Overlapping domain decomposition methods for elliptic quasi-variational inequalities related to impulse control problem with mixed boundary conditions

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1 Proc. Indian Acad. Sci. (Mat. Sci.) Vol. 121, No. 4, November 2011, pp c Indian Academy of Sciences Overlapping domain decomposition metods for elliptic quasi-variational inequalities related to impulse control problem wit mixed boundary conditions MOHAMED HAIOUR 1 SALAH BOULAARAS 2 1 Department of Matematics, Faculty of Science, University of Annaba, B.P. 12 Annaba 23000, Algeria 2 Hydrometeorological Institute of Formation Researc, B.P Seddikia, Oran 31025, Algeria aiourm@yaoo.fr; sale_boulaares@yaoo.fr MS received 23 April 2010; revised 6 July 2011 Abstract. In tis paper we provide a maximum norm analysis of an overlapping Scwarz metod on non-matcing grids for quasi-variational inequalities related to impulse control problem wit mixed boundary conditions. We provide tat te discretization on every sub-domain converges in uniform norm. Furtermore, a result of approximation in uniform norm is given. Keywords. Domain decomposition; geometrical convergence; quasi-variational inequalities; impulse control; error analysis. 1. Introduction Scwarz metod as been invented by Herman Amus Scwarz in Tis metod as been used to solve te stationary or evolutionary boundary value problems on domains wic consists of two or more overlapping subdomains (see [1 3, 6, 8 12]). Te solution is approximated by an infinite sequence of functions wic results from solving a sequence of stationary or evolutionary boundary value problems in eac of te subdomain. In tis work we provide a maximum norm analysis of an overlapping Scwarz metod on non-matcing grids for quasi-variational inequalities related to impulse control problem wit respect to te mixed boundary conditions. We can state our problem as follows: Find u H 1 ( ) solution of u f 0, u Mu 0 Mu 0, ( u f )(u Mu) = 0in, u η = ϕ in Ɣ 0 u = 0inƔ/ Ɣ 0, 481

2 482 Moamed Haiour Sala Boulaaras were is a smoot bounded domain of R 2 wit boundary Ɣ f is a regular function M is an operator given by Mu = k +inf ξ 0,x+ξ u (x + ξ) were k > 0 ξ 0 means tat ξ = (ξ 1,ξ 2,...,ξ n ) wit ξ i 0, Ɣ 0 is te part of te boundary defined by Ɣ 0 ={x = Ɣ suc tat ξ > 0, x + ξ / }. Finally, u η = u η, suc tat η is te normal vector. Te symbol (.,.) sts for te inner product in L 2 ( ), (.,.) Ɣ0 sts for te inner product in L 2 (Ɣ 0 ). On te analytical side, quasi-variational inequalities ave been extensively studied in te last tree decades (see [1, 2, 7 11]) for te numerical approximations computational aspects we ave a few results (see [3 6, 12]). In te present paper, we provide a new approac for te finite element approximation of an overlapping Scwarz metod on non-matcing grids for te quasi-variational inequalities related to impulse control problem. We consider a domain wic is te union of two overlapping subdomains, were eac sub-domain as its own generated triangulation. Te grid points on te subdomain boundaries need not matc te grid points from te oter sub-domains. Under a discrete maximum principle [10], we sow tat te discretization on eac sub-domain converges quasi-optimally in te L -norm. Te paper is organized as follows: In 2, we state te continuous alternating Scwarz sequence for quasi-variational inequalities define teir respective finite element counterparts in te context of overlapping grids. In 3, we provide te error analysis of te overlapping domain decomposition metods, were simple proofs to main fundamental teorems are proved. Ten te geometrical convergence of te problem is establised. Furtermore, an error estimate for eac sub-domain is derived in a uniform norm. 2. Te Scwarz metod for te elliptic quasi-variational inequalities. In tis section, we introduce some definitions some classical results related to quasivariational inequalities. 2.1 Elliptic quasi-variational inequalities Let be a convex domain in R 2 wit sufficiently smoot boundary. We consider te following obstacle problem u f 0, u H 1 ( ), in, u Mu 0, Mu 0, Mu = k + inf ξ 0,x+ξ u (x + ξ), (2.1) u η = ϕ in Ɣ 0 u = 0inƔ/ Ɣ 0. We are also given te rigt- side f suc tat g L ( ).

