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420

Sp given by

,s>,l

^f):'

Finally, the problem (P^)N is numerically solved via a mixed-hybrid finite element

method which involves the introduction of other approximation parameter h correspond-

ing to the step-size of the mesh. This method is particularly suitable since it provides

a direct way to approach the normal derivative of u

(even if it obliges us to write the

boundary condition on E in the way —Tdu

/dv = u

As a consequence of the numerical approximation, the operator K(u>,/3) is approxi-

mated by

(W,

{Si

&)-

)„ (n

)

(21)

and finally the numerical problem to solve is

J For a given

/3,

find w > 0 (w

g Gi{P)), with (w, $) € E,

' 1 such that -1 is an eigenvalue of K^

(u},/3).

Thus,

the numerical algorithm is as follows: For a given p, we look for a guided

mode u in an interval [wj,a;,] without singular frequencies. To do that, we compute the

operators

(a;j,

P) and K^

(v,, 0) (which involves the resolution of problems (Vi) and

then (V

) for each basis function on T

) and their corresponding eigenvalues X^

(uji, 0)

and

(UJ

0). If (A^

AfR

,0) + l)(A^

, 0) +1) < 0, then there exists

€ [ui,u

]

such that \^[

(co,P) = — 1 and we compute it by using a fixed point procedure. Then

we compute an associated eigenfunction ip and we solve problems (Pi) and (P

) for the

Dirichlet condition

on T. This allows us to compute uf and finally to represent it

on f2;,.

5 Numerical results

The first test corresponds to a step-index optical fiber whose core is a circle of radious

0.45 with refraction index n

= 1.7 and cladding index equal to n

= 1 (see Fig. 3). This

device can be considered as an "artificial" stratified medium composed of three layers

(with hi = 0.5,

Yi2

= 0.75, h

= 1, see Fig. 1) where the refraction index of all layers is

constant and equal to 1. The interest of this test lies in the fact the dispersion relation

between

(OJ,

0) has an analytical formula in terms of Bessel functions

iC(0-45Q_ -4(0-45

)

(0.45/Co)

(0A5

where

(22)

Koo =

A//3

—

n^ui

and

= \Jn\u

By solving numerically equation (22) we can obtain some reference solutions. Thus,

for instance, we obtain that for the value /? = 2 there exists a guided mode associated

421

:m?ty

Figure 3: Cross section for the optical fiber test

Table 1: Eigenvalues of the operator K^

,iv

(1.8534,2). h = 1/33, N = 100.

R=l R=2

R=3

-0.9952 -0.9989 -0.9990

to u! =

1.8534.

A first test was to compute the operator K

' (ui,/3) associated to this

pair (w,/?) and to check that

—1

is an associated eigenvalue. Tables 1, 2 and 3 report the

eigenvalue of K

,JV

(tj,/9) which is closest to

—1

for different values of h, N and R in order

to show the influence of these parameters in the numerical procedure. The abscises of the

boundaries S

and S~ are located respectively at a

= 0.5 and a~ = —0.5.

The second test corresponds to an integrated optics device whose cross section has

been sketched in Fig. 4. We have considered that the interior layer fij is a three-layered

medium (with boundaries hi = 0.25, h

= 1.25,

^13

= 1.5) where the refraction index is

the same in all layers and equal to 3.38. The refraction index profile is given by

n(xi,x

)

1 if x

> 1.5,

3.17 if x

< 0,

3.44 if (xi, x

) e

(-1,1)

x (0.25,1.25),

3.38 otherwise.

l*»

l-k

jn.=a.3a.

m&mmm^^

Figure 4: Cross section for the integrated optics test

In this case, analytical solutions are no available and this is why the solutions have

been compared with the ones obtained with the method proposed by Mahe[7]. Following

422

Table 2: Eigenvalues of the operator 1^(1.8534,2). h = 1/33, R = 3.

N = 3 N = 30

= 100

-0.9920 -0.9930 -0.9990

Table 3: Eigenvalues of the operator 1(^(1.8534, 2).

= 100, R = 3.

ft = 1/11 ft =1/33

-0.9985 -0.9990

Mahe's method, we can obtain that there exists a guided mode for the values 8 = 13.5

and w = 4. When computing the eigenvalues of the operator K^

,Ar

(4,13.5) for the values

N = 100, h = 1/20 and R = 1 we obtain -1.000067 as eigenvalue closest to -1. The

associated guided mode has been represented (restricted to domain fi(,) in Figure 5.

