Mathematical Modelling and Numerical Analysis ERROR ESTIMATE FOR A FINITE VOLUME SCHEME IN A GEOMETRICAL MULTI-SCALE DOMAIN

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Mathematical Modelling and Numerical Analysis

ERROR ESTIMATE FOR A FINITE VOLUME

SCHEME IN A GEOMETRICAL MULTI-SCALE

DOMAIN

Marie-Claude Viallon

To cite this version:

multi-s ale domain.

Marie-Claude Viallon

Université de Lyon, UMR CNRS 5208,Université Jean Monnet, Institut CamilleJordan.

Abstra t. We study a nite volume s heme, introdu ed in a previous paper [36℄, to solve an ellipti linear partial dierential equation in a rod stru ture. The rod-stru ture is two-dimensional (2D) and onsists of a entralnode and several outgoingbran hes. The bran hes areassumedtobeone-dimensional(1D).Sothedomainispartially1D,andpartially2D.We all su hastru tureageometri almulti-s aledomain.Weestablishadis retePoin aréinequalityin termsofaspe i

H

1

normdened onthis geometri almulti-s ale1D-2Ddomain,thatisvalid for fun tionsthat satisfy a Diri hlet ondition onthe boundary of the 1D part of the domain andaNeumann onditionontheboundaryofthe2Dpartofthedomain.Wederivean

L

2

error estimate between the solutionof the equation and itsnumeri alnite volumeapproximation.

Résumé. Nous étudions un s héma de type volumes nis, introduit dans un pré édent arti le [36℄, pour résoudre une équation aux dérivées partielles elliptique linéaire dans une stru ture-tube. Lastru ture-tube est bi-dimensionnelle (2D)et onstituée d'un noeud d'où partent plu-sieurs bran hes. Les bran hes sont supposées uni-dimensionnelles (1D). Le domainesur lequel l'équation est posée est ainsi 1D en partie, et 2D en partie. Il sera qualié de géométrique-ment multi-é helle. Nous établissons une inégalité de Poin aré dis rète exprimée en fon tion d'une norme

H

1

spé ique dénie sur e domaine 1D-2D géométriquement multi-é helle, qui est valablepour desfon tionssatisfaisantune onditionauxlimitesde Diri hletsur lafrontière de la partie 1D du domaine et une ondition de Neumann sur la frontière de la partie 2D du domaine. Nousétablissons unemajorationd'erreur en norme

L

2

entre la solutiondel'équation et son approximation par le s héma volumesnis.

Mathemati s Subje t Classi ation: 35J25, 74S10, 65N12, 65N15,65N08

Mots lés : nite volume s heme, ellipti problem, dis retePoin aré inequality, error estimate, multi-s ale domain

1 Introdu tion

This paperis on erned with anite volume s heme for a geometri al multi-s ale domain. We obtain a spe i dis rete Poin aré inequality for that type of stru ture. This inequality is then used to improve from

O(

√

h)

to

O(h)

a rst error estimate obtained in [36℄ for a simple modelproblem.

H

1

and

L

2

normsusing the nitevolumes hemeintrodu edin[36℄, isdis ussed. Theseresults improveapreviousestimatethatisremindedinSubse tion1.5.Subse tion1.6isdevotedtothe review of dierent ways to state interfa e onditions between domains of dierent dimensions, Subse tion1.7tothereviewofdis retePoin aréinequalities,espe iallyforfun tionsthatvanish only on a part of the boundary. In Subse tion 1.8, from a numeri al standpoint, we ompare the dimension redu tion of the domain to the use of non-mat hing grids, taking a row of big ells. Then some remarks on the domain de omposition approa h end Se tion 1. In Se tion2, we present our hybrid s heme to solve (1). We prove that the s heme gives a unique dis rete solution. In Se tion3, following [19℄, we denea

H

1

dis retenorm and weestablish adis rete Poin aré inequality. Last in Se tion 4, we derive the error estimates previously announ ed in Subse tion 1.4.

1.1 The dimensionally-heterogeneous modelling

However, inthis paper, wesolvea modelproblemin asimple 2D rodstru ture. Wedo not onsider arealisti modelsu hasdes ribedabove.Itisarststep.Extensionstomorerealisti problems are possible.

1.2 Des ription of the geometri al multi-s ale 1D-2D domain

Before introdu ing the 1D-2D domain on whi h our model problem is set, let us look at the following example of nite rod stru tures. It onsists of one node and

n

bran hes. This onstru tion isdone in [36℄ and is dis ussed below.

Let

e

j

= [O, O

j

], j = 1, ..., n,

be

n

losed segments in

IR

2

, having a ommon end point denoted by

O

, with length

l

j

= OO

j

, j = 1, ..., n

.

Let

(x, y)

denotethe oordinates inthe anoni albasis of

IR

2

,and

(x

e

j

_{, y}

e

j

₎

denotethelo al oordinates asso iated with the segment

e

j

, j = 1, ..., n

. This lo al system is orthonormal and su h that

x

e

j

is the oordinatein the dire tion

e

j

.

Let

ε > 0.

Let

θ

1

, ..., θ

n

be positive numbers independent of

ε

. Let

B

ε

j

=

{(x, y) | x

e

j

∈ (0, l

j

), y

e

j

∈ (−

εθ

₂

j

,

εθ

₂

j

)

},

and

β

ˆ

ε

j

=

{(x, y) | x

e

j

= l

j

, y

e

j

∈

(

₋

εθ

j

2

,

εθ

j

2

)

}.

Let

ω

0

be a bounded domain in

IR

2

with smooth boundary ontaining

O

(see [36℄). Let

ω

ε

0

=

{(x, y) |

(x,y)−O

ε

∈ ω

0

}

. Weassume that

B

ε

j

\ ω

ε

0

∩ B

i

ε

\ ω

0

ε

=

∅, i 6= j

.The domain

ω

ε

0

(see the dottedlineinFigure1(a))is addedinordertosmooth theboundaryof thenal stru ture by removing the orners.

Let

Ω

ε

=

∪

n

j=1

B

j

ε

∪ ω

ε

0

.

The domain

Ω

ε

is thus the

1/ε

−

homotheti ontra tion of a xed domain

Ω

, as depi ted inFigure 1(a) with

n = 5

. The thi kness of the bran hes is the ratio of the diameter tothe height,and is proportionalto

ε

.

x

y

A

x

x

x

x

y

l

O

Ω

ε

ω

0

δ

δ

δ

δ

δ

5

O

l

l

₃

l

₂

4

l

β

1

ε

θ

ε

1

1

e

e

e

e

x

e

1

2

3

4

5

e1

1

ε

x

y

A

x

x

x

x

y

O

δ

4

l

e

e

e

e

x

e

2

3

4

5

e1

1

γ

γ

γ

γ

1

’

2

γ

₃

4

5

’

’

’

’

δ

δ

δ

δ

D

_ε

O

₁

l

1

l

₂

3

l

5

l

S

S

S

S

S

1

2

3

4

5

Figure 1 (a)The initialdomain

Ω

ε

and (b) The geometri almulti-s aledomain

D

ε

.

