B. Louro

Jacques Henry

M.C. Soares

Comptes Rendus Mathématique

n-arabolicdurésenssystèmeslesillee sectiontesC. R. Acad. Sci. Paris, Ser. I 339 (2004) 175–180Partial Differential EquationsA factorization method for ellipticproblems in a circular domainJacques Henrya, B. Lourob, M.C. Soaresba MAB, INRIA Futurs – Bordeaux, université Bordeaux 1,351, cours de la libération, 33405 Talence cedex, Franceb Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Monte de Caparica,2829-516 Caparica, PortugalReceived 17 July 2003; accepted after revision 1 June 2004Presented by Alain BensoussanAbstractWe present a method to factorize a second order elliptic boundaryvalue problem in a circular domain, in a system of ucoupled first order initial valueproblems. We use a space invariant embeddingtechnique along the radius of the circle, in adecreasing way. This technique is inspired in the temporal invariant embedding used by J.-L. Lions for the control of psystems. The singularity at the origin for the initial value problems is studied. A formal calculation for more general star-shapedomains is presented.To cite this article: J. Henry et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméUne méthode de factorisation de problèmes elliptiques dans un domaine circulaire. Nous présentons une méthode pofactoriser un problème aux limites elliptiquedu second ordre dans un domaine circulaire, en un système de problèmes découplaux valeurs initiales du premier ordre.Nous utilisons une technique de plongement invariant en espace radiale, dans le sdécroissant. Cette technique s’inspire du plongement invariant temporel employé par J.-L. Lions pour le contrôle deparaboliques. La singularité à l’origine pour les problèmes aux valeurs initiales est étudiée. On prèsenteaussi un calcul formedans une situation plus générale de domaines étoilés.Pour citer cet article : J. Henry et al., C. R. Acad. Sci. Paris, Ser. I 339(2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégéeOn se propose d’appliquer spatialement la technique duplongement invariant de Bellman [2] à des problèmaux limites elliptiques. Dans [3] la méthode est appliquée à l’équation de Poisson dans un cylindre. La famde problèmes de la méthode de plongement invariant est définie par des sous-cylindres limités par unvariable et une condition aux limites additionnelle sur cettesection. L’opérateur reliant les conditions aux limiE-mail addresses:jacques.henry@inria.fr (J. Henry), bjl@fct.unl.pt (B. Louro), mcs@fct.unl.pt (M.C. Soares).1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2004.06.001176 J. Henry et al. / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 175–180eites enunenir seiresuchycto-)pasaire poursur cette section (Dirichlet–Neumann ou Neumann–Dirichlet) vérifie uneéquation de Riccati en fonction d’uncoordonnée le long de l’axe du cylindre. A l’aide de cet opérateur on peut transformer le problème aux limdeux problèmes de type parabolique découplés dans des sens opposés. Cette factorisation peut être vue commegénéralisation en dimension infinie de la factorisation de GaussLU par blocs de matrices bloc-triangulaires. Oveut étudier ici le cas où la famille de surfaces qui limite les sous-domaines part du bord du domaine pour venréduire en un point. On considère donc le problème de Dirichlet pour l’équation de Poisson sur le disqueΩ centréà l’origine et de rayona, de bordΓa(P) − u = f, dansΩ; u|Γa = u0. (1)SoitΩs le disque centré à l’origine et de rayons,0 < s < a de bordΓs . On supposeu0 ∈ H 1/2(Γa), f ∈ L2(Ω), etune régularité supplémentaire sur un voisinageO de l’origine,f ∈ C0,α(O). Après passage en coordonnées