Figure 5: The guided mode for 8 = 13.5, w = 4.

References

[1] A. S. Bonnet-Ben Dhia. Analyse mathematique de la propagation de modes guides

dans les fibres optiques. Technical Report 229, Ecole Nationale Superieure de Tech-

niques Avancees, 1989.

[2] C. Vassallo. Theorie des guides d'ondes electromagnetiques. CNET-ENST, Paris,

1985.

423

[3] A. S. Bonnet-Ben Dhia, G. Caloz, and F. Mahe. Guided modes of integrated optical

guides, a mathematical study. IMA Journal of Applied Mathematics, 60:225-261,

1998.

[4] M. D. Gomez Pedreira. A numerical method for the computation of guided waves in

integrated optics. PhD thesis, Universidad de Santiago de Compostela, 1999.

[5] D. Gomez Pedreira and P. Joly. A method for computing guided waves in integrated

optics. Part 1: Mathematical analysis. To appear in SIAM J. Numer. Anal.

[6] P. Joly and C. Poirier. A numerical method for the computation of electromagnetic

modes in optical fibres. Math. Meth in Appl. Sciences, 22:389-447, 1999.

[7] F. Mahe. Etude Mathematique et numerique de la propagation d'ondes electro-

magnetiques dans les microguides de I'optique integree. PhD, Universite de Rennes

I, 1993.

[8] D. Gomez Pedreira and P. Joly. A method for computing guided waves in integrated

optics. Part 2: Numerical approximation and error analysis. To appear in SIAM J.

Numer. Anal.

[9] D. Gomez Pedreira and P. Joly. Mathematical analysis of a method to compute

guided waves in integrated optics. Technical Report

3933,

Institut National de

Recherche en Informatique et Automatique (I.N.R.I.A), May 2000.

[10] D. Gomez Pedreira and P. Joly. In preparation.

A non-stationary model for catalytic converters with

cylindrical geometry

Jean-David HOERNEL

Laboratoire de Mathematiques et Applications

Universite de Haute-Alsace, 4 rue des Freres Lumiere

F-68093 MULHOUSE Cedex FRANCE

Email : j-d.hoernel@wanadoo.fr

Abstract

We prove some existence and uniqueness results and some qualitative properties

for the solution of a system modelling the catalytic conversion in a cylinder. This

model couples parabolic partial differential equations posed in a cylindrical domain

and on its boundary.

1 Introduction

A gas containing N

—

1 different chemical species is flowing through a cylindrical passage

with a parabolic speed profile. Chemical species are diffusing in the cylinder and are

reacting only on the boundary of the cylinder.

We investigate the existence, uniqueness and qualitative properties of the solution of

a non-stationary system of partial differential equations describing the evolution of the

concentrations of the N

—

1 chemical species and of the temperature, both with respect

to the time variable and along the cylinder. This model of catalytic converters starts

with the contribution of Ryan, Becke and Zygourakis [5]. However, we have added an

axial diffusion term on the boundary in the present model, see [1] for a more detailed

description of this model.

Due to its internal symmetry, the cylinder may be reduced to the domain Q =

[0,1[ x ]0,1[, the boundary of which is S = {1} x ]0,1[. The concentrations (resp. the

temperature) inside the cylinder fi are named C,f, i =

1,...,N

—

1 (resp.

CJV/)-

The

concentrations (resp. the temperature) on the boundary E are named Ci

(resp. C/v

The problem is written in a normalized way as

t-x

I *\ Pit

(

-^{z,t) = -

.^(l

)

z,t) + ^(C£

)

...

)

C+.)(

t) (1)

424

425

for i 6

{1,...

N} and with C£ (z,t)

—

sup (C

0).

The initial and boundary conditions

are :

(r,0,t)

= C

(r),

(l,z,t)

= C

(z,t),

«^(l,t) = 0,

(0,z,t) = 0,

(z,0) =

is0

(z)

.^(o,t)

= o.

(2)

The functions r*, i e

{1,...

, JV},

are supposed to be Lipschitz continuous

i[Cis,•••,c

)

-r

,...,c

fcj2_^|c

-C;

and verify the following hypotheses :

(HI) \fx e {R

f : I, {x

... ,x

) > 0.

