Thatis, we have the two results

lim

N −→∞

ǫ→0

lim

ku

⋆

_(x)

_{− u}

⋆

N

(x)

k

L

2

_(Ω)

= 0

and

lim

N −→∞

ǫ→0

lim

u

⋆

(x)

− u

⋆

N

(x) + ǫ

X

1≤i≤n



w

i



x,

x

ǫ



∂

i

u

⋆

(x)

− w

iN



x,

x

ǫ



∂

i

u

⋆

N

(x)



H

1

_(Ω)

= 0

where the two su essive limits annot be inverted.

Remark 4 As explained in se tion 3.3, the assumption (42) an barely be satised a priori. But in pra ti e, the error bounds derived in the a posteriori analysis of se tion 3.1 allow to he k this assumption. The numeri al results of Se tion 4 even show that the onvergen e withrespe t to

N

in (42) is exponential.