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TFETI for Scalar Problems

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Scalable Algorithms for Contact Problems

Part of the book series: Advances in Mechanics and Mathematics ((AMMA,volume 36))

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Abstract

We shall first illustrate the ideas of scalable domain decomposition algorithms for contact problems by describing the solution of two scalar problems governed by elliptic boundary variational inequalities.

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References

  1. Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  2. Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J.: Solution of Variational Inequalities in Mechanics. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  3. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer Series in Computational Physics. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  4. Dostál, Z., Gomes, F.A.M., Santos, S.A.: Solution of contact problems by FETI domain decomposition with natural coarse space projection. Comput. Methods Appl. Mech. Eng. 190(13–14), 1611–1627 (2000)

    Article  MATH  Google Scholar 

  5. Bochev, P.: On the finite element solution of the pure neumann problem. SIAM Rev. 47(1), 50–66 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dostál, Z., Gomes, F.A.M., Santos, S.A.: Duality based domain decomposition with natural coarse space for variational inequalities. J. Comput. Appl. Math. 126(1–2), 397–415 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bramble, J.H., Pasciak, J.E., Schatz, A.H.: The construction of preconditioners for elliptic problems by substructuring I. Math. Comput. 47, 103–134 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Pechstein, C.: Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems. Sringer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  9. Farhat, C., Mandel, J., Roux, F.-X.: Optimal convergence properties of the FETI domain decomposition method. Comput. Methods Appl. Mech. Eng. 115, 365–385 (1994)

    Article  MathSciNet  Google Scholar 

  10. Glowinski, R.: Variational Inequalities. Springer, Berlin (1980)

    MATH  Google Scholar 

  11. Glowinski, R., Lions, J.L., Tremolieres, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)

    MATH  Google Scholar 

  12. Sofonea, M., Matei, A.C.: Variational Inequalities with Applications. A Study of Antiplane Frictional Contact Problems. Springer, New York (2009)

    MATH  Google Scholar 

  13. Migorski, S., Ochal, A., Sofonea, M.: Modeling and analysis of an antiplane piezoelectric contact problem. Math. Models Methods Appl. Sci. 19, 1295–1324 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hackbusch, W., Mittelmann, H.: On multi-grid methods for variational inequalities. Numerische Mathematik 42, 65–76 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hoppe, R.H.W.: Multigrid algorithms for variational inequalities. SIAM J. Numer. Anal. 24(5), 1046–1065 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mandel, J.: Étude algébrique d’une méthode multigrille pour quelques problèmes de frontière libre. Comptes Rendus de l’Académie des Sciences Series I(298), 469–472 (1984)

    MATH  Google Scholar 

  17. Kornhuber, R.: Monotone Multigrid Methods for Nonlinear Variational Problems. Teubner, Stuttgart (1997)

    MATH  Google Scholar 

  18. Gräser, C., Kornhuber, R.: Multigrid methods for obstacle problems. J. Comput. Math. 27(1), 1–44 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Farhat, C., Roux, F.-X.: A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32, 1205–1227 (1991)

    Article  MATH  Google Scholar 

  20. Farhat, C., Roux, F.-X.: An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems. SIAM J. Sci. Comput. 13, 379–396 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dostál, Z., Horák, D., Kučera, R.: Total FETI - an easier implementable variant of the FETI method for numerical solution of elliptic PDE. Commun. Numer. Methods Eng. 22, 1155–1162 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Of, G.: BETI - Gebietszerlegungsmethoden mit schnellen Randelementverfahren und Anwendungen. Ph.D. Thesis, University of Stuttgart (2006)

    Google Scholar 

  23. Of, G., Steinbach, O.: The all-floating boundary element tearing and interconnecting method. J. Numer. Math. 17(4), 277–298 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Park, K.C., Felippa, C.A., Gumaste, U.A.: A localized version of the method of Lagrange multipliers. Comput. Mech. 24, 476–490 (2000)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  26. Simo, J.C., Laursen, T.A.: An augmented Lagrangian treatment of contact problems involving friction. Comput. Struct. 42, 97–116 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dostál, Z., Friedlander, A., Santos, S.A.: Solution of contact problems of elasticity by FETI domain decomposition. Domain decomposition methods 10. AMS, Providence. Contemp. Math. 218, 82–93 (1998)

    Article  MATH  Google Scholar 

  28. Dostál, Z., Horák, D.: Scalable FETI with optimal dual penalty for a variational inequality. Numer. Linear Algebra Appl. 11(5–6), 455–472 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dostál, Z., Horák, D., Stefanica, D.: A scalable FETI-DP algorithm for a coercive variational inequality. IMACS J. Appl. Numer. Math. 54(3–4), 378–390 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dostál, Z., Horák, D., Stefanica, D.: A scalable FETI-DP algorithm for semi-coercive variational inequalities. Comput. Methods Appl. Mech. Eng. 196(8), 1369–1379 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Dostál, Z., Horák, D., Stefanica, D.: A scalable FETI-DP algorithm with non-penetration mortar conditions on contact interface. J. Comput. Appl. Math. 231(2), 577–591 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Dostál, Z.: Box constrained quadratic programming with proportioning and projections. SIAM J. Optim. 7(3), 871–887 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Dostál, Z., Horák, D.: Scalability and FETI based algorithm for large discretized variational inequalities. Math. Comput. Simul. 61(3–6), 347–357 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dostál, Z., Horák, D.: Theoretically supported scalable FETI for numerical solution of variational inequalities. SIAM J. Numer. Anal. 45(2), 500–513 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Badea, L., Tai, X.C., Wang, J.: Convergence rate analysis of a multiplicative Schwarz method for variational inequalities. SIAM J. Numer. Anal. 41(3), 1052–1073 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lee, J.: Two domain decomposition methods for auxiliary linear problems for a multibody variational inequality. SIAM J. Sci. Computi. 35(3), 1350–1375 (2013)

    Article  MathSciNet  Google Scholar 

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Correspondence to Zdeněk Dostál .

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Dostál, Z., Kozubek, T. (2016). TFETI for Scalar Problems. In: Scalable Algorithms for Contact Problems. Advances in Mechanics and Mathematics, vol 36. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6834-3_10

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