(H2) If at least one of the x

, 1 < i < N, is equal to 0, then

(xi,...

0,...

XN)

= 0.

(H3) For every (x,y) G (R+)

x (R+)

-~^2

i— {rt(xi,... ,x

) -Ti(y

... ,y

))(xi~yi) > 0.

i=l 7is

Remark 1.1 We observe that :

Hypotheses (H2) and (H3) imply :

Vz 6 (K+)

: - ^ 8p*-r* (x

...

) x

{

> 0.

We have:

(1,

z, t) = C

(z, t) ; C,, (r,

t) = C

(r) ; C

(z, 0) =

is0

(z).

In order to ensure the continuity of the concentrations in Q and on the boundary £

at z = 0, we must have at t = 0, C,/ (1,0,0) = Ci

(0,0), which implies : Qo (1) =

is0

(0) ;Ci.(0,t) = C«,(l).

2 Existence of the solution

We establish the existence of the solution using the diagram :

Cif —> Cif

Cis

Indeed, the proof of the existence is decomposed in two steps

426

Existence of a solution in £2 (given the solution on the boundary E) and existence

of a solution on the boundary E (given the solution in

SI)

;

F = \P o $ is a contraction in some appropriate functional space.

2.1 Preliminary results

We have the following continuous embeddings (see [3, p. 103])

(Si) C L

(SI)

r(1

r2)

(SI)

; W

(SI)

C W?-

(Si); W

(Si)

(

r2)

(SI).

Because

-§^-(1,

z,t) = j- J

-gjf-r (I

—

) dr, we can write the problem as

= Q

^LL^f

' dz

PiS

rdr \ dr

lis

pad

' dz

/?,

y»

2£n

Wo dz

(l-r3)dr + to(C£,...,C+,),

with igjl,..., N}, and the initial or boundary conditions

{r,0,t) = C

(r),

(l,z,t) = C

(z,t),

^(M) = 0,

dr <°'*'*>

Ci.(z,0)

^(o,«)

o (z)

(3)

2.2 Existence in the cylinder

Assuming that the C

are known on the boundary, and performing a change of function

in order to have homogeneous boundary conditions at r = 1, we obtain the following

problem in which we omitted the time variable t

(4)

with i € {1,

We set

' (l-r

)

«.

N} and

w, = '(d,,..

Wf —U

OUf

~dz"

-*;

• ,

Ctff),

d 1 duf \ du

dr \ dr J dz

(r,0) = u

(r),

(l,z) = 0,

7?<o,*)

- o,

= (Cls,

• •

• , CMS) I

= diag(j3

,--- ,(3

= Le(LU0,i)f\~e(L

(QA))

= l

[

u€(L

(0A)f\^e(L

(0,l))

,u(l)

and let

be the dual space of W

427

Definition 2.1 Assume that u

€ (L

(0,1))

and ^ (z) € (L

(0,1))^. A function u

is a weak solution of (4) if and only ifuf G

(0,1;

Wo), -g^- €

(0,1;

W^) and if for

every

€ L

(0,1; Wo

), we /im;e ;

/:/:(??-K'w:/>^>«

J 0

J C

<9z

) r(l

—

)drdz.

Proposition 2.2 Let ug and 4^ be as in the preceding definition. Then there exists at

least one weak solution of (4). This solution verifies

HII

^(^(^u-^cwr)

< c,

[

dui

(L?(0,1))

dz < C,

(5)

< C,

(0,1:W'

where C is a positive constant which only depends on the data of the problem.

Proof.

We use a Galerkin approximation of u

for which we establish the three above

estimates, and pass to the limit in order to prove the Proposition, see [1] for the details.

Proposition 2.3 Under the conditions given in the Definition 2.1, the solution of (4) is

such that :

€ L

(0,1;

Wor)

n C (o,

(0,1))") .

Proof.

From the embedding W

C (L

(0,1)) C Wg

, and because u

belongs to

(0,l;W

) and -^f belongs to L

(0, l;W^.), we deduce the result using Proposition

23.23 of [6, p. 422]. *•

Remark 2.4 In order to take into account the time variable t, all the above expressions

v (.) G H have to be understood as v(.,t) £l

(0,T; H).