3 Quasi-variational inequalities related to impulse control problem 483 M is an operator given by Mu = k + inf ξ 0,x+ξ u (x + ξ). (2.2) Applying Green s formula, (2.1) can be transformed to te following elliptic variational inequalities a (u,v u) ( f, (v u)) (ϕ,(v u)) Ɣ0 0, on, v V u Mu 0 Mu 0, Mu = k + inf u (x + ξ), ξ 0,x+ξ u η = ϕ in Ɣ 0 u = 0inƔ/ Ɣ 0,. (2.3) were V is te non empty convex set defined by { V = v H 1 ( ) : v } η = ϕ in Ɣ 0,v Mu on v = 0inƔ/ Ɣ 0, (2.4) were ϕ is a regular function defined on Ɣ 0. Tus, it can be easily deduce tat a (u,v) = u vdx = ( u, v), ( f,v) = f vdx (ϕ,v) Ɣ0 = ϕ vdσ. Ɣ 0 Teorem 1 [7]. Under te previous assumptions, te problem (2.1) as an unique solution. Moreover it satisfies: u W 2,p, 2 p. Teorem 2 [7]. If ϕ W 2, ( ) Mϕ W 2, loc ( ) (resp. ϕ is locally semi-concave). Ten u W 1, loc ( ) (resp. u W 2, loc ( )). Let V be te space of finite elements consisting of continuous piecewise linear functions K ϕ te non-empty discrete convex set associated to K ϕ defined as K ϕ ={u V : u =π ϕ on Ɣ 0, u r Mu on u =0 inɣ/ Ɣ 0 }, (2.5)

4 484 Moamed Haiour Sala Boulaaras were π is an interpolation operator on Ɣ 0, r is te usual finite element restriction operator on. Te discrete counterpart of (2.3) consists of finding u K ϕ suc tat a (u,v u ) ( f, (v u )) (ϕ,(v u )) Ɣ0 0, Mu = k + inf ξ 0,x+ξ u r Mu 0, u (x + ξ), (2.6) u η = ϕ in Ɣ 0 u = 0inƔ/ Ɣ 0. Te lemma below establises a monotonicity property of te solution of (2.3) wit respect to te obstacle defined as an impulse control problem. Lemma 1[7]. If we ave u ũ in te K ϕ, ten Mu Mũ we ave u K ϕ,λ R, M (u + λ) = Mu + λ. (2.7) Remark 1. Under te previous ypoteses we ave te following inequality Mu Mũ L ( ) u ũ L ( ), u, ũ K ϕ. (2.8) Proof. We ave u ũ + u ũ L ( ). Using te Lemma 1, we get Mu M ( ũ + u ũ L ( )) = Mũ + u ũ L ( ), tus Mu Mũ u ũ L ( ). Similarly, intercanging te roles of Mu Mũ, we get Mũ Mu u ũ L ( ). Hence Mu Mũ L ( ) u ũ L ( ). Notation 1. Let (Mξ,ϕ), (M ξ, ϕ) be a pair of data, ξ = σ (Mξ,ϕ), ξ = σ(m ξ, ϕ) be te corresponding solutions to te following quasi-variational inequalities (QVI): a (ξ,v ξ) ( f,v ξ) + (ϕ,(v ξ)) Ɣ0, v K ϕ

5 Quasi-variational inequalities related to impulse control problem 485 a( ξ,v ξ) ( f,v ξ) + ( ϕ,(v ξ)) Ɣ0, v K ϕ. Ten, te following comparison result olds. Lemma 2. If ϕ ϕ, ten σ (Mξ,ϕ) σ(m ξ, ϕ). Proof. Let v = min(0,ξ ξ). In te region were v is negative (v <0), we ave ξ ξ M ξ Mξ, wic means tat te obstacle is not active for u. So, for v, weave a (ξ,v) = ( f,v) + (ϕ,v) Ɣ0, (2.9) ξ + v M ξ. (2.10) So a( ξ,v) ( f,v) + (ϕ,v) Ɣ0. (2.11) Subtracting (2.9) (2.11) from eac oter, we obtain a(ξ ξ,v) 0. (2.12) But a (v, v) = a(ξ ξ,v) = a( ξ ξ,v) 0, (2.13) so v = 0 consequently, ξ ξ wic completes te proof. Te proof for te discrete case is similar. PROPOSITION 1 Under te previous ypoteses, we ave te following inequality u ũ L ( i ) Mu Mũ L ( i ) + ϕ ϕ L ( i j), suc tat i = j, i, j = 1, 2. (2.14)