pola(û(ρ, θ) = u(x1, x2) verifie (8)), on considère la famille de problèmes(P̂s,h) définis sur l’anneauΩ \ Ωs avec lacondition aux limites additionnelle surΓs∂ûs∂ρ|Γs = h. SoientI =]0,2π[ , et H 1ρ,P (I) = {v | v ∈ L2(I), 1ρ ∂v∂θ ∈L2(I), v(0) = v(2π)}. On choisith dansH 1/2ρ,P (I)′, oùH 1/2ρ,P (I) = [H 1ρ,P (I),L2(I)]1/2. Par linéarité de (8) on aûs |Γs = P(s)h + r(s), (2)où P(s) est l’opérateur Neumann–Dirichlet. SoitX = {v̂ | v̂ ∈ L2ρ(0, a;H 1ρ,P (I)) ∩ ∂û∂ρ ∈ L2ρ(0, a;L2(I))} oùL2ρ(0, a;Y ) est l’espace des fonctions deρ à valeurs dansY de carré integrable dans(0, a) avec le poidsρ. A partirde l’identité (2) écrite en suivant une solution de(P̂s0,h), on obtient la factorisation du problème aux limites(P̂) parThéorème 0.1. La solutionû du problèmêP est la solution unique du système suivant de problèmes de Cadécouplés, du premier ordre enρ1. pour touth, h̄ dansL2(I), l’opérateur autoadjointP(ρ) ∈ L(L2(I),H 1ρ,P (I)) ∩ L(H 1/2ρ,P (I)′,H 1/2ρ,P (I)) ∩L(H 1ρ,P (I)′,L2(I)), P  0, borné enρ, vérifie l’équation de Riccati(dPdρh, h̄)+(1ρ2∂∂θ◦ Ph, ∂∂θ◦ P h̄)−(1ρh,P h̄)= (h, h̄) (3)dansD′(0, a), avec la condition initialeP(a) = 0 ;2. pour touth dansL2(I), r ∈ X vérifie l’équation(∂r∂ρ,h)+(1ρ2∂r∂θ,∂∂θ◦ Ph)= (f̂ ,Ph) (4)dansD′(0, a), avec la condition initialer(a) = û0 ;3. pour touth dansH 1ρ,P (I)′, û ∈ X vérifie l’équation−(∂û∂ρ,Ph)+ 〈û, h〉H1ρ,P (I),H1ρ,P (I)′ = 〈r, h〉H1ρ,P (I),H1ρ,P (I)′ (5)dansD′(0, a), avec la condition initialêu(0) = limρ→0 r(ρ) dansL2(I) qui est constant enθ .P et r sont ainsi définis de façon unique. Les Éqs.(3) et (4) sont bien posées pourρ décroissant dea à 0 et (5)pourρ croissant de0 à a.Pour étudier le comportement deu solution de (5) au voisinage de l’origine, on commence par étudier la farisation du problème(Pε) − uε = f, dansΩ \ Ωε; uε|Γa = u0;∫Γε∂uε∂ndΓ = 0; uε|Γε est constant (6où Ωε est de rayonε, 0 < ε < a et l’on fait tendreε vers 0. Afin de montrer que le résultat précédent n’estrestreint aux domaines circulaires et aux coordonnées polaires on donne aussi le résultat d’un calcul similun domaine étoilé.J. Henry et al. / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 175–180 177Angelficationontrold-ed onarabolica–l.e wherealarinates,ect1. IntroductionThe technique of invariant embedding was first introduced by Bellman [2] and was formally used byand Bellman [1] in the resolution of Poisson’s problem defined over a rectangle. J.L. Lions [4] gave a justifor this invariant embedding for the computation of the optimal feedback in the framework of Optimal Cof evolution equations of parabolic type. Henry and Ramos[3] presented a justification for the invariant embeding of Poisson’s problem in a cylindrical domain. Embedding the problem in a family of similar ones definsubcylinders bounded by a variable section, they obtained a factorization in two uncoupled problems of ptype, in opposite directions. The factorization uses a family of operators on functions of the section satisfyingRiccati equation. These operators relate the boundary conditi s on the section (Dirichlet–Neumann or NeumannDirichlet). They showed that this can be seen as a generalization, up to infinite dimension, of theLU block factoriza-tion of matrices: solving the Riccati equation is analogous to computing theL andU factors for a block tridiagonamatrix and solving the two parabolic problems is related to solving the lower and upper triangular systemsIn this Note, we want to generalize this method to other types of geometries and, in particular, to the casthe family of surfaces which limits the sub-domains, starts on the outsideboundary of the domain and shrinks topoint. We present here the simple situation whereΩ (respΩs ) is a disk ofR2 with radiusa (resp.s) and centeredon the origin and where the sub-domains defined by the invariant embedding are the annuliΩ \ Ωs , s ∈ (0, a).2. Formulation of the problem and main resultWe consider the Dirichlet problem for the Poisson equation defined overΩ .