Now, let us des ribe the 1D-2D domain under onsideration. Let

δ > 0

, su h that

δ <

min

{l

j

, j = 1, ..., n

}

and su h that

ω

ε

Let

S

j

=

{(x, y) | y

e

j

_{= 0, x}

e

j

∈ (δ, l

j

)

}, j = 1, ..., n,

be segments su hthat

S

j

⊂ e

j

. Wedenote

γ

′

j

=

{(x, y) | x

e

j

= δ, y

e

j

∈ (−

εθ

j

2

,

εθ

j

2

)

}, j = 1, ..., n,

the interfa es between

Ω

′

ε

and

Ω

ε

\ Ω

′

ε

. (Forthe sakeof simpli ity,we donot make the dependen e on

ε

of

γ

′

j

.) Let us dene

D

ε

= Ω

′

ε

∪ ∪

n

j=1

S

j

. The set

D

ε

is what we all a geometri al multi-s ale domain. We assume that

ω

ε

0

\ ∪

n

j=1

B

j

ε

is not too large. More pre isely, we assume that

m(Ω

′

ε

)

is of the same magnitude as

m(

∪

n

j=1

B

′ε

j

)

, so as to have

m(Ω

′

ε

) = O(εδ)

, where

m

is the 2D Lebesgue measure.

Inthis paper, we onsider both the ase of a geometri almulti-s aledomainwhere

ε

and

δ

are xed, and the ase where

ε

tends tozero and

δ

depends on

ε

. The twostudies are madeat the same time,and Theorem 8and Theorem 11 are stated inSe tion 4related toea h ase.

1.3 The model problem

The boundary value problem in the domain

D

ε

, that we onsider in this paper, is the following:















































v

′′

j

(x

e

j

) = f

j

(x

e

j

), x

e

j

∈ (δ, l

j

), j = 1, ..., n (a)

v

j

(l

j

) = 0, j = 1, ..., n

△u(x, y) = 0, (x, y) ∈ Ω

′

ε

(b)

∂u

∂n

(x, y) = 0, (x, y)

∈ ∂Ω

′

ε

\(∪

n

j=1

γ

j

′

)

u(x, y) = v

j

(δ), (x, y)

∈ γ

j

′

, j = 1, ..., n

v

′

j

(δ) =

1

θ

j

ε

Z

γ

′

j

∂u

∂n

dγ, j = 1, ..., n

(c)

(1)

We assume that the fun tions

f

j

are independent of

ε

and vanish in some neighborhood of

O

j

, j = 1, ..., n

. For the sake of simpli ity,as in [36℄, the right-hand side is taken equalto zero in

Ω

′

ε

, but this ondition ould be relaxed. However, it is well known that the error estimates forthe onvergen erateof thenumeri almethodsrequiresomeregularityoftheexa t solution. Soweassume that the right-handside is su h that

u

∈ C

2

_(Ω

′

ε

)

and

v

j

∈ C

2

_{([δ, l}

j

]), j = 1, ..., n

.

More pre isely, we dene aglobal solution

u

d

of (1)by letting

u

d

(x, y) =



u(x, y)

if

(x, y)

∈ Ω

′

ε

v

j

(x

e

j

)

if

(x, y)

∈ B

ε

j

, x

e

j

∈ (δ, l

j

), j = 1, ..., n

(2) The solution

u

d

is dened in

Ω

ε

but

u

d

_{(x, y)}

does not depend on

y

e

j

when

(x, y)

∈ B

ε

j

\ B

′ε

j

. Dening the solution on

Ω

ε

will allowus to use a standard

L

2

norm in a 2D domain to write the error estimate of Theorem 8.

Problem (1) has been introdu ed in [36℄ in the framework of the method of asymptoti partial domain de omposition (MAPDD) (see [35℄). The following lemma has been proved in [37℄ (see estimate(6)) and [36℄

Lemma 1 For any

J > 0

, there is

M

, independent of

ε

, su h that if

δ = Mε

|

ln

ε

|,

then

ku

ε

− u

d

k

H

1

_(Ω

ε

)

= O(ε

where

u

ε

isthe solution of the following ellipti linear model equation











△u

ε

= f,

in

Ω

ε

u

ε

= 0,

on

β

ˆ

ε

j

, j = 1, ..., n

∂u

ε

∂n

= 0,

on

∂Ω

ε

\(∪

n

j=1

β

ˆ

j

ε

)

(3)

where

f

is a smoothfun tion dened in

Ω

ε

su hthat

f (x, y) = f

j

(x

e

j

₎

, if

(x, y)

∈ B

ε

j

\ B

′ε

j

, j =

1, ..., n

, and

f (x, y) = 0

if

(x, y)

∈ Ω

′

ε

. There exists afun tion

u

ε

∈ C

2

_(Ω

ε

)

solutionof (3), if

f

issu iently smooth [27℄. Itis proved in [36℄ that the followingestimates hold

Lemma 2 If

δ

is of order

ε

ln

ε

then

kv

′

j

k

∞

= O(1)

and

kv

′′

j

k

∞

= O(1), j = 1, ..., n,

k∇uk

∞

= O(1),

k∇

2

u

k

∞

= O



1

ε



These bounds willbeuseful to provethe error estimate of Theorem 11.

1.4 Comments on the numeri al approximation and the error esti-mate

Anhybrid(inthe sense that itsolves aproblemina geometri almulti-s aledomain)nite volumes hemeisproposedin[36℄tosolve(1).To onstru tthes heme,themethodologywhi h was proposed in [45℄ is rst explained. In [45℄, the authors give a numeri al methodology to address the solution of the 3D Navier-Stokesequations and its ouplingwith some 1D models (see[6℄,[7℄,[33℄,[32℄also).Tofollowthispathtosolve(1),letusremarkthat(1) anberewritten















v

′′

j

(x

e

j

) = f

j

(x

e

j

), x

e

j

∈ (δ, l

j

), j = 1, ..., n

v

j

(l

j

) = 0, j = 1, ..., n

v

j

(δ) = α

j

, (x, y)

∈ γ

j

′

, j = 1, ..., n

v

′

j

(δ) = β

j

(4)































△u(x, y) = 0, (x, y) ∈ Ω

′

ε

∂u

∂n

(x, y) = 0, (x, y)

∈ ∂Ω

′

ε

\(∪

n

j=1

γ

j

′

)

u(x, y) = α

j

, (x, y)

∈ γ

j

′

, j = 1, ..., n

1

θ

j

ε

Z

γ

′

j

∂u

∂n

dγ = β

j

(5)

The basi idea in [45℄ is to onsider the numeri al resolution of the 2D problem (5) on one hand, and of the 1D problems (4) on the other hand, as bla k-boxes whi h re eive the input data (

α

j

, j = 1, ..., n

) and give ba k

(β

j

, j = 1, ..., n)

as outputdata. A system inthe interfa e unknowns (

α

j

, β

j

, j = 1, ..., n

) is obtained, whi h is solved by an iterative method. This te h-nique, whi h is a domainde omposition approa h, willnot be dealt with here. Instead, in the present paper, a dire tmethodisused, and (4)and (5)are not understoodas bla k-boxes but related by (1- ) (reminded below foreasy referen e and guidan e) :

Here,

β

j

, j = 1, ..., n

, are nolonger unknowns and onlythe interfa e unknowns

α

j

, j = 1, ..., n

, are kept. We use nite volume s hemes toapproa h (4'), (5'), and (6), where (4') (resp. (5')) is the system(4)(resp. (5)) withitslastequation removed. The unknowns orrespondingwith

α

j

, j = 1, ..., n

,are

v

j,0

, j = 1, ..., n

,intheresultings hemethatisre alledin(11) inSubse tion 2.2.