6 486 Moamed Haiour Sala Boulaaras Proof. Setting we ave Hence β = Mu Mũ L ( i ) + ϕ ϕ L ( i j), suc tat i = j, i, j = 1, 2, (2.15) Mu Mũ + Mu Mũ Mũ + Mu Mũ Mũ + Mu Mũ L ( i ) Mũ + Mu Mũ L ( i ) + ϕ ϕ L ( i j). Mu Mũ + β. On te oter, we ave tat is to say, ϕ ϕ + ϕ ϕ ϕ + ϕ ϕ ϕ + ϕ ϕ L ( i j) ϕ + ϕ ϕ L ( i j) + Mu Mũ L ( i ), ϕ ϕ + β. (2.16) Since σ is increasing in L ( ), weave Hence σ (Mu,ϕ) σ (Mũ + β, ϕ + β) σ (Mũ, ϕ) + β. σ (Mu,ϕ) σ (Mũ, ϕ) β. Similarly, intercanging te roles of te couples (Mu,ϕ) (Mũ, ϕ), we get σ (Mũ, ϕ) σ (Mu,ϕ) β. Te proof for te discrete case is similar. Teorem 3 [3, 4]. Under te previous ypoteses te maximum principle [5], tere exists a constant C independent of suc tat u u L ( ) C 2 log Te continuous Scwarz sequences Let be a bounded open domain in R 2 we assume tat is smoot connected. Ten we decompose into two sub-domains 1, 2 suc tat = 1 2 (2.17)

7 Quasi-variational inequalities related to impulse control problem 487 u satisfies te local regularity condition u i W 2,p ( i ) (2.18) we denote by Ɣ =, Ɣ 1 = 1,Ɣ 2 = 2,γ 1 = 1 2, = 2 1, 1,2 = 1 2. We consider te model obstacle problem. Find u K ϕ suc tat a (u,v u) ( f, (v u)) (ϕ,(v u)) Ɣ0 0, v K ϕ, u Mu 0 Mu 0, Mu = k + inf u (x + ξ), ξ 0,x+ξ (2.19) u η = ϕ in Ɣ 0 u = 0inƔ/ Ɣ 0, were f is a given function in L ( ), we will assume in tis section tat f 0, ϕ 0. Indeed tis enables us to make suc an assumption by adding constants to u ϕ a positive function to f. We define te following process: Let u 0 K ϕ be given, we define te alternating Scwarz sequences ( +1) on 1 suc tat +1 K ϕ solves a(+1,v +1 ) ( f,(v +1 )) 1 (ϕ, (v +1 )) Ɣ0 0, v V, +1 M 1 0 Mu 0, (2.20) +1 = on 1,v= on 1 ( ) on 2 suc tat K ϕ solves a(,v ) ( f,(v )) 2 (ϕ, (v )) Ɣ0 0, v V, M 2 0, M 2 0, = 1 on 2,v= 1 on 2. (2.21) 2.3 Discretization For i = 1, 2, let τ i be a stard regular quasi-uniform finite element triangulation in i ; i ( 1 = 2 = ) being te mes size. We assume tat te two triangulations are mutually independent on 1,2 in te sense tat a triangle belonging to one triangulation does not necessarily belong to te oter.

8 488 Moamed Haiour Sala Boulaaras Let V i be te space of continuous piecewise linear functions on τ i wic vanis on i j, i = j, i, j = 1, 2. For w C ( i ), we define V i w ={v V i : v = π i (w) on i j ; v = 0inƔ/ Ɣ 0 ; i = j, i, j = 1, 2}, (2.22) were π i denotes te interpolation operator on i. We assume tat te respective matrices resulting from te discretization of problems (2.20) (2.21) are M-matrix [5]. We define te discrete counterparts of te continuous Scwarz sequences defined in (2.20) (2.21), respectively by +1 V( ) suc tat a(+1,v +1 ) ( f,(v +1 )) 1 (ϕ, (v = r M 1 )) Ɣ0 0, v V ( on 1,v = on 1, ), (2.23) V suc tat ( 1 ) a(,v ) ( f,(v )) 2 (ϕ, (v )) Ɣ 0 0, v V ( 1 ), = u2n 1 on 2,v = 1 on 2, (2.24) r M Error analysis Tis section is devoted to te proof of te main result of te present paper. We need to introduce an auxiliary sequence of discrete quasi-variational inequalities first ten prove te two fundamental teorems. For ζ 0 = u0 K ϕ, we define te sequences (ζ 2n+1 ) suc tat ζ 2n+1 V ( ) solves a(ζ 2n+1,v ζ 2n+1 ) ( f,(v ζ 2n+1 )) 1 (ϕ, (v ζ 2n+1 ζ 2n+1 ζ 2n+1 = r M 1 )) Ɣ0 0, v V ( on 1,v = on 1, ), (3.1)