(P) − u = f, in Ω; u|Γa = u0, (7)whereΓs denotes the circle of radiuss and center at the origin, andf ∈ L2(Ω), u0 ∈ H 1/2(Γa). We assumethe additional regularity around the originf ∈ C0,α(O), O being a neighborhood of the origin. Introducing pocoordinates,̂u(ρ, θ) = u(x1, x2) satisfies(P̂) − 1ρ∂∂ρ(ρ∂û∂ρ)− 1ρ2∂2û∂θ2= f̂ , in ]0, a[×]0,2π[; û|Γa = û0, û(ρ,0) = û(ρ,2π). (8)However, by doing this, we introduce a new difficulty on the problem since, with this new system of coordwe have a singularity at the origin. We embed problem (P) in a family of similar problems (Ps,h) defined on theannulusΩ \ Ωs , s ∈]0, a[ and satisfying an additional Neumann boundary condition inΓs ; in terms of problem(P̂), this can be written as∂ûs∂ρ|Γs = h, whereûs is the solution of̂Ps,h. DefiningI =]0,2π[, H 1ρ,P (I) as the spaceof periodic functionsv of θ , verifying v ∈ L2(I) and 1ρ∂v∂θ∈ L2(I), we takeh ∈ H 1/2ρ,P (I)′, whereH 1/2ρ,P (I) =[H 1ρ,P (I),L2(I)]1/2. By linearity of (8) we haveûs |Γs = P(s)h + r(s), (9)whereP(s) is the Neumann to Dirichlet map. LetX beX = {v̂ | v̂ ∈ L2ρ(0, a;H 1ρ,P (I)) ∩ ∂û∂ρ ∈ L2ρ(0, a;L2(I))},whereL2ρ(0, a;Y ) means functions ofρ with values inY square integrable on(0, a) with the weightρ. Formallythe factorization of problem (8) is obtained by takings = ρ > s0 andh = ∂ûs0∂ρ in (9) for a solutionûs0 of P̂s0,h.Then multiplying (9) byρ, taking the derivative with respect toρ, substituting the second derivative with respto ρ from (8) andu from (9) yields(dPdρ− P 1ρ2∂2∂θ2P − 1ρP − I)∂ûs0∂ρ+ ∂r∂ρ− P 1ρ2∂2r∂θ2− P f̂ = 0, (10)for anyh and so for any∂ûs0∂ρ. From the boundarycondition onΓa in (8) we obtainP(a) = 0 andr(a) = û0. Theresult is synthesized by the following theorem178 J. Henry et al. / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 175–180nryng thelowingllyTheorem 2.1. The solutionû of (8) is the unique solution of the following system of uncoupled, first order iρ,initial value problems1. for everyh, h̄ in L2(I), the selfadjoint operatorP(ρ) ∈ L(L2(I),H 1ρ,P (I)) ∩ L(H 1/2ρ,P (I)′,H 1/2ρ,P (I)) ∩L(H 1ρ,P (I)′,L2(I)), P  0, bounded as a function ofρ, satisfies the Riccati equation(dPdρh, h̄)+(1ρ2∂∂θ◦ Ph, ∂∂θ◦ P h̄)−(1ρh,P h̄)= (h, h̄) (11)in D′(0, a), with the initial conditionP(a) = 0;2. for everyh in L2(I), r ∈ X satisfies the equation(∂r∂ρ,h)+(1ρ2∂r∂θ,∂∂θ◦ Ph)= (f̂ ,Ph) (12)in D′(0, a), with the initial condition r(a) = û0;3. for everyh in H 1ρ,P (I)′, û ∈ X satisfies the equation−(∂û∂ρ,Ph)+ 〈û, h〉H1ρ,P (I),H1ρ,P (I)′ = 〈r, h〉H1ρ,P (I),H1ρ,P (I)′ (13)in D′(0, a), with the initial condition û(0) = limρ→0 r(ρ) in L2(I) which is constant.P andr thus defined are also unique. Eqs.(11)and (12) are well posed forρ decreasing froma to 0 and (13)is well posed forρ increasing from0 to a.This formulation of problem(P) furnishes the Neumann to Dirichlet operatorP(s) for the annulusΩ \ Ωswhich is of interest for various kindsof problems as domain decomposition or the definition of transparent boundaconditions. Furthermore Eq. (11) is intrinsic and only Eqs. (12) and (13) depend on the dataf,u0. For multipleresolution of(P) with various data, only (12) and (13) need to be recomputed which is faster than solviinitial boundary value problem.3. Factorization by invariant embedding (scheme of the proof)In order to avoid the singularity created at the origin by polar coordinates