Numerical Algorithms

, Volume 75, Issue 4, pp 879–908 | Cite as

Grid solution of problem with unilateral constraints

  • M. Chau
  • A. Laouar
  • T. Garcia
  • P. SpiteriEmail author
Original Paper


The present study deals with the solution of a problem, defined in a three-dimensional domain, arising in fluid mechanics. Such problem is modelled with unilateral constraints on the boundary. Then, the problem to solve consists in minimizing a functional in a closed convex set. The characterization of the solution leads to solve a time-dependent variational inequality. An implicit scheme is used for the discretization of the time-dependent part of the operator and so we have to solve a sequence of stationary elliptic problems. For the solution of each stationary problem, an equivalent form of a minimization problem is formulated as the solution of a multivalued equation, obtained by the perturbation of the previous stationary elliptic operator by a diagonal monotone maximal multivalued operator. The spatial discretization of such problem by appropriate scheme leads to the solution of large scale algebraic systems. According to the size of these systems, parallel iterative asynchronous and synchronous methods are carried out on distributed architectures; in the present study, methods without and with overlapping like Schwarz alternating methods are considered. The convergence of the parallel iterative algorithms is analysed by contraction approaches. Finally, the parallel experiments are presented.


Variational inequality Parallel iterative algorithms Asynchronous iterations Unilateral constraints problem Grid computing Fluid mechanics 

Mathematics Subject Classification (2010)