9 (ζ 2n Quasi-variational inequalities related to impulse control problem 489 ) suc tat ζ 2n V ( 1 ) solves a(ζ 2n,v ζ 2n) ( f,(v ζ 2n)) 2 ζ 2n ζ 2n (ϕ, (v ζ 2n = 1 r M 2. )) Ɣ 0 0, v V ( 1 ), on 2,v = 1 on 2, (3.2) 3.1 Convergence proof via te maximum principle We introduce te sets T 2n = V ( 1 ) : u2n r M 2 f, u2n = u2n 1 on 2, on 2 = 0inƔ/ Ɣ 0 T 2n+1 = +1 V ( ) : u2n+1 f, +1 = on 1, +1 r M 1 on 1 +1 = 0inƔ/ Ɣ 0.. Lemma 3 [12]. (2.24)), ten If A is te M-matrix (resp. u2n+1 (resp. u2n+1 ) is te solution (2.23), (resp. ) is te minimal of T 2n (resp. T 2n+1 ). Teorem 4. Let u beasolutionof(2.6). Ten te iterative sequence { } (resp. {+1 }) is monotone; tat is, T 2n (resp. +1 T 2n+1 ) u +2 u0. Proof. We take u 0 = u 2 suc tat u 0 = f. We know tat if u0 r Mu ten ( u 0 f ) 2 0, i.e. ( u 0, (v u 0 )) 2 ( f, (v u 0 )) 2 (ϕ, (v u 0 )) Ɣ 0 0. Terefore u 0 T 0. From Lemma 2 we know tat u 2 is te minimal element of T 0. So u 2 r Mu 0 yields u2 u0. By induction, for index n we obtain u2n 2 u 2 u0 = u. We know tat if u 3 r Mu 1, ten ( u 3 f ) 1 0, i.e. ( u 3, (v u 3 )) 1 ( f, (v u 3 )) 1 (ϕ, (v u 0 )) Ɣ 0 0.

10 490 Moamed Haiour Sala Boulaaras Terefore u 3 T 3. From Lemma 4 we know tat u 3 is te minimal element of T 3 wic yields u 3 u1. By induction, for index n we obtain +1 1 u 1. Lemma 4. If A = (a ij ) i, j={1...n} is te M-matrix, ten tere exists two constants k 1, k 2, i.e., k 1 = sup {w (x), x } (0, 1), k 2 = sup {w (x), x γ 1 } (0, 1) suc tat sup u +1 k 1 sup u (3.3) γ 1 γ 1 sup u +1 k 2 sup u. (3.4) Proof. Setting M 1 = sup γ1 +1 u M = sup γ1 u, we may suppose tat M 1 = 0. We prove M 1 < M. (3.5) If (3.3) is not true ten tere exists x i0 γ 1 suc tat +1 (x i0 ) u(x i0 ) =M 1 M. Hence, we ave (noting a ii > 0, a ij 0fori = j because A is te M-matrix) N N 0 a ii0 (+1 u ) a ii0 0. i=1 i=1 We know by Teorem 3 tat +1 u wic implies tat a ii0 +1 u M 1 = 0. i =i 0 Terefore +1 u =M 1, if a ii0 = 0. (3.6) Since (a ij ) i, j={1...n} = A is irreducible, tere exists x i1, x i2,...,x is suc tat a i0 i 1, a i1 i 2,..., a is l k = 0. We know by (3.6) tat +1 ( ) ( ) xi1 u xi1 =M1 Similarly, we get +1 ( ) ( ) x i2 u xi2 = = u 2n+1 ( ) ( ) xis u xis ( ) ( ) xlk u xlk =M1. Hence we ave 0 = +1 N N a li +1 u a li M 1 > 0, i=1 i=1