The aimof the present paper isto re onsider this s heme to solve (1), and in parti ularto improve the order of onvergen e obtained in [36℄. In [36℄, we get an error estimate of order

√

h

, where

h

is the size of the mesh. In Theorem 8 below we get abetter estimate

O(h)

. This is one of the main results of the paper. Here

ε

and

δ

are xed given parameters and we don't have toexpress the bound with respe t tothese parameters.

However, inaddition,(4')-(5')-(6)may alsobeused tosolve (3).In viewofLemma1,(1)is areasonable approximationfor(3) if

ε

issmalland

δ

of order

ε

ln

ε

.So, anumeri al approxima-tion of the solution of (1) is also a numeri al approximation of the solution of (3). In Se tion 4,anerror estimatebetween thesolutionof (3)anditsnumeri alapproximation isobtainedin Theorem 11in onjun tion with Theorem 8. Sin e both

h

and

ε

tend to zero in this ase, the error estimateis alsoexpressed interms of

ε

. Notethat anite element implementationof (3) is studiedin [21℄ with

n = 1

, and anerror estimateis obtained.

We obtain a better error estimate than in [36℄ be ause (1) is really onsidered as a geo-metri al multi-s ale problem.We dene dis rete

L

2

and

H

1

norms for fun tions on

D

ε

. A

H

1

dis retenormhas been introdu edin[42℄,inthe ase ofastru ture withasinglebran h.Here, we propose a generalizationto stru tures with

n

bran hes. It involves the onvex ombination of the values of the fun tions on both sides of ea h interfa e

γ

′

j

, j = 1, ..., n

. To the best of our knowledge,thereisnoerrorestimateintheliterature whenusing ageometri almulti-s ale nitevolumes heme.Moreover,theproblem(1)issu hthatNeumannboundary onditionsare imposed onthe 2D part of the domain,and Diri hletboundary onditions are imposedon the boundaryofthe1Dpartofthedomain.Asno lassi alPoin aréinequalityisdire tlyappli able onanissueofthisnature,ithasbeenne essarytoestablishadis retePoin aréinequalityin

D

ε

.

1.5 About the estimate in [36℄ We re all here how the estimate

O(

√

h)

is obtained in [36℄. Let

v

j

, ev

j

, e

u

j

, j=1,...,n, be the solutions of the following independent sub-problems, some of them being 1D, and the other being 2D



v

′′

j

= f

j

,

on

S

j

v

j

(δ) = 0



e

v

′′

j

= 0,

on

S

j

e

v

j

(δ) = 1







△eu

j

= 0,

on

Ω

′

ε

e

u

j

|

γ

′

j

= 1,

e

u

j

|

γ

′

k

= 0,

if

k

6= j, k = 1, ..., n

(7)

The solution of (1) an then be written



v

j

= v

j

+ α

j

ev

j

, j = 1, ..., n

u =

P

n

_j=1

α

j

e

u

j

The auxiliaryvariables

α

j

, j = 1, ..., n

, are then dened by

1

θ

j

ε

n

X

k=1

α

k

Z

γ

′

j

∂e

u

k

∂n

dγ

− α

j

ev

′

j

(δ) = v

′

j

(δ), j = 1, ..., n

(9)

so that the interfa e onditions (1- ) are satised.

Weremark that

α

j

= u

j

|

γ

′

j

= v

j

(δ)

,j=1,...,n,are the valuesof the solutiononthe interfa es

γ

′

j

. Thanks to the linearity of (1), the problem has been ompletely split in [36℄. The authors rst derived the errorsforea hlinearsub-problems(7)separatelyby using lassi alte hniques for nite volume s hemes, on one hand on the domains

S

j

, j = 1, ..., n,

and on the other hand on the domain

Ω

′

ε

. They then dedu ed the error on the re onstru ted solution(8). This is not optimalbe ausetheapproximationof

α

j

, j = 1, ..., n,

isnot.Ultimately,undertheassumptions ofLemma1and Lemma2,theyget anerror estimate

O

r

hδ

ε

!

+ O(ε

J

)

between the solution

of(3)anditsapproximation.To ontroltheerrorsontheinterfa es,theauthorsneedtoassume that

h

|

ln

ε

|

ε

tendsto zero when

h

and

ε

tend tozero, and some regularityfor the mesh. Theerrorestimatefor(1)isnot learlygivenin[36℄. However,the approximationof(1)isa ne essary steptogetthe oneof(3),thenitiseasytodedu e from[36℄ anerrorestimate

O(

√

h)

between the solution of (1) and its approximation (in this ase

ε

and

δ

are xed onstants). The estimate isobtained under some regularity for the mesh.

So the present ase is quite dierent sin e we do not need to estimate the error between

α

j

, j = 1, ..., n,

and their approximations.

1.6 Some remarks about the interfa e onditions In the present work, the interfa e onditions on

γ

′

1.7 About Poin aré inequalities

Error estimates for numeri al methods are obtained thanks to fun tional analysis tools, su h as dis rete Sobolev inequalities. Con erning the nite volume framework, and the two-dimensional ase, arst dis rete Poin aré inequality for pie ewise onstant fun tionshas been a hieved for Diri hlet boundary onditions in [14℄, following [29℄, in a polygonal onvex do-main. In [19℄, the authors generalize this inequality in a polygonal domain. Dis rete Sobolev inequalities (estimating the

L

p

norm) are presented in [19, 13, 17, 18℄. In [19℄ and [25℄, the authorsestablisha"meanPoin aré"(Poin aré-Wirtinger)inequality(estimatingthe

L

2

norm) for Neumannboundary onditions inapolygonal domain.Adis rete "meanPoin aré" inequa-lity (estimating the

L

p

norm) is obtained in [26℄ and [12℄ on Voronoi nite volume meshes. A Sobolev-Poin aréinequality(embeddingof

W

1,q

into

L

p

)wasstatedusingaproofbasedonthe spa e offun tions ofbounded variationin[20℄ and [5℄(also in[18℄ forthe zero boundary value ase).The previousresultsweremostly presented inthe frameworkofadmissiblemesheswhi h satisfy the following orthogonality property : thereexists apoint asso iatedwith ea helement ofthemeshsu hthatthestraightline onne tingthesepointsfortwoneighboring ellsis ortho-gonaltothe ommonsideofthesetwo ells(seethedenitionin[19℄and(10)below),butmore general meshes are possible (see [19℄). In [43℄ the author presents both dis rete Poin aré and "meanPoin aré"inequalitiesforfun tionsdened onameshwherethe orthogonalityproperty is not ne essarilysatised (otherreferen es inthe nite element framework are given therein), as well as in [4℄ and [31℄ in the dis rete duality nite volume ontext. Previously a dis rete Poin aréinequality onnon-mat hing gridshas been establishedin[11℄. In allthe papers listed above dealing with non onforming meshes, itis ne essary to dene a spe i

H

1

norm that is appropriate for the mesh.