68U10 65F10 65Y05 65N22 65C20 65Y20 65M12 


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  1. 1.
    Axelson, O., Barker, V.A.: Finite element solution of boundary value problems. Theory and computation. Academic press (1984)Google Scholar
  2. 2.
    Badea, L., Tai, X.C., Wang, J.: Convergence rate analysis of a multiplicative Schwarz method for variational inequalities. SIAM J. on Numer. Anal. 41(3), 1052–1073 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Badea, L., Wang, J.: An additive Schwarz method for variational inequalities. Math. Comp. 69, 1341–1354 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bai, Z.Z., Evans, D.J.: Matrix multisplitting methods with applications to linear complementary problems: parallel asynchronous methods. Intern J. Computer Math. 79(2), 205–232 (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bai, Z.Z., Evans, D.J.: Chaotic iterative methods for the linear complementary problems: parallel asynchronous methods. J. Comput. Appl. Math. 96, 127–138 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barbu, V.: Nonlinear semigroups and differential equations in Banach spaces. Noordhoff International Publishing (1976)Google Scholar
  7. 7.
    Baudet, G.: Asynchronous iterative methods for multiprocessors. J. Assoc. Comput. Mach. 25, 226–244 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bertsekas, D.P., Tsitsiklis, J.: Parallel and Distributed Computation. Numerical Methods. Athena Scientific, Englewood Cliffs (1997)zbMATHGoogle Scholar
  9. 9.
    Bolze, R., Cappello, F., Caron, E., Daydé, M., Desprez, F., Jeannot, E., Jégou, Y., Lanteri, S., Leduc, J., Melab, N., Mornet, G., Namyst, R., Primet, P., Quetier, B., Richard, O., Talbi, E.-G., Touche, I.: GRID 5000: A large scale and highly reconfigurable experimental grid testbed. International Journal of High Performance Computing Applications 20(4), 481–494 (2006)CrossRefGoogle Scholar
  10. 10.
    Chandy, K.M., Lamport, L.: Distributed snapshots : determining global states of distributed systems. ACM Trans. Comput. Syst. 3(1), 63–75 (1985)CrossRefGoogle Scholar
  11. 11.
    Chau, M., El Baz, D., Guivarch, R., Spitéri, P.: MPI Implementation of parallel subdomain methods for linear and nonlinear convection-diffusion problems. J. Parallel Distrib. Comput. 67, 581–591 (2007)CrossRefzbMATHGoogle Scholar
  12. 12.
    Chau, M., Couturier, R., Bahi, J., Spiteri, P.: Parallel solution of the obstacle problem in grid environment. International Journal of High Performance Computing Applications 25(4), 488–495 (2011)CrossRefGoogle Scholar
  13. 13.
    Chau, M., Couturier, R., Bahi, J., Spiteri, P.: Asynchronous grid computation for American options derivative. Adv. Eng. Softw. 60-61, 136–144 (2013)CrossRefGoogle Scholar
  14. 14.
    Chau, M., Garcia, T., Spiteri, P.: Asynchronous Schwarz methods applied to constrained mechanical structures in grid environment. Adv. Eng. Softw. 74, 1–15 (2014)CrossRefGoogle Scholar
  15. 15.
    Chazan, D., Miranker, W.: Chaotic relaxation. Linear Algebra Appl. 2, 199–222 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Comte, P., Miellou, J.C., Spiteri, P.: La notion de h-accrtivité, applications. CRAS 283, 655–658 (1976)zbMATHGoogle Scholar
  17. 17.
    Dahlquist, G.: On matrix majorants and minorants, with applications to differential equations. Linear Algebra Appl. 52/53, 199–216 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Duvaut, G., Lions, J.L.: Les inéquations en mécanique. Dunod (1972)Google Scholar
  19. 19.
    El Tarazi, M.: Some convergence results for asynchronous algorithms. Numer. Math. 39, 325–340 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Evans, D.J., Deren, W.: An asynchronous parallel algorithm for solving a class of nonlinear simultaneous equations. Parallel Comput. 17, 165–180 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Feingold, D., Varga, R.S.: Block diagonally dominant matrices and generalizations of the Gerschgorin theorem, Pacific. J. Math. 12, 1241–1250 (1963)zbMATHGoogle Scholar
  22. 22.
    Fiedler, M., Ptak, V.: On matrices with non-positive off-diagonal elements and positive principal minors. Math. Jounal 12(87), 382–400 (1962)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Giraud, L., Spiteri, P.: Résolution parallèle de problèmes aux limites non linéaires. M2 AN 25, 579–606 (1991)zbMATHGoogle Scholar
  24. 24.
    Glowinski, R., Lions, J.L., Tremolieres, R.: Analyse numérique des inéquations variationnelles, Dunod, tome 1 and 2 (1976)Google Scholar
  25. 25.
    Hoffman, K.H., Zou, J.: Parallel efficiency of domain decomposition methods. Parallel Comput. 19, 1375–1391 (1993)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kuznetsov, Y.A., Neittaanmaki, P., Tarvainen, P.: Block relaxation methods for algebraic obstacle problems with M-matrices. East-West J. Numer. Math. 4, 69–82 (1996)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kuznetsov, Y.A., Neittaanmaki, P., Tarvainen, P.: Schwarz methods for obstacle problems with convection-diffusion operators. In: Keyes, D.E., Xu, J.C. (eds.) Proceedings of Domain Decomposition Methods in Scientifical and Engineering Computing, pp 251–256. AMS (1995)Google Scholar
  28. 28.
    Li, C., Zeng, J., Zhou, S.: Convergence analysis of generalized Schwarz algorithms for solving obstacle problems with T-monotone operator. Computers & Mathematics with Applications 48(3-4), 373–386 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lions, J.L.: Sur les problémes unilatéraux, Séminaire N. Bourbaki 350, 55–77 (1969)Google Scholar
  30. 30.
    Lions, J.L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. XX, 493–519 (1967)CrossRefzbMATHGoogle Scholar
  31. 31.
    Lions, P.L., Mercier, B.: Approximation numérique des équations de Hamilton-Jacobi-Bellman, R.A.I.R.O Analyse numérique 14, 369–393 (1980)Google Scholar
  32. 32.
    Miellou, J.C.: Méthodes de Jacobi, Gauss-Seidel, sur-relaxation par blocs, appliquées à une classe de problèmes non linéaires. CRAS 273, 1257–1260 (1971)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Miellou, J.C.: Sur une variante de la méthode de relaxation, appliquée à des problèmes comportant un opérateur somme d’un opérateur différentiable et d’un opérateur maximal monotone diagonal. CRAS 275, 1107–1110 (1972)zbMATHGoogle Scholar
  34. 34.
    Miellou, J.C.: Algorithmes de relaxation chaotique à retards. RAIRO Analyse numé,rique R1, 55–82 (1975)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Miellou, J.C., El Baz, D., Spiteri, P.: A new class of asynchronous iterative algorithms with order interval. Math. Comput. 67(221), 237–255 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Miellou, J.C., Spiteri, P.: Un critère de convergence pour des méthodes générales de point fixe. M2AN 19, 645–669 (1985)CrossRefzbMATHGoogle Scholar
  37. 37.
    Miellou, J.C., Spiteri, P., El Baz, D.: Stopping criteria, forward and backward errors for perturbed asynchronous linear fixed point methods in finite precision. IMA J. Numer. Anal. 25, 429–442 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)zbMATHGoogle Scholar
  39. 39.
    Ostrowski, A.M.: On some metrical properties of operator matrices and matrices partitioned into blocks. J. Math. Anal. Appl. 2, 161–209 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Robert, F.: Recherche d’une M-matrice parmi les minorantes d’un opérateur linéaire. Numer. Math. 9, 189–199 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Schröder, J.: Nichtlineare Majoranten beim Verfahren der Schrittweisen Näherung. Arch. Math. (Bused) 7, 471–484 (1956)CrossRefzbMATHGoogle Scholar
  42. 42.
    Spiteri, P., Miellou, J.C., El Baz, D.: Parallel asynchronous Schwarz and multisplitting methods for a nonlinear diffusion problem. Numerical Algorithms 33, 461–474 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Spiteri, P., Miellou, J.C., El Baz, D.: Asynchronous Schwarz alternating method with flexible communications for the obstacle problem, réseaux et systèmes répartis - Calculateurs parallèles. Hermes 13(1), 47–66 (2001)Google Scholar
  44. 44.
    Varga, R.: Matrix Iterative Analysis. Prentice Hall (1962)Google Scholar
  45. 45.
    Ziane-Khodja, L., Chau, M., Couturier, R., Bahi, J., Spiteri, P.: Parallel solution of American option derivatives on GPU clusters. Computers and Mathematics with Applications 65(11), 1830–1848 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Méthodes de décomposition, Méthodes numériques d’analyse de systèmes, tome 2, cahier de l’IRIA 11 (1972). (see

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.IRT SystemXPalaiseauFrance
  2. 2.Faculté des Sciences, Département de MathématiquesUniversité d’Annaba, Laboratoire LANOSAnnabaAlgérie
  3. 3.IRIT - ENSEEIHTToulouseFrance
  4. 4.UVSQ-PRISMVersailles CedexFrance

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