Advertisement

On Scalable Algorithms for Numerical Solution of Variational Inequalities Based on FETI and Semi-monotonic Augmented Lagrangians

  • Zdeněk Dostál
  • David Horák
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)

Summary

Theoretical and experimental results concerning a new FETI based algorithm for numerical solution of variational inequalities are reviewed. A discretized model problem is first reduced by the duality theory of convex optimization to the quadratic programming problem with bound and equality constraints. The latter is then optionally modified by means of orthogonal projectors to the natural coarse space introduced by Farhat and Roux in the framework of their FETI method. The resulting problem is then solved by a new variant of the augmented Lagrangian type algorithm with the inner loop for the solution of bound constrained quadratic programming problems. Recent theoretical results are reported that guarantee scalability of the algorithm. The results are confirmed by numerical experiments.

Keywords

Variational Inequality Contact Problem Domain Decomposition Multigrid Method Quadratic Programming Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Z. Dostál. On preconditioning and penalized matrices. Num. Lin. Alg. Appl., 6:109–114, 1999.MATHCrossRefGoogle Scholar
  2. Z. Dostál. Inexact semi-monotonic augmented Lagrangians with optimal feasibility convergence for quadratic programming with simple bounds and equality constraints. submitted to SIAM J. Num. Anal., 2003.Google Scholar
  3. Z. Dostál, A. Friedlander, and S. Santos. Augmented Lagrangians with adaptive precision control for quadratic programming with simple bounds and equality constraints. SIAM J. Opt., 13:1120–1140, 2003.CrossRefMATHGoogle Scholar
  4. Z. Dostál, F. Gomes, and S. Santos. Duality based domain decomp. with natural coarse space for variat. ineq. J. Comput. Appl. Math., 126:397–415, 2000a.MathSciNetCrossRefMATHGoogle Scholar
  5. Z. Dostál, F. Gomes, and S. Santos. Solution of contact problems by FETI domain decomp. Comput. Meth. Appl. Mech. Eng., 190:1611–1627, 2000b.CrossRefMATHGoogle Scholar
  6. Z. Dostál and D. Horák. Scalability and FETI based algorithm for large discretized variational inequalities. Math. and Comput. in Simul., 61:347–357, 2003a.CrossRefMATHGoogle Scholar
  7. Z. Dostál and D. Horák. Scalable FETI with optimal dual penalty for a variational inequality. to appear in Num. Lin. Alg. and Appl., 2003b.Google Scholar
  8. Z. Dostál and J. Schoeberl. Minimizing quadratic functions subject to bound constraints with the rate of convergence and finite termination. to appear in Comput. Optimiz. and Appl., 2004.Google Scholar
  9. C. Farhat and D. Dureisseix. A numerically scalable domain dec. meth. for solution of frictionless contact problems. to appear in Int. J. Num. Meth. Eng., 2002.Google Scholar
  10. C. Farhat, J. Mandel, and F. Roux. Optimal convergence properties of the FETI domain decomp. method. Comp. Meth. Appl. Mech. Eng., 115:367–388, 1994.MathSciNetCrossRefGoogle Scholar
  11. C. Farhat and F. Roux. An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems. SIAM J. Sc. Stat. Comput., 13:379–396, 1992.MathSciNetCrossRefMATHGoogle Scholar
  12. I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovíšek. Solution of Variational Inequalities in Mechanics. Springer Verlag Berlin, 1988.Google Scholar
  13. A. Klawonn and O. Widlund. FETI and Neumann-Neumann iterative substructuring methods: connections and new results. Communic. on Pure and Appl. Math., LIV:57–90, 2001.MathSciNetCrossRefGoogle Scholar
  14. R. Kornhuber. Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems. Teubner, Stuttgart, 1997.MATHGoogle Scholar
  15. R. Kornhuber and R. Krause. Adaptive multigrid methods for Signorini's problem in linear elasticity. Comput. Visualiz. in Science, 4:9–20, 2001.MathSciNetCrossRefMATHGoogle Scholar
  16. J. Mandel. Étude algébrique d'une méthode multigrille pour quelques problèmes de frontière libre. Compt. Rendus de l'Acad. des Scien., pages 469–472, 1984.Google Scholar
  17. J. Mandel and R. Tezaur. Convergence of a Substructuring Method with Lagrange Multipliers. Numer. Math., 73:473–487, 1996.MathSciNetCrossRefMATHGoogle Scholar
  18. J. Schoeberl. Efficient contact solvers based on domain decomposition techniques. Comput. and Math., 42:1217–1228, 1998a.Google Scholar
  19. J. Schoeberl. Solving the Signorini problem on the basis of domain decomposition techniques. Part. Diff. Eqs. in Physics and Biology, 60:323–344, 1998b.MATHGoogle Scholar
  20. B. Wohlmuth and R. Krause. Monotone methods on nonmatching grids for nonlinear contact problems. Technical report, 2002. Research Report No. 2002/02 of the Stuttgart University, Sonderforsungsbereich 404.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Zdeněk Dostál
    • 1
  • David Horák
    • 1
  1. 1.Applied Mathematics Tr17.listopaduVŠB-Technical University OstravaOstravaCzechRepublic

Personalised recommendations