In the present work, we use an admissible mesh in

Ω

′

ε

, but the global mesh of

D

ε

is in some ways "non onforming". We a tually dene a spe i

H

1

norm for fun tions dened on

D

ε

(see Subse tion 1.4). In (1), we impose zero boundary value on the 1D part of the domain and thereisaNeumannboundary onditiononthe 2D partof thedomain,sowe needtostate rst a dis rete "mean boundary Poin aré" inequality (inequality that involves a mean value on a part of the boundary), and then to dedu e a Poin aré inequality for fun tions with zero value on a part of the boundary. Su h an inequality is obtained in [19, 43℄, and in [5℄ for a onvexdomain.Adis reteSobolev-Poin aréinequality (estimatingthe

L

p

norm)isestablished for fun tions with nonzero boundary values in [3℄. But, these results annot be applied to a dimensionally-heterogeneousdomain. In Se tion 3, we follow the proof in [19℄, evaluating pre- isely the onstant bounds as in [43℄, to get the suitable dis rete Poin aré inequality that is used in Se tion4 todedu e the

L

2

error estimates.

1.8 To hoose a oarse grid instead of redu ing the dimension?

de omposition : what are the interfa e onditions on the non-mat hing grids? There isa wide literatureonthistopi ,andeven,morespe i ally,usingnitevolumes hemes(seeforinstan e [1, 40, 11℄). A omparison between an hybrid s heme used on a dimensionally-heterogeneous (1D-2D) domain and the so- alled TPFA s heme (dened in[19℄) used on a full 2D non mat- hing nite volume mesh, solving the Poisson equation in a rod stru ture with a single node and a single bran h, an be found in [42℄. The bran h is of thi kness

ε

, and meshed with a row of re tangular ells

ε

high by

h

wide, where

h

is the size of the mesh of the remaining part of the domain ( orresponding to the node). The a priori estimate on the error whi h is a hieved in[42℄forthe TPFAs heme,following[11℄,dependson

ε

for several reasons:the size of the globalmesh depends on the size of the re tangles, the sum of the lengthof the atypi al edges is equal to

ε

, and the se ond derivative of the solution is of the order

1/ε

(see Lemma 2). Under the assumption that

h < ε

, the most signi ant term is

O(

√

ε)

, and it isimpossible to get a bound with respe t to

h

. Quite the ontrary, the error estimate obtained in [42℄ for the hybrids heme an beexpressed asa fun tion of

h

(this result isgeneralized in this paper, see Theorem 8 and Theorem 11), as well as a fun tion of

ε

. This is a main advantage of the geometri al multi-s ale domain.Though, the numeri alexperiments in[42℄ show that the two s hemes provide similar performan es. On the other hand, a dis rete Poin aré inequality for non-mat hinggridsisobtained, forinstan ein[11℄,underthe assumptionofquasi-uniformness ofthe mesh.Adis retePoin aréinequalityisused in[42℄whi hdoesnot requireanyrestri tive assumption onthe mesh.The proofof thisinequalityis not given in[42℄, itisaparti ular ase of the one that isprovided inthe present paper (see Lemma7).

1.9 The domain de omposition approa h

In [6℄, the authors introdu e a spe ialized vo abulary to name the s heme that dis retizes (4')-(5')-(6):themonolithi s heme.Alternately,ade ouplednumeri als hememaybedevised in ase of workingwith stand-alone1D and2D (or3D) odes, su hasbla k boxes.In this ase we an split the omputations by performing iterations between the 1D and 2D (or 3D) sub-problems.In[6℄,theauthors alledthesess hemes:thesegregated ouplings hemes.Duetothe heterogeneous feature of the geometri al multi-s ale problems, the monolithi s heme gives a linearsystem thatisill onditioned.Forthisreason,manyauthorsadoptaniterativeapproa h by solving separatelythe sub-problems. For instan e, the te hnique presented in [32℄ and [33℄ an be understood asa domain de omposition approa hwhere the partitioningtakespla e at the oupling interfa es among models of dierent dimensions. This allows to parallelize the omputations into the sub-domains. However, this splitting strategy, in whi h the sub-models are solved separatelyanditeratively,willnotbe overed here.The monolithi s hemeishereby explored.

2 Numeri al s heme

2.1 The mesh

Letus dene a mesh of the intervals

(δ, l

j

)

onthe axis

Ox

e

j

_{, j = 1, ..., n.}

Forea h value of

j

, we hoose

N

j

∈ IN

∗

_,

and

N

j

+ 1

distin tand in reasing values

x

e

j

x

e

j

1/2

= δ, x

e

j

N

j

+1/2

= l

j

. Denote

I

e

j

i

= (x

e

j

i−1/2

, x

e

j

i+1/2

)

,and

h

e

j

i

= x

e

j

i+1/2

− x

e

j

i−1/2

, i = 1, . . . , N

j

. Set

h

e

j

₌

max

{h

e

j

i

, i = 1, ..., N

j

}

the size of the mesh of the interval

(δ, l

j

)

. Then we hoose

N

j

points

x

e

j

i

, i = 1, ..., N

j

,

su h that

x

e

j

i

∈ I

e

j

i

. Set

x

e

j

0

= δ, x

e

j

N

j

+1

= l

j

, and

h

e

j

i+1/2

= x

e

j

i+1

− x

e

j

i

, i = 0, ..., N

j

.

Letus onstru tanadmissiblemeshover

Ω

′

ε

denoted by

T

.Weassumeinthe followingthat

Ω

′

ε

is polygonal. We remind (see the denition in [19℄) that su h a mesh onsists in a family of open polygonal onvex subsets

K

of

Ω

′

ε

(with positive measures) alled ontrol volumes, a family of edges

σ

(with stri tly positivemeasures) of the ontrolvolumes denoted by

E

,and a family of points

x

K

hosen in ea h ontrolvolume

K

denoted by

P

. The mesh

T

satises the followingproperties

1)The losure of the union of allthe ontrolvolumes is

Ω

′

ε

.

2)

For any

K

∈ T ,

there isa subset

E

K

of

E

su h that

∂K =

[

σ∈E

K

σ,

and

[

K∈T

E

K

=

E.

