Restricted Additive Schwarz Method for Some Inequalities Perturbed by a Lipschitz Operator

  • Lori BadeaEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)


We introduce and analyze a restricted additive Schwarz method for some inequalities perturbed by a Lipschitz operator. An existence and uniqueness result concerning the solution of the inequalities we consider is given. Also, we introduce the method as a subspace correction algorithm, prove the convergence and estimate the error in a general framework of a finite dimensional Hilbert space. By introducing the finite element spaces, we get that our algorithm is really a restricted additive Schwarz method and conclude that the convergence condition and convergence rate are independent of the mesh parameters and of both the number of subdomains and the parameters of the domain decomposition, but the convergence condition is a little more restrictive than the existence and uniqueness condition of the solution.


  1. 1.
    L. Badea, Additive Schwarz method for the constrained minimization of functionals in reflexive banach spaces, in Domain Decomposition Methods in Science and Engineering XVII, ed. by U. Langer et al. Lecture Notes in Computational Science and Engineering, vol. 60 (Springer, Berlin, 2008), pp. 427–434Google Scholar
  2. 2.
    L. Badea, Schwarz methods for inequalities with contraction operators. J. Comput. Appl. Math. 215(1), 196–219 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    L. Badea, Convergence rate of some hybrid multigrid methods for variational inequalities. J. Numer. Math. 23, 195–210 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    X.-C. Cai, M. Sarkis, A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21, 792–797 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    X.-C. Cai, C. Farhat, M. Sarkis, A minimum overlap restricted additive Schwarz preconditioner and applications to 3D flow simulations, in Contemporary Mathematics, vol. 218 (American Mathematical Society, Providence, 1998), pp. 479–485zbMATHGoogle Scholar
  6. 6.
    V. Dolean, P. Jolivet, F. Nataf, An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation (SIAM, Philadelphia, 2015)CrossRefGoogle Scholar
  7. 7.
    E. Efstathiou, M.J. Gander, Why restricted additive Schwarz converges faster than additive Schwarz. BIT Numer. Math. 43, 945–959 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Frommer, D.B. Szyld, An algebraic convergence theory for restricted additive Schwarz methods using weighted max norms. SIAM J. Numer. Anal. 39, 463–479 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Frommer, R. Nabben, D.B. Szyld, An algebraic convergence theory for restricted additive and multiplicative Schwarz methods, in Domain Decomposition Methods in Science and Engineering, ed. by N. Debit et al. (CIMNE, UPS, Barcelona, 2002), pp. 371–377zbMATHGoogle Scholar
  10. 10.
    R. Glowinski, J.L. Lions, R. Trémolières, Numerical Analysis of Variational Inequalities, ed. by J.L. Lions, G. Papanicolau, R.T. Rockafellar. Studies in Mathematics and its Applications, vol. 8 (North-Holland, Amsterdam, 1981)Google Scholar
  11. 11.
    R. Nabben, D.B. Szyld, Convergence theory of restricted multiplicative Schwarz methods. SIAM J. Numer. Anal.. 40(6), 2318–2336 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    H.-R. Xu, K.-K. Huang, S.-L. Xie, Restricted additive Schwarz method for nonlinear complementarity problem with an m-function. Commun. Comput. Inform. Sci. 158, 46–50 (2011)CrossRefGoogle Scholar
  13. 13.
    H. Xu, K. Huang, S. Xie, Z. Sun, Restricted additive Schwarz method for a kind of nonlinear complementarity problem. J. Comput. Math. 32(5), 547–559 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    H. Yang, Q. Li, Overlapping restricted additive Schwarz method applied to the linear complementarity problem with an h-matrix. Comput. Optim. Appl. 51, 223–239 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    L.-T. Zhang, T.-X. Gu, X.-P. Liu, Overlapping restricted additive Schwarz method with damping factor for h-matrix linear complementarity problem. Appl. Math. Comput. 271, 1–10 (2015)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics of the Romanian Academy, Francophone Center for Mathematics in BucharestBucharestRomania

Personalised recommendations