11 Quasi-variational inequalities related to impulse control problem 491 wic is a contradiction wit (3.6). Te proof for (3.4) case is similar. Remark 2. Te demonstration of Lemma 4 is an adaptation of te one in [12] given for te problem of variational inequality. Tis lemma remains true for te problem introduced in tis paper. Te main convergence result is given by te following teorem. Teorem 5. Te sequences (+1 ), ( ), n 0 produced by te Scwarz alternating metod converge geometrically to te solution u of te obstacle problem (2.3). More precisely, tere exist k 1, k 2 (0, 1) wic depend on ( 1, ) ( 2,γ 1 ) suc tat for all n 0, sup u +1 k1 n kn 2 sup u u 0 (3.7) 1 γ 1 sup u kn 1 kn 1 2 sup u u 0. (3.8) 2 Proof. Under Lemma 4, we ave u +1 w (x) sup u. γ 1 Hence Tus sup u +1 k 1 sup u. γ 1 γ 1 sup u +1 k n+1 1 k2 n sup u u 0 γ 1 γ 1 we also ave Hence tat is to say, u w (x) sup u 1. sup u k 2sup u 1, sup u kn 1 kn 2 sup u u 0. Equations (3.7) (3.8) follow from te maximum principle wic yields sup u +1 =sup u +1 =sup u for n 0 1 γ 1 γ 1

12 492 Moamed Haiour Sala Boulaaras Hence sup u +1 =sup u +1 =sup u for n 0. 2 sup u +1 k1 n kn 2 sup u u 0 1 γ 1 sup 2 u kn 1 kn 1 2 sup u u Error estimate for te quasi-variational inequalities Teorem 6. Let u be a solution of problem (2.3). Ten tere exists a constant C independent of bot n suc tat u +1 L ( 1) C2 log 3 (3.9) u L ( 2) C2 log 3. (3.10) Proof. Setting k = k 1 = k 2, using Teorems 2 4 we ave u +1 L ( 1) u u L ( 1) + u +1 L ( 1) u u L ( 1) + k2n u u 0 L (γ 1 ) C 2 log 2 + k 2n u u 0 L (γ 1 ) C 2 log 2 + k 2n ( u u L (γ 1 ) + u u 0 L (γ 1 )) C 2 log 2 + C 2 k 2n log 2 also setting k 2n log, we get u +1 L ( 1) C2 log 3. Te proof for te (3.10) case is similar.

13 Acknowledgement Quasi-variational inequalities related to impulse control problem 493 Te autors would like to tank te referee for er/is careful reading for relevant remarks wic elped tem in improving teir paper. References [1] Badea L, On te Scwarz alternating metod wit more tan two sub-domains for monotone problems, SIAM J. Numer. Anal. 28(1) (1991) [2] Bensoussan A Lions J L, Contrôle impulsionnel et Inéquations Quasi-variationnelles (Dunod) (1982) [3] Boulbracene M Saadi S, Maximum norm analysis of an overlapping nonmatcing grids metod for te obstacle problem (Hindawi Publising Corporation) (2006) pp [4] Boulbracene M Haiour M, Te finite element approximation of Hamilton Jacobi Belman equations, Comput. Mat. Appl. 41 (2001) [5] Ciarlet P Raviart P, Maximum principle uniform convergence for te finite element metod. Commun. Mat. Appl. Mec. Eng. 2 (1973) 1 20 [6] Haiour M Hadidi E, Uniform convergence of Scwarz metod for noncoercive variational inequalities. Int. J. Contemp. Mat. Sci. 4(29) (2009) [7] Kuznetsov Yu A, Neitaanmaki P Tarvainen P, Scwarz metods for obstacl problems wit convection-diffusion operator, Domain decomposition Metods in Scientific Engineering Computing (University Park. Pa. 1993) (eds) D E Keyes J Xu. Contemp. Mat. (Rode Isl: American Matematical Society) (1994) vol. 180, pp [8] Lions P L Pertame B, Une remarque sur les opérateurs non linéaires intervenant dans les inéquations quasi-variationnelles. Annales de la faculté des sciences de Toulouse 5 e serie, tome 5, n (1983) pp [9] Lions P L, On te Scwarz alternating metod İ, First International Symposium on Domain Decomposition Metods for Partial Differential Equations (Paris, 1987) (SIAM, Piladelpia) (1988) pp [10] Lions P L, On te Scwarz alternating metod İİ, Stocastic interpretation order properties, Domain Decomposition Metods (Los Angeles, California, 1988) (SIAM, Piladelpia) (1989) pp [11] Pertame B, Some remarks on quasi-variational inequalities te associated impulsive control problem, Annales de l I. H. P. Section C. Tome 2. n 0 3 (1985) pp [12] Zeng Jinping Zou Suzi, Scwarz algoritm for te solution of variational inequalities wit nonlinear source terms, Appl. Mat. Comput. 97 (1998) 23 35