3)

For any

(K, L)

∈ T

2

_{, K}

_{6= L,}

one of three followingassertions holds: either

K

∩ L = ∅,

or

K

∩ L

is a ommon vertex of Kand L,

or

K

∩ L = σ, σ

being a ommon edge of Kand L denoted by

σ

K/L

.

4)The family

P = (x

K

)

K∈T

is su h that for any

K

∈ T , x

K

∈ K.

For any

(K, L)

∈ T

2

_{, K}

_{6= L,}

it is assumed that

x

K

6= x

L

and thatthe straight linegoing through

x

K

and

x

L

is orthogonal to

σ

K/L

.

5)

For any

σ

∈ E,

if

σ

⊂ ∂Ω

′

ε

, σ

∈ E

K

and

x

K

∈ σ,

/

the orthogonal

proje tion of

x

K

onthe straightline ontainingthe edge

σ,

belongs to

σ.

(10) Let

E

int

=

{σ ∈ E, σ 6⊂ ∂Ω

′

ε

}

. For any

(K, L)

∈ T

2

_{, K}

_{6= L}

, if

σ = σ

K/L

, let

d

σ

be the distan e between

x

K

and

x

L

. For any

K

∈ T

,if

σ

∈ E

K

and if

σ

⊂ ∂Ω

′

ε

,

let

d

σ

bethe distan e between

x

K

and

σ

. Weassume that for any

σ

∈ E, d

σ

6= 0.

For any

K

∈ T ,

let

m(K)

be the area of

K

. For any

σ

∈ E,

let

m(σ)

be the length of

σ

. Let

h

0

be the size ofthe mesh

T

,

h

0

=

max

{

diam

(K), K

∈ T }

,wherediam isthe abbreviation for diameter.

We denote by

T S

the global 1D-2D mesh of

D

ε

. Let

h

be the size of the 1D-2D mesh of

D

ε

:

h =

max

{h

0

, h

e

j

_{, j = 1, ..., n}

_}

.

2.2 The hybrid s heme

The s heme is obtained by integrating

v

′′

j

= f

j

on ea h ell

I

e

j

i

, i = 1, ..., N

j

, and

△u = 0

over ea h ontrolvolume

K

∈ T

. The numeri al ux

F

j,i+1/2

is anapproximationof

v

′

j

(x

e

j

i+1/2

)

of nite dieren e type;

v

j,i

is an approximation of

v

j

(x

e

j



































































































F

j,i+1/2

− F

j,i−1/2

= h

e

i

j

f

e

j

i

, i = 1, ..., N

j

, j = 1, ..., n

(a)

F

j,i+1/2

=

v

j,i+1

− v

j,i

h

e

j

i+1/2

, i = 0, . . . , N

j

, j = 1, ..., n

f

e

j

i

=

1

h

e

j

i

Z

x

ej

_i+1/2

x

ej

_i−1/2

f

j

(x)dx, i = 1, . . . , N

j

, j = 1, ..., n

v

X

j,N

j

+1

= 0, j = 1, ..., n

σ∈E

K

F

K,σ

= 0,

∀K ∈ T

(b)

F

K,σ

=











m(σ)

d

σ

(u

L

− u

K

)

,

∀σ ∈ E

int

,

if

σ = σ

K/L

m(σ)

d

σ

(v

j,0

− u

K

) ,

∀σ ⊂ γ

′

j

, σ

∈ E

K

, j = 1, ..., n

0

,

∀σ ⊂ ∂Ω

′

ε

\(∪

n

j=1

γ

′

j

)

v

j,1

− v

j,0

h

e

j

1/2

=

1

θ

j

ε

X

σ∈E

K

,σ⊂γ

j

′

m(σ)

d

σ

(v

j,0

− u

K

), j = 1, ..., n

(c)

(11)

Let us noti e that

v

j,0

is a onvex ombination of the approximated values of the solution on ea h side of

γ

′

j

, j = 1, ..., n,

sin e

v

j,0

=





v

j,1

h

e

j

1/2

+

1

θ

j

ε

X

σ∈E

K

,σ⊂γ

j

′

m(σ)

d

σ

u

K









1

h

e

j

1/2

+

1

θ

j

ε

X

σ⊂γ

′

j

m(σ)

d

σ





−1

(12)

Forthe sakeof simpli ity,in(11 ) and(12),the summationisdonefor

σ

⊂ γ

′

j

,andforea h of them,

K

is the ontrolvolumesu h that

σ

∈ E

K

.

The approximate solution of (1)isdened by

u

d

T

(x, y) =



u

T

(x, y) , (x, y)

∈ Ω

′

ε

v

jT

(x

e

j

) , (x, y)

∈ B

j

ε

, x

e

j

∈ (δ, l

j

), j = 1, ..., n

with



u

T

(x, y) = u

K

, (x, y)

∈ K, K ∈ T

v

jT

(x

e

j

) = v

ji

, x

e

j

∈ (x

e

_i−1/2

j

, x

e

_i+1/2

j

), i = 1, ..., N

j

, j = 1, ..., n.

(13)

2.3 Existen e and uniqueness of the nite volume approximation The s heme (11) leads to alinear system of the form

AU = B

in whi h

U

is the unknown, where

U

T

_{= (}

_{{v

ji

, i = 1, ..., N

j

}, j = 1, ..., n}, {u

K

, K

∈ T })

.

Lemma 3 Thereisaunique solution

(

{{v

ji

, i = 1, ..., N

j

}, j = 1, ..., n}, {u

K

, K

∈ T })

to equa-tions (11).

Proof. We assume that

B = 0

. Let us prove that

U = 0

. We multiply (11a) by

v

j,i

and sum over

i

,thenmultiplyby

θ

j

ε

andsumover

j

.Wemultiply(11b)by

u

K

andsumoverK.Weobtain

n

X

j=1

θ

j

ε

N

j

X

i=1

(F

_i+1/2

j

− F

i−1/2

j

)v

j,i

+

X

K∈T

X

σ∈E

K

n

X

j=1

θ

j

ε





N

j

X

i=1

F

_i+1/2

j

v

j,i

−

N

_X

j

−1

i=0

F

_i+1/2

j

v

j,i+1



+

X

σ∈E

int

σ=σ

K|L

F

K,σ

(u

K

−u

L

)+

n

X

j=1

X

σ∈E

K

σ⊂γ

′

j

m(σ)

d

σ

(v

j,0

−u

K

)u

K

= 0

On the other hand,the denition of the numeri aluxes leads to

n

X

j=1

θ

j

ε





N

j

X

i=1

−

(v

j,i+1

− v

j,i

)

2

h

e

j

i+1/2

−

v

j,1

− v

j,0

h

e

j

1/2

v

j,1



−

X

σ∈E

int

σ=σ

K|L

m(σ)

d

σ

(u

K

−u

L

)

2

+

n

X

j=1

X

σ∈E

K

σ⊂γ

′

j

m(σ)

d

σ

(v

j,0

−u

K

)u

K

= 0

Multiplying (11 ) by

θ

j

εv

j,0

, summingover

j

, and addingto the above equality,we get

n

X

j=1

θ

j

ε





N

j

X

i=1

−

(v

j,i+1

− v

j,i

)

2

h

e

j

i+1/2

−

(v

j,1

− v

j,0

)

2

h

e

j

1/2



−

X

σ∈E

int

σ=σ

K|L

m(σ)

d

σ

(u

K

−u

L

)

2

−

n

X

j=1

X

σ∈E

K

σ⊂γ

′

j

m(σ)

d

σ

(v

j,0

−u

K

)

2

= 0

Hen e, allthe omponents of

U

are equal, and sin e

v

j,N

j

+1

= 0, j = 1, ..., n

,we have

U = 0

. Remark 4 Theprevious linereads

−ks

d

T

k

2

1,T

= 0

where

k.k

1,T

is dened below (see Denition 5), and

s

d

T

is a fun tion onstant over ea h ontrol volume of the mesh

T S

whi h oin ides with

u

d

T

. The proof of the existen e and uniqueness of the solution of (11) is also done in [36℄ using another method.

