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A three-level BDDC algorithm for a saddle point problem

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Abstract

BDDC algorithms have previously been extended to the saddle point problems arising from mixed formulations of elliptic and incompressible Stokes problems. In these two-level BDDC algorithms, all iterates are required to be in a benign space, a subspace in which the preconditioned operators are positive definite. This requirement can lead to large coarse problems, which have to be generated and factored by a direct solver at the beginning of the computation and they can ultimately become a bottleneck. An additional level is introduced in this paper to solve the coarse problem approximately and to remove this difficulty. This three-level BDDC algorithm keeps all iterates in the benign space and the conjugate gradient methods can therefore be used to accelerate the convergence. This work is an extension of the three-level BDDC methods for standard finite element discretization of elliptic problems and the same rate of convergence is obtained for the mixed formulation of the same problems. Estimate of the condition number for this three-level BDDC methods is provided and numerical experiments are discussed.

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References

  1. Börgers C.: The Neumann–Dirichlet domain decomposition method with inexact solvers on the subdomains. Numer. Math. 55, 123–136 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bramble, J.H., Pasciak, J.E., Vassilev, A.: Non-overlapping domain decomposition preconditioners with inexact solves. In: Domain Decomposition Methods in Sciences and Engineering, vol. IX. DDM.org, Augsburg, London pp. 40–52 (1998)

  3. Brenner S.C., Sung L.-Y.: Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions. Math. Comp. 69, 1319–1339 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brenner S.C., Sung L.-Y.: BDDC and FETI-DP without matrices or vectors. Comput. Methods Appl. Mech. Eng. 196(8), 1429–1435 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezzi F., Fortin M.: Mixed and hybrid finite element. Springer Series in Computational Mathematics, vol. 15. Springer, Berlin (1991)

    Google Scholar 

  6. Dohrmann C.R.: A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25(1), 246–258 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dohrmann, C.R.: A substructuring preconditioner for nearly incompressible elasticity problems. Technical Report SAND2004-5393, Sandia National Laboratories, Albuquerque, New Mexico (2004)

    Google Scholar 

  8. Dohrmann C.R.: An approximate BDDC preconditioner. Numer. Linear Algebra Appl. 14(2), 149–168 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Goldfeld P., Pavarino L.F., Widlund O.B.: Balancing Neumann–Neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity. Numer. Math. 95(2), 283–324 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Haase G., Langer U., Meyer A.: The approximate Dirichlet domain decomposition method. I. An algebraic approach. Computing 47(2), 137–151 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Haase G., Langer U., Meyer A.: The approximate Dirichlet domain decomposition method. II. Applications to 2nd-order elliptic BVPs. Computing 47(2), 153–167 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hwang F.-N., Cai X.-C.: Parallel fully coupled Schwarz preconditioners for saddle point problems. Electron. Trans. Numer. Anal. 22, 146–162 (2006) (electronic)

    MathSciNet  MATH  Google Scholar 

  13. Kim H.H., Tu X.: A three-level BDDC algorithm for mortar discretization. SIAM J. Numer. Anal. 47, 1576–1600 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Klawonn A., Rheinbach O.: Inexact FETI–DP methods. Int. J. Numer. Methods Eng. 69(2), 284–307 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Klawonn A., Widlund O.B.: A domain decomposition method with Lagrange multipliers and inexact solvers for linear elasticity. SIAM J. Sci. Comput. 22(4), 1199–1219 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Klawonn A., Widlund O.B.: Dual-primal FETI methods for linear elasticity. Comm. Pure Appl. Math. 59(11), 1523–1572 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li J.: A dual-Primal FETI method for incompressible Stokes equations. Numer. Math. 102, 257–275 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li J., Widlund O.B.: BDDC algorithms for incompressible Stokes equations. SIAM J. Numer. Anal. 44(6), 2432–2455 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li J., Widlund O.B.: FETI–DP, BDDC, and block Cholesky methods. Int. J. Numer. Methods Eng. 66, 250–271 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li J., Widlund O.B.: On the use of inexact subdomain solvers for BDDC algorithms. Comput. Methods Appl. Mech. Eng. 196(8), 1415–1428 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mandel J., Dohrmann C.R.: Convergence of a balancing domain decomposition by constraints and energy minimization. Numer. Linear Algebra Appl. 10(7), 639–659 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mandel J., Dohrmann C.R., Tezaur R.: An algebraic theory for primal and dual substructuring methods by constraints. Appl. Numer. Math. 54(2), 167–193 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mandel J., Sousedik B., Dohrmann C.R.: Multispace and multilevel BDDC. Computing 83(2–3), 55–85 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pavarino L.F., Widlund O.B.: Balancing Neumann–Neumann methods for incompressible Stokes equations. Comm. Pure Appl. Math. 55(3), 302–335 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Smith B.F.: A parallel implementation of an iterative substructuring algorithm for problems in three dimensions. SIAM J. Sci. Comput. 14(2), 406–423 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Toselli, A.: Domain decomposition methods for vector field problems. PhD thesis, Department of Computer Science, Courant Institute of Mathematical Sciences. TR-785 (1999)

  27. Toselli A., Widlund O.B., Wohlmuth B.I.: An iterative substructuring method for Maxwell’s equations in two dimensions. Math. Comp. 70(235), 935–949 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Tu X.: A BDDC algorithm for a mixed formulation of flows in porous media. Electron. Trans. Numer. Anal. 20, 164–179 (2005)

    MathSciNet  MATH  Google Scholar 

  29. Tu, X.: BDDC Domain Decomposition Algorithms: Methods with Three Levels and for Flow in Porous Media. PhD thesis, Department of Computer Science, Courant Institute, New York University. TR2005-879 (2006). http://cs.nyu.edu/csweb/Research/TechReports/TR2005-879/TR2005-879.pdf

  30. Tu X.: A BDDC algorithm for flow in porous media with a hybrid finite element discretization. Electron. Trans. Numer. Anal. 26, 146–160 (2007)

    MathSciNet  MATH  Google Scholar 

  31. Tu, X.: Three-level BDDC. In: Domain decomposition methods in science and engineering XVI. Lect. Notes Comput. Sci. Eng., vol. 55, pp. 437–444. Springer, Berlin (2007)

  32. Tu X.: Three-level BDDC in three dimensions. SIAM J. Sci. Comput. 29(4), 1759–1780 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tu X.: Three-level BDDC in two dimensions. Int. J. Numer. Methods Eng. 69, 33–59 (2007)

    Article  MATH  Google Scholar 

  34. Wohlmuth B.I.: Discretization Methods and Iterative Solvers Based on Domain Decomposition. Lecture Notes in Computational Science and Engineering, vol. 17. Springer, Berlin (2001)

    Book  Google Scholar 

  35. Wohlmuth B.I., Toselli A., Widlund O.B.: Iterative substructuring method for Raviart-Thomas vector fields in three dimensions. SIAM J. Numer. Anal. 37(5), 1657–1676 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Lawrence, KS, 66045-7594, USA

    Xuemin Tu

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  1. Xuemin Tu
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Correspondence to Xuemin Tu.

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This work was supported in part by the Director, Office of Science, Computational and Technology Research, US Department of Energy under Contract No. DE-AC02-05CH11231.

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Tu, X. A three-level BDDC algorithm for a saddle point problem. Numer. Math. 119, 189–217 (2011). https://doi.org/10.1007/s00211-011-0375-2

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  • DOI: https://doi.org/10.1007/s00211-011-0375-2

Mathematics Subject Classification (2000)