2.3 Existence on the boundary

Assuming that -^f is known, we have

= MO-r.

du„

W~°'~dz

(z, 0)

= u

(z),

e^(o,t) = o,

>/>"

•

) dr,

%t) = 0,

(6)

428

with:

r =

*{v

...,T

IV = diag(^,---,^±

\Pif PN{

S = (Si,-• •

,6ff),

= (O

, • • • ,6NS) •

Let

'(0,l) = {

€(L

(0,lf||e(L

(0,l))

}

and let H*(0,1) be the dual space of tf](0,1).

Definition 2.5 Suppose that ^

belongs

to L

(o, T; L

(o, 1; (L

(0,1))

}\ and that u

belongs to (I? (0,1)) . A function u

is a weak solution of (6) if and only if u

€

(0,T;Hi (0,1)), ^ e L

(0,T;H*

(Q,l)), satisfies u

(z,0) = u

(z), and if for all

tp<=L

(0,

T; Hi (0,1))

;

we have :

= 11 (6r(u+),tp)dzdt- f f J (r

^,^\r(l-r

)drdzdt.

Proposition 2.6 Let Uf and u

as in the preceding definition. Then there exists at least

one weak solution u

of (6).

Proof.

We prove the existence of a solution for the linearized weak formulation of the

problem using Theorem 2.2 of [4, p. 286] and then we use some fixed point argument. •

Proposition 2.7 Under the conditions given in the Definition 2.5, the solution of (6) is

such that :

u. €

(0,r;ff

(0,l))nc(o,T;(L

(0,l))").

Proof.

Prom the embeddings Hi (0,1) C (L

(0,1))

(0,1) and the fact that

6 L

(0,

T; Hi (0,1)), ^ e L

(0,

(0,1)), we deduce the result using Proposition

23.23 of [6, p. 422]. •

2.3.1 T = $ o $ is a contraction

Consider the mapping Y

i—>

—•

Uf, and let fi = swp

{

(lulP'if) /inf; 9

Proposition 2.8 1. The mapping

: Uf

—>

£/, is such that if fia

< 4, then :

/"* 3U

An f

/ -^ dzdt < * sup / / MA] (r,s,t)r(l-r

)drdt. (7)

oio dz a

(4-fia

)

s€

[

,i]J

J o" "

429

The

mapping

<£

:{/„—>

Uf is

such that

SUP

/ /

\\U

f(r,s,t)r(l-r

)drdt<-

f f

»e[0,l]

J 0 J 0

Define

(0,r;L

(0,l;^o

))nL

(o,r;C(o,l;(L

(0,l)f)),

(0,T;#j(0,l))nc(o,T;(L

(0,l)f),

{(u,T>)eW)xW

,,|u(l

)

z,t)

«(*,*)}•

Theorem

2.9

Under

the

hypothesis

{HZ) the

problem

(1)

admits

solution

if:

This solution belongs

to W

Proof.

Using

(7) and (8), we

obtain

'/- \/^7T^T

;

llUfl

SUP

f\\Ui\\

{r,s,t)r{l-r

)drdt.

The

which minimizes

the

Lipschitz constant

given

by a

= 2/n,

which leads

11^/11/

jU-v/e

K^f /2-

This proves that

F : (7/ i—> [// is a

contraction

if and

only

/i<2/

•

3 Uniqueness

of the

solution

Theorem

3.1

Assuming that

the

solution

of (1) is

smooth enough,

the

system

(1) has at

most

one

solution.

Proof.

suppose

the

existence

of two

couples

solutions {C}j,C}

)

and

{Cf

,Cl).

of (1), and

define:

= C}

- Cf

; W

= C}

- Cf

. We

multiply

the

i-th equation

of the

system verified

by Wif by

rWif,

integrate

on [0,1] x [0,1] x

[0,

and

use

the

equation verified

by Wi

on the

boundary. Thanks

to {H3), one can

deduce that

Wif

equal

zero

in the

cylinder because

Wif is

equal

zero

at the

inlet

{z = 0) and

the

outlet

of the

cylinder

{z = 1) and its

partial derivative with respect

to r is

equal

to zero

too,

which implies, thanks

to the

equation verified

by W

on the

boundary, that

the partial derivative

of Wif

with respect

to z is

equal

zero

in the

cylinder.

also

deduce that

^gf- is

equal

zero

on the

boundary.

equal

zero

time

T and at

time

because

of the

initial conditions. Because

T is

arbitrarily chosen,

Wu is

equal

zero

on the

boundary

E at any

time.

•

dUs

dzdt.

(8)