3 The dis rete Poin aré inequality The proof of an

L

2

error estimate requires a dis rete Poin aré inequality. We remind that

D

ε

= Ω

′

ε

∪ ∪

n

j=1

S

j

. We introdu e the spa e of pie ewise onstant fun tions asso iated with the 1D-2Dmeshof

D

ε

,and adis rete

H

1

normfor thisspa e.Thedis retePoin aréinequality, that is established inLemma 7, isexpressed interms of this dis rete

H

1

norm. Denition 5 a) We dene

X(

T )

the set of fun tions from

Ω

′

ε

to

R

whi h are onstant over ea h ontrol volume of

T

.

b) We dene

X(

T S)

the set of fun tions from

D

ε

to

R

whi h are onstant over ea h ontrol volume of

T S

.

) Let

w

∈ X(T S)

, su h that

w(x, y) =



w

K

, (x, y)

∈ K, K ∈ T

w

j,i

, (x, y)

∈ S

j

, x

e

j

∈ (x

_i−1/2

e

j

, x

e

_i+1/2

j

), i = 1, ..., N

j

, j = 1, ..., n.

We dene and we denote

(i)

kwk

2,T

=





X

K∈T

m(K)w

_K

2

+

n

X

j=1

θ

j

ε

N

j

X

i=1

h

e

j

i

w

j,i

2





1/2

(ii)

kwk

1,T ,∗

=

P

σ∈E

int

m(σ)d

σ



D

σ

w

d

σ



2

!

1/2

(iii)

kwk

1,T

=

P

σ∈E

int

,σ⊂(∪

n

j=1

γ

j

′

)

m(σ)d

σ



D

σ

w

d

σ



2

+

P

n

_j=1

θ

j

ε

P

N

j

i=0

(w

j,i+1

− w

j,i

)

2

h

e

j

i+1/2

!

1/2

where

D

σ

w =



| w

K

− w

L

|, σ ∈ E

int

, σ = σ

K|L

| w

K

− w

j,0

|, σ ⊂ γ

j

′

, σ

∈ E

K

, j = 1, ..., n

w

j,N

j

+1

= 0, j = 1, ..., n,

and

w

j,0

=





w

j,1

h

e

j

1/2

+

1

θ

j

ε

X

σ∈E

K

,σ⊂γ

′

j

m(σ)

d

σ

w

K









1

h

e

j

1/2

+

1

θ

j

ε

X

σ⊂γ

′

j

m(σ)

d

σ





−1

.

Remark 6 Thefun tions

k.k

2,T

and

k.k

1,T

are norms, and

k.k

1,T ,∗

is semi-norm, on

X(

T S)

. On theother hand, we an explain

kwk

2,T

and

kwk

1,T

as lassi al dis retenorms of a fun tion

˜

w

dened a.e. on

Ω

ε

and su h that

w

˜

|

D

ε

= w

. Let us dene

w

˜

by

˜

w(x, y) =



w

K

, (x, y)

∈ K, K ∈ T

w

j,i

, (x, y)

∈ B

j

ε

, x

e

j

∈ (x

e

j

i−1/2

, x

e

j

i+1/2

), i = 1, ..., N

j

, j = 1, ..., n.

then we have

kwk

2,T

=

k ˜

w

k

L

2

(Ω

ε

)

. We an onsider a mesh of

Ω

ε

in luding

T

and a row of re tangular ells

ε

highby

h

wide on

B

ε

j

\ B

′ε

j

, j = 1, ..., n

. Thefun tion

w

˜

is pie ewise onstant on this mesh, and

kwk

1,T

is equal to a 2D lassi aldis rete

H

1

norm of

w

˜

on this mesh.

Lemma 7 Let

w

∈ X(T S)

, there is a onstant

c

independentof

h

su h that

kwk

2

2,T

≤ ckwk

2

1,T

Proof. Let

w

∈ X(T S)

su h that

where

l

max

=

max

{l

j

, j = 1, ..., n

}.

Though,provingLemma7amountstoprovingthe existen eof a onstant

c

independentof

h

su h that

kwk

2

L

2

_(Ω

′

ε

)

≤ ckwk

2

1,T

.

Now, wefollowthe path of Lemma 10.2in [19℄ toprove a"dis rete mean Poin aré inequa-lity".Theauthorsassumethatthedomain,inwhi htheproblemisset,isanopenbounded poly-gonal onne tedsubsetof

R

2

:

Ω

′

ε

satisesthisrequirementallowingtheresultstobeused.Then, following the proof in [19℄, there is a nite number of disjoint onvex polygonal sets, denoted by

{Ω

1

, ..., Ω

p

}

,su h that

Ω

′

ε

=

∪

p

i=1

Ω

i

.Here, itmakessense to assume that

Ω

1

= B

′ε

1

be ause

B

′ε

1

is onvex,and

γ

′

1

⊂ ∂Ω

1

is lo ated onthe interfa e. Let

I

ij

= Ω

i

∩ Ω

j

, i

6= j, i, j ∈ {1, ..., p}

as in[19℄.Let usrememberthat only the set of index su hthat

m(I

ij

) > 0

is onsidered.

Now, let usdene the stri tly positivesquantities

µ

and

λ

:

min



m(I

ij

)

ε

, i, j

∈ {1, ..., p}



= µ

min



m(Ω

i

)

m(Ω

′

ε

)

, i

_{∈ {1, ..., p}}



= λ

(15)

Why tointrodu e

ε

above todene

µ

?The domain

Ω

ε

hasbeen onstru ted sothatthe width of ea hbran his the imageof agiven segment obtained by a

1/ε

−

homotheti ontra tion. In-deed, the thi kness of

Ω

1

isequalto

θ

1

ε

.That isthe reason why wedonot assume that

m(I

ij

)

isgreaterthanastri tly positive onstant(asin[19℄),butratherthat theratio

m(I

ij

)ε

−1

is so.