we start by defining the folintermediate problem:(Pε) − uε = f, in Ω \ Ωε; uε|Γa = u0;∫Γε∂uε∂ndΓ = 0; uε|Γε is constant (14)whereΩε is a circular domain of radius 0< ε < a and concentric withΩ . It’s easy to see that this problem is weposed.Proposition 3.1. Whenε → 0, ũε, defined as̃uε ={uε, in Ω \ Ωε,uε = uε|Γε , in Ωε, whereuε is the solution of problem(14)converges tou, solution of problem(7), in H 10 (Ω).We can write problem (14) in polar coordinates restricting problem (8) over]ε, a[×I and joining the boundarconditionsûε|Γε constant,∫Γε∂ûε∂ρdθ = 0. Since∂ûε∂ρ|Γε is well determined through the conditions “ûε|Γε constant”and “∫Γε∂ûε∂ρdθ = 0”, it’s clear that(P̂ε) belongs to the family(P̂s,h) for s = ε. From (9), the solution̂uε of (14)in polar coordinates, satisfies the relationûε(ρ) = P(ρ) ∂ûε∂ρ |Γρ + r(ρ),∀ρ ∈ [ε, a[ . Equations forP and r arederived as in (10). From these equations above, and respective initial conditions, we can obtainP andr. KnowingP(ε) andr(ε) we can determine uniquelŷuε(ε) satisfying “ûε| constant” and “∫Γ∂ûε∂ρdθ = 0”. Let M andN beΓε εJ. Henry et al. / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 175–180 179.escalily ofxactly-sionaldefined byM = {v ∈ (H 1/2ρ,P (I))′ |∫ 2π0 v dθ = 0} andN = M⊥ = {v ∈ H 1/2ρ,P (I) | v is constant}. One hasL2(I) =(M ∩ L2(I)) ⊕ N and letΠM andΠN be the projection, for theL2(I) metrics, on each subspace respectivelyProposition 3.2. Givenr(ε) ∈ L2(I), there exists a unique solution̂uε(ε) ∈ N and ∂ûε∂ρ (ε) ∈ M for the equationûε(ε) = P(ε) ∂ûε∂ρ (ε) + r(ε). In particular, ûε(ε) = ΠNr(ε).Using the Galerkin method as in [4,3] and adequate properties on the operatorP nd functionr, we can justifythe preceding formal calculation and weobtain, after passing to the limit when the dimension tends to infinity, thfollowing result:Proposition 3.3. For everyh, h̄ in L2(I), the operatorP ∈ L∞(ε, a;L(H 1/2ρ,P (I)′,H 1/2ρ,P (I))) satisfies the followingequation(∂P∂ρh, h̄)+(1ρ2∂∂θ◦ Ph, ∂∂θ◦ P h̄)−(1ρh,P h̄)= (h, h̄), (15)in D′(ε, a), withP(a) = 0. The functionr belongs toX|Ω\Ωε , satisfiesr(a) = û0 and for everyh in L2(I), satisfiesin D′(ε, a) the following equation(∂r∂ρ,h)+(1ρ2∂r∂θ,∂∂θ◦ Ph)= (f̂ ,Ph). (16)SinceP and r do not depend onε and thanks to the estimates onP(ρ) and r(ρ), we can takeε arbitrar-ily small and consequently consider the previous equalities defined onD′(0, a). Let ‖ · ‖ρ denote the norm inL(H 1/2ρ,P (I)′,H1/2ρ,P (I)). The following theorem gives the behaviour ofP andr around the origin which providethe regularity claimed in Theorem 2.1:Proposition 3.4. For P satisfying(15)andP(a) = 0, we havelimρ→0 ‖P(ρ)‖ρ = 1. Furthermorelimρ→0 ‖P(ρ)−ρ(P∞ oΠM)‖ρ = 0, whereP∞ is the negative selfadjoint operator satisfying−P∞ ∂2∂θ2 P∞ = I .The solutionr of (16)andr(a) = û0, has a limitr(0) constant with respect toθ : limρ→0 ‖r(ρ)−r(0)‖L2(I) = 0.It should be also remarked that, measured with the fixed norm‖ · ‖L(L2(I),L2(I)), P(ρ) goes to 0 asρ goes to 0.Concerning the equation on̂uε we haveProposition 3.5. For everyh in H 1ρ,P (I)′, ûε satisfies inD′(ε, a) the following equation−(∂ûε∂ρ,Ph)+ 〈ûε, h〉H1ρ,P (I),H1ρ,P (I)′ = 〈r, h〉H1ρ,P (I),H1ρ,P (I)′, (17)with the initial condition ûε(ε) = ΠNr(ε).Using Proposition 3.1 we obtain the convergence inX of ûε to û satisfying (13). Furthermore, thanks to the loregularity assumption onf near the origin which implies thatu ∈ C2,α(O), one can prove that limε→0 ûε(ε) = û(0)(the proof for the uniqueness of the solution of (13) uses the determination ofû(0)).Remark 1. (a) Another factorization could be obtained by using an invariant embedding defined by the famdisksΩs . Here the main difficulty is to define the initial conditions forP andr at the origin.