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

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Domain Decomposition Methods in Science and Engineering

Part of the book series: Lecture Notes in Computational Science and Engineering ((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.

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References

  • Z. Dostál. On preconditioning and penalized matrices. Num. Lin. Alg. Appl., 6:109–114, 1999.

    Article  MATH  Google Scholar 

  • 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 

  • 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.

    Article  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Dostál, F. Gomes, and S. Santos. Solution of contact problems by FETI domain decomp. Comput. Meth. Appl. Mech. Eng., 190:1611–1627, 2000b.

    Article  MATH  Google Scholar 

  • Z. Dostál and D. Horák. Scalability and FETI based algorithm for large discretized variational inequalities. Math. and Comput. in Simul., 61:347–357, 2003a.

    Article  MATH  Google Scholar 

  • 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 

  • 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 

  • 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 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • I. Hlaváček, J. Haslinger, J. Nečas, and J. Lovíšek. Solution of Variational Inequalities in Mechanics. Springer Verlag Berlin, 1988.

    Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • R. Kornhuber. Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems. Teubner, Stuttgart, 1997.

    MATH  Google Scholar 

  • R. Kornhuber and R. Krause. Adaptive multigrid methods for Signorini's problem in linear elasticity. Comput. Visualiz. in Science, 4:9–20, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • J. Mandel and R. Tezaur. Convergence of a Substructuring Method with Lagrange Multipliers. Numer. Math., 73:473–487, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Schoeberl. Efficient contact solvers based on domain decomposition techniques. Comput. and Math., 42:1217–1228, 1998a.

    Google Scholar 

  • J. Schoeberl. Solving the Signorini problem on the basis of domain decomposition techniques. Part. Diff. Eqs. in Physics and Biology, 60:323–344, 1998b.

    MATH  Google Scholar 

  • 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 

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Author information

Authors and Affiliations

  1. Applied Mathematics Tr17.listopadu, VŠB-Technical University Ostrava, CZ-70833, Ostrava, CzechRepublic

    Zdeněk Dostál & David Horák

Authors
  1. Zdeněk Dostál
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  2. David Horák
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Editor information

Editors and Affiliations

  1. Moffett Field, CA

    Timothy J. Barth

  2. Bonn

    Michael Griebel

  3. New York

    David E. Keyes  & Tamar Schlick  & 

  4. Espoo

    Risto M. Nieminen

  5. Leuven

    Dirk Roose

  6. Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 2-6, 14195, Berlin, Germany

    Ralf Kornhuber

  7. Department of Mathematics, University of Houston, Philip G. Hoffman Hall 651, 77204-3008, Houston, TX, USA

    Ronald Hoppe

  8. Dassault Aviation, 78 Quai Marcel Dassault Cedex 300, 92552, St. Cloud, France

    Jacques Périaux

  9. Laboratoire Jacques-Louis Lions, Université Paris VI, 175 Rue de Chevaleret, 75013, Paris, France

    Olivier Pironneau

  10. Courant Institute of Mathematical Science, New York University, Mercer Street 251, 10012, New York, USA

    Olof Widlund

  11. Department of Mathematics Eberly College of Science, Pennsylvania Sate University, McAllister Building 218, 16802, University Park, PA, USA

    Jinchao Xu

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© 2005 Springer-Verlag Berlin Heidelberg

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Dostál, Z., Horák, D. (2005). On Scalable Algorithms for Numerical Solution of Variational Inequalities Based on FETI and Semi-monotonic Augmented Lagrangians. In: Barth, T.J., et al. Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol 40. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26825-1_50

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