Now, we ontinue as in[19℄, dening

m

1

(w)

the meanvalue of

w

over

Ω

1

,and

m

Ω

′

ε

(w)

the mean value of

w

over

Ω

′

ε

, that is

m

1

(w) =

1

m(Ω

1

)

Z

Ω

1

w(x, y)dxdy,

m

Ω

′

ε

(w) =

1

m(Ω

′

ε

)

Z

Ω

′

ε

w(x, y)dxdy.

Sin e

kwk

2

L

2

_(Ω

′

ε

)

≤ 3kw − m

Ω

′

ε

(w)

k

2

L

2

_(Ω

′

ε

)

+ 3m(Ω

′

ε

)

|m

Ω

′

ε

(w)

− m

1

(w)

|

2

_{+ 3m(Ω}

′

ε

)m

1

(w)

2

(16)

proving Lemma 7 amounts a tually to provingthe existen e of three onstants

c

1

, c

2

, c

3

,

inde-pendentof

h

su h that

a)

kw−m

Ω

′

ε

(w)

k

2

L

2

_(Ω

′

ε

)

≤ c

1

kwk

2

1,T

b)

|m

Ω

′

ε

(w)

−m

1

(w)

|

2

≤ c

2

kwk

2

1,T

)

m

1

(w)

2

≤ c

3

kwk

2

1,T

(17) The proof of Lemma 10.2 in[19℄ givesthe existen eof

c

1

, c

2

,only dependingon

Ω

′

ε

, su h that

kw − m

Ω

′

ε

(w)

k

2

L

2

_(Ω

′

ε

)

≤ c

1

kwk

2

1,T ,∗

|m

Ω

′

ε

(w)

− m

1

(w)

|

2

≤ c

2

kwk

2

1,T ,∗

The proof of (17a) and (17b) follows sin e

kwk

2

1,T ,∗

≤ kwk

2

1,T

.

easilyappliedinthe urrent ontext. Thatiswhy wefollownowthe proofin[44℄, taking

Ω

1

for the onvex domainand

γ

′

1

⊂ ∂Ω

1

for the part of the boundarywith a null Diri hlet ondition. Of ourse, the fun tion

w

is not null on

γ

′

1

. It is the dieren e between the result obtained in [19℄ or [44℄, and Lemma 7. Introdu ing

kwk

2

1,T

instead of

kwk

2

1,T ∗

allows to over ome this

di ulty.

As in[44℄, we begin the proof of (17 ) by hoosing a ve tor

b

1

, su h that, for ea h pointin

Ω

1

, ea h line dened by this point and

b

1

interse ts

γ

′

1

. We take

b

1

= e

1

. We need here only one ve tor, while the author need afamily of ve tors in[44℄. Now, we adapt this proof to our geometri al multi-s aledomain.

Forall

(x, y)

∈ Ω

1

, D((x, y), e

1

)

designates the semi-linedened by itsorigin

(x, y)

and the ve tor

e

1

; let

P (x, y) = γ

′

1

∩ D((x, y), e

1

)

. For

σ

∈ E

,

χ

σ

isafun tionfrom

R

2

_×R

2

to

{0, 1}

su hthat

χ

σ

(r, z)

isequalto

1

if

σ

∩[r, z] 6= ∅

and equal to

0

otherwise.

Let

K

∈ T

su h that

K

∩ Ω

1

6= ∅

. Then we have for a.e.

(x, y)

∈ K ∩ Ω

1

:

| w

K

|≤

X

σ∈E

int

,σ⊂γ

1

′

(D

σ

w) χ

σ

((x, y), P (x, y)) +

N

1

X

i=0

| w

1,i

− w

1,i+1

|

sin e

w

1,N

1

+1

= 0

. This requirement is essential to ensure the inequality above. Let us remark that there is

σ

⊂ γ

′

1

su h that

P (x, y)

∈ σ

, then

D

σ

w =

|w

L

− w

1,0

|

for some

L

(see Denition 5) su h that

σ

∈ E

L

. The use of

w

1,0

allows to get out of

Ω

′

ε

and join the boundary of the 1D domain

S

1

.

By the Cau hy S hwarz inequality, wehave

w

2

K

≤









X

σ∈E

int

σ⊂γ

′

1

(D

σ

w)

2

d

σ

c

σ

χ

σ

((x, y), P (x, y))+

N

1

X

i=0

(w

1,i

− w

1,i+1

)

2

h

e

1

i+1/2

















X

σ∈E

int

σ⊂γ

′

1

d

σ

c

σ

χ

σ

((x, y), P (x, y))+

N

1

X

i=0

h

e

j

i+1/2









(18) where

c

σ

=

| e

1

· n

σ

|

.

Sin e

e

1

is the axis of the rst bran h(where

Ω

1

is found), we have

X

σ∈E

int

,σ⊂γ

′

1

d

σ

c

σ

χ

σ

((x, y), P (x, y))

≤ δ

Integrating (18) over

K

∩ Ω

1

and summingoverall

K

∈ T

su hthat

K

∩ Ω

1

6= ∅

yields

X

K∈T

w

2

_K

m(K

∩Ω

1

)

≤ l

1









X

σ∈E

int

σ⊂γ

′

1

(D

σ

w)

2

d

σ

c

σ

Z

Ω

1

χ

σ

((x, y), P (x, y))dxdy



+ m(Ω

1

)

N

1

X

i=0

(w

1,i

− w

1,i+1

)

2

h

e

1

i+1/2









(19) Sin e, following[19℄, we have

Z

Ω

1

kwk

2

L

2

_(Ω

1

)

≤ l

max



δ

X

σ∈E

int

,σ⊂γ

1

′

m(σ)d

σ



D

σ

w

d

σ



2

+ m(Ω

1

)

N

1

X

i=0

(w

1,i

− w

1,i+1

)

2

h

e

1

i+1/2





≤ l

max

δ





X

σ∈E

int

,σ⊂γ

1

′

m(σ)d

σ



D

σ

w

d

σ



2

+ θ

1

ε

N

1

X

i=0

(w

1,i

− w

1,i+1

)

2

h

e

1

i+1/2





≤ l

max

δ

kwk

2

1,T

As we have

m

1

(w)

2

≤

1

m(Ω

1

)

kwk

2

L

2

_(Ω

1

)

≤

l

max

δ

m(Ω

1

)

kwk

2

1,T

this proves (17 ).

With(14) and (16), we dedu e that there is a onstant

c

depending only on

D

ε

su h that

kwk

2

2,T

≤ ckwk

2

1,T

,soLemma7isproved.ThislemmaisusedtostateTheorem8andTheorem 11 below. Theorem 8 gives an error estimate for (1) assuming

ε

and

δ

are xed. Theorem 11 relates to(3) assuming

ε

tends tozero.

Ifweare justinterestedintheresolutionof(1)thenamorepre isedenitionofthe onstant

c

doesnot matter.To get the estimateof Theorem 8it isenough toknowthat

c

depends only on

D

ε

.