(b) From the numerical viewpoint, a particular finite difference discretization of (11), (12) and (13) gives ethe Gauss LU factorization of the matrix of the finite difference discretization of problem(P̂). The same observation holds for finite element discretization. This is why this factorization can be viewed as an infinite dimen180 J. Henry et al. / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 175–180d (13),ghera formalnlogianciaadeor.,extension of the Gauss factorization. Further, it may give clues to alternative discretization of (11), (12) anresulting in new numerical schemes for(P̂).4. Star-shaped domainsWe believe that this method is much more general than presented here and that it can be extended to hidimensions, to other operators than the Laplacian and more general domains. For example we present nowcalculation in the case of star-shaped domains. LetΩ be an open set containing the originO , star-shaped withrespect toO , with boundaryΓ = ∂Ω . We want to solve (P) in Ω . Let α be the angle( OM, n) whereM is a pointon Γ andn is the outward normal toΩ at M. We assume that−π/2 < α0  α  α1 < π/2 and that the equatioof Γ in polar coordinates is given byρ = ϕ(θ). We consider the homothety of centerO and ratio 0< τ < 1, whichtransformsΩ to Ωτ with boundaryΓτ . We consider the following system of curvilinear coordinates: forM ∈ Ω ,(τ, t) are such thatM ′, the image ofM by a 1/τ homothety, belongs toΓ and t,0  t < t0, is the curvilinearabscissa ofM ′ on Γ (t0 is the length ofΓ ); û(τ, t) = u(x1, x2). The family of problemsPs,h is now defined onΩ \ Ωs with the Neumann boundary condition∂û∂n |Γs = h. By linearity, there existP(s) andr(s) such thatû|Γs = P(s)h + r(s). (18)I now denotes]0, t0[ and(·, ·) the scalar product inL2(I). ThenP , r andû satisfy1. for every h, h̄ in L2(I), the selfadjoint operatorP(τ) ∈ L(L2(I),H 1τ,P (I)) ∩ L(H 1/2τ,P (I)′,H 1/2τ,P (I)) ∩L(H 1τ,P (I)′,L2(I)), P  0, bounded as a function ofτ , satisfies the Riccati equation(dPdτh, h̄)−(ϕ̂ sinατh,∂∂t◦ P h̄)−(∂∂t◦ Ph, ϕ̂ sinατh̄)+(ϕ̂ cosατ2∂∂t◦ Ph, ∂∂t◦ P h̄)−(1τh,P h̄)= (ϕ̂ cosα h, h̄) (19)in D′(0,1), with the initial conditionP(1) = 0;2. for everyh in L2(I), r satisfies the equation(∂r∂τ,h)−(ϕ̂ sinατ∂r∂t, h)+(ϕ̂ cosατ2∂r∂t,∂∂t◦ Ph)= (ϕ̂ cosαf̂ ,Ph) (20)in D′(0,1), with the initial conditionr(1) = û0;3. for everyh in H 1τ,P (I)′, û satisfies the equation−(1ϕ̂ cosα∂û∂τ,Ph)+(tanατ∂û∂t,Ph)+ 〈û, h〉H1τ,P (I),H1τ,P (I)′ = 〈r, h〉H1τ,P (I),H1τ,P (I)′ (21)in D′(0,1), with the initial conditionû(0) = limτ→0 r(τ ) in L2(I) which is constant.AcknowledgementsThe third author would like to acknowledge the financial support of Fundação para a Ciência e a Tecnowithin the grant Praxis XXI, BD/ 21443/99. The second author was partially supported by Fundação para a Ciêe a Tecnologia (program POCTI/FEDER) via Centro de Matemática e Aplicações Fundamentais da Universidde Lisboa.References[1] E. Angel, R. Bellman, Dynamic Programming and Partial Differential Equations, Academic Press, 1972.[2] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, 1957.[3] J. Henry, A.M. Ramos, Factorization of second order elliptic boundaryv lue problems by dynamic programming, Nonlinear Anal. Thein press.[4] J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Dunod, 1968.

A factorization method for elliptic problems in a circular domain

Henry, Jacques

Louro, B.

Soares, M.C.

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/j.crma.2004.06.001.pdf

A factorization method for elliptic problems in a circular domain

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