A strongly convergent primal–dual method for nonoverlapping domain decomposition

Abstract

We propose a primal–dual parallel proximal splitting method for solving domain decomposition problems for partial differential equations. The problem is formulated via minimization of energy functions on the subdomains with coupling constraints which model various properties of the solution at the interfaces. The proposed method can handle a wide range of linear and nonlinear problems, with flexible, possibly nonlinear, transmission conditions across the interfaces. Strong convergence in the energy spaces is established in this general setting, and without any additional assumption on the energy functions or the geometry of the problem. Several examples are presented.

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References

  1. 1.

    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)

    Google Scholar 

  2. 2.

    Alotaibi, A., Combettes, P.L., Shahzad, N.: Best approximation from the Kuhn–Tucker set of composite monotone inclusions. Numer. Funct. Anal. Optim. arXiv:1401.8005. (to appear)

  3. 3.

    Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and PDE’s. J. Convex Anal. 15, 485–506 (2008)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces, 2nd edn. SIAM, Philadelphia (2014)

    Google Scholar 

  5. 5.

    Attouch, H., Cabot, A., Frankel, P., Peypouquet, J.: Alternating proximal algorithms for linearly constrained variational inequalities. Applications to domain decomposition for PDE’s. Nonlinear Anal. 74, 7455–7473 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Attouch, H., Damlamian, A.: Application des méthodes de convexité et monotonie à l’étude de certaines équations quasi-linéaires. Proc. R. Soc. Edinb. Sect. A 79, 107–129 (1977)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Attouch, H., Picard, C.: Variational inequalities with varying obstacles: the general form of the limit problem. J. Funct. Anal. 50, 329–386 (1983)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Attouch, H., Soueycatt, M.: Augmented Lagrangian and proximal alternating direction methods of multipliers in Hilbert spaces. Applications to games, PDE’s and control. Pac. J. Optim. 5, 17–37 (2009)

  9. 9.

    Badea, L.: Convergence rate of a Schwarz multilevel method for the constrained minimization of nonquadratic functionals. SIAM J. Numer. Anal. 44, 449–477 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Bank, R., Holst, M., Widlund, O., Xu, J. (eds.): Domain Decomposition Methods in Science and Engineering XX. Lect. Notes Comput. Sci. Eng., vol. 91. Springer, Berlin (2013)

  11. 11.

    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Google Scholar 

  12. 12.

    Beirão da Veiga, H., Crispo, F.: On the global regularity for nonlinear systems of the \(p\)-Laplacian type. Discret. Contin. Dyn. Syst. Ser. S 6, 1173–1191 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Bercovier, M., Gander, M.J., Kornhuber, R., Widlund, O. (eds.): Domain Decomposition Methods in Science and Engineering XVIII. Lect. Notes Comput. Sci. Eng., vol. 70. Springer, Berlin (2009)

  14. 14.

    Boţ, R.I., Hendrich, C.: A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators. SIAM J. Optim. 23, 2541–2565 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Brézis, H.: Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations. In: Zarantonello, E. (ed.) Contributions to Nonlinear Functional Analysis. Academic Press, New York (1971)

    Google Scholar 

  16. 16.

    Briceño-Arias, L.M., Combettes, P.L.: A monotone \(+\) skew splitting model for composite monotone inclusions in duality. SIAM J. Optim. 21, 1230–1250 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Chan, T.F., Glowinski, R. (eds.): Proceedings of Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, Houston (1989). [SIAM, Philadelphia (1990)]

  18. 18.

    Chan, T.F., Mathew, T.P.: Domain decomposition algorithms. Acta Numer. 3, 61–143 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Combettes, P.L.: Strong convergence of block-iterative outer approximation methods for convex optimization. SIAM J. Control Optim. 38, 538–565 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Combettes, P.L.: Systems of structured monotone inclusions: duality, algorithms, and applications. SIAM J. Optim. 23, 2420–2447 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Combettes, P.L., Pesquet, J.-C.: Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued Var. Anal. 20, 307–330 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Condat, L.: A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms. J. Optim. Theory Appl. 158, 460–479 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Drábek, P., Milota, J.: Methods of Nonlinear Analysis—Applications to Differential Equations. Birkhäuser, Basel (2007)

    Google Scholar 

  24. 24.

    Ekeland, I., Temam, R.: Analyse Convexe et Problèmes Variationnels. Dunod, Paris (1974). [English translation: Convex Analysis and Variational Problems. SIAM, Philadelphia (1999)]

  25. 25.

    Fornasier, M., Langer, A., Schönlieb, C.-B.: A convergent overlapping domain decomposition method for total variation minimization. Inverse Probl. 116, 645–685 (2010)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Fornasier, M., Schönlieb, C.-B.: Subspace correction methods for total variation and \(\ell _1\)-minimization. SIAM J. Numer. Anal. 47, 3397–3428 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Frehse, J.: On the regularity of the solution of a second order variational inequality. Boll. Un. Mat. Ital. 4, 312–315 (1972)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)

    Google Scholar 

  29. 29.

    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)

    Google Scholar 

  30. 30.

    Haugazeau, Y.: Sur les Inéquations Variationnelles et la Minimisation de Fonctionnelles Convexes. Thèse, Université de Paris, Paris (1968)

  31. 31.

    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

    Google Scholar 

  32. 32.

    Le Tallec, P.: Domain decomposition methods in computational mechanics. Comput. Mech. Adv. 1, 121–220 (1994)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Lions, P.-L.: On the Schwarz alternating method-III. A variant for nonoverlapping subdomains. In: Chan, T.F., Glowinski, R., Periaux, J., Widlund, O.B. (eds.): Third international symposium on domain decomposition methods for partial differential equations. SIAM, Philadelphia, pp. 202–223 (1989)

  35. 35.

    Liu, W.B., Barrett, J.W.: A remark on the regularity of the solutions of the \(p\)-Laplacian and its application to their finite element approximation. J. Math. Anal. Appl. 178, 470–487 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Moreau, J.J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. C. R. Acad. Sci. Paris Sér. A 255, 2897–2899 (1962)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Nečas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques. Masson, Paris (1967)

    Google Scholar 

  38. 38.

    Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, New York (1999)

    Google Scholar 

  39. 39.

    Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Toselli, A., Widlund, O.: Domain Decomposition Methods-Algorithms and Theory. Springer, Berlin (2005)

    Google Scholar 

  41. 41.

    Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38, 667–681 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)

    Google Scholar 

  43. 43.

    Zeidler, E.: Nonlinear Functional Analysis and its Applications II/A-Linear Monotone Operators. Springer, New York (1990)

    Google Scholar 

  44. 44.

    Zeidler, E.: Nonlinear Functional Analysis and its Applications II/B-Nonlinear Monotone Operators. Springer, New York (1990)

    Google Scholar 

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Correspondence to Patrick L. Combettes.

Additional information

The work of H. Attouch was supported by ECOS under Grant C13E03, and by Air Force Office of Scientific Research, USAF, under Grant FA9550-14-1-0056. The work of L. M. Briceño-Arias and P. L. Combettes was supported by MathAmSud under Grant N13MATH01. L. M. Briceño-Arias was also supported by Conicyt under Grants Fondecyt 3120054 and 11140360, and under Grant Anillo ACT1106.

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Attouch, H., Briceño-Arias, L.M. & Combettes, P.L. A strongly convergent primal–dual method for nonoverlapping domain decomposition. Numer. Math. 133, 443–470 (2016). https://doi.org/10.1007/s00211-015-0751-4

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Mathematics Subject Classification

  • 49M29
  • 49M27
  • 65K10
  • 65N55