The estimate of Theorem 11 requires pre ise informations on the dependen e of

c

1

, c

2

, c

3

with respe t to

ε

and

δ

. Evaluating the onstants from the proof of Lemma 10.2 in [19℄, one has

c

1

= O



diam

(Ω

′

ε

)

4

m(Ω

k

)

m(Ω

i

)

2

+

diam

(Ω

′

ε

)

diam

(Ω

i

)

2

_m(Ω

k

)

m(I

ij

)m(Ω

i

)

+

diam

(Ω

′

ε

)

4

m(Ω

i

)

, i, j, k

_{∈ {1, ..., p}}



c

2

= O



diam

(Ω

′

ε

)

4

m(Ω

i

)

2

+

diam

(Ω

′

ε

)

diam

(Ω

i

)

2

m(I

ij

)m(Ω

i

)

, i, j

_{∈ {1, ..., p}}



(20) Weremind that we assume inthis ase that

δ

isof order

ε

ln

ε

. With(15), we dedu e that

c

1

= O



diam

(Ω

′

ε

)

4

m(Ω

′

ε

)

+

diam

(Ω

′

ε

)

3

ε



= O



δ

3

ε



c

2

= O



diam

(Ω

′

ε

)

4

m(Ω

′

ε

)

2

+

diam

(Ω

′

ε

)

3

ε m(Ω

′

ε

)



= O



δ

2

ε

2



(21) Last we have

c

3

=

l

max

δ

m(Ω

1

)

= O



1

ε



(22)

And then, wesee from(14), (16) and (17)that thereis a onstant

c

, namely

kwk

2

2,T

≤ ckwk

2

1,T

Moreover, we on lude with (21) and (22) that

c = O



δ

3

ε



+ O(δ) + O(1) = O(1)

when

ε

tendstozero, assumingthat

δ

isof order

ε

ln

(ε)

. So, alsointhis ase, the onstant

c

in Lemma 7 depends neither on

h

nor on

ε

.

4 The error estimate

The error estimate between the solutionof (1) and its nite volume approximation, whi h is obtained in [36℄, uses the linearity of the problem to prevent the oupling between its 1D and its 2D parts. So in [36℄, a standard

H

1

norm on the 1D domains

S

j

, j = 1, ..., n

, and a standard

H

1

norm onthe2D domain

Ω

′

ε

are used. Thedisadvantageof this methodisthat the errors between the values

α

j

, j = 1, ..., n,

of thesolutiononthe interfa es between the domains of dierent dimensions and the approximate values

v

j,0

, play an importantrole in al ulating the global error. Andthese errors are not optimized (see Subse tion 1.6).

To over ome this di ulty, we use here the spe i dis rete

H

1

norm dened in the pre-vious se tion on

D

ε

. Using (12), the approximate values

v

j,0

, j = 1, ..., n,

of the solution on the interfa es are related to ( onvex ombinations of) the other unknowns : the approximate values of the solution on both sides of the interfa es between the 1D parts and the 2D part. So

v

j,0

, j = 1, ..., n,

may be removed from the s heme (11) by expressing

v

j,0

in terms of

v

j,1

and

u

K

su h that there is

σ

∈ E

K

, σ

⊂ γ

′

j

, a ording to(12). In the same way,

kwk

1,T

may be rewritten without

w

j,0

, j = 1, ..., n,

in Denition 5. The global error

e

T

is dened just below, an estimateof

ke

T

k

1,T

is obtained withoutusing any estimate on

| α

j

− v

j,0

|, j = 1, ..., n,

that allows toimprove the result obtained in[36℄.

We remind that the solution of (1) is assumed to be regular, that means that

u

∈ C

2

_(Ω

′

ε

)

and

v

j

∈ C

2

_{([δ, l}

j

]), j = 1, ..., n

.

We state below the main result ofthe paper. Theorem 8 If

u

d

T

isthenitevolumeapproximationof(1)denedby(13),if

u

d

isthesolution of (1) dened by(2) and is assumed to be regular, and if

e

T

∈ X(T S)

is dened by

e

T

(x, y) =



e

K

= u(x

K

)

− u

K

, (x, y)

∈ K, K ∈ T

e

j,i

= v

j

(x

e

i

j

)

− v

j,i

, (x, y)

∈ S

j

, x

e

j

∈ (x

e

_i−1/2

j

, x

e

_i+1/2

j

), i = 1, ..., N

j

, j = 1, ..., n.

then, there are two onstants

c

1

and

c

2

depending only on

u

d

and

D

ε

su h that

ke

T

k

1,T

≤ c

1

h

(23)

and

ku

d

− u

d

T

k

L

2

_(Ω

ε

)

≤ c

2

h

(24)

with

h

the size of the mesh of

D

ε

. Proof.

We prove an estimate for

ke

T

k

1,T

, and on lude thanks to the Poin aré inequality. This proof is not lassi al be ause of the interfa e terms relating to the onsisten y error on the diusion ux when

σ

⊂ γ

′

j

, j = 1, ..., n

.

We onsider rst the ontinuousproblem(1) .Weintegrate(1a) overea h1D elland (1b) over ea h

K

∈ T

.We obtain























F

j,i+1/2

− F

j,i−1/2

= h

e

i

j

f

e

j

i

, i = 1, ..., N

j

, j = 1, ..., n

F

j,i+1/2

= v

j

′

(x

e

j

i+1/2

), i = 0, . . . , N

j

, j = 1, ..., n

X

σ∈E

K

F

K,σ

= 0,

∀K ∈ T

F

K,σ

=

R

σ

∂u

∂n

dγ,

∀σ ∈ E

K

(25) We dene



















F

∗

j,i+1/2

=

v

j

(x

e

i+1

j

)

− v

j

(x

e

i

j

)

h

e

j

i+1/2

, i = 1, . . . , N

j

, j = 1, ..., n

F

∗

j,1/2

=

v

j

(x

e

1

j

)

− u

∗

j

(δ)

h

e

j

1/2

, j = 1, ..., n

(26) with

u

∗

_j

(δ) =





v

j

(x

e

1

j

)

h

e

j

1/2

+

1

θ

j

ε

X

σ∈E

K

,σ⊂γ

′

j

m(σ)

d

σ

u(x

K

)









1

h

e

j

1/2

+

1

θ

j

ε

X

σ⊂γ

′

j

m(σ)

d

σ





−1

, j = 1, ..., n

(27)

In the same spirit, we introdu e

F

∗

K,σ

=











m(σ)

d

σ

(u(x

L

)

− u(x

K

)) ,

∀σ ∈ E

int

,

if

σ = σ

K/L

m(σ)

d

σ

(u

∗

j

(δ)

− u(x

K

))

,

∀σ ⊂ γ

′

j

, σ

∈ E

K

, j = 1, ..., n

0

,

∀σ ⊂ ∂Ω

′

ε

\(∪

n

j=1

γ

j

′

)

(28)

The onsisten y errors are dened by

(

R

j,i+1/2

= F

j,i+1/2

∗

− F

j,i+1/2

, i = 0, ..., N

j

, j = 1, ..., n