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Iterative Substructuring Methods for Spectral Element Discretizations of Elliptic Systems in Three Dimensions

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Parallel Solution of Partial Differential Equations

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 120))

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Abstract

Spectral element methods are considered for symmetric elliptic systems of second-order partial differential equations, such as the linear elasticity and the Stokes systems in three dimensions. The resulting discrete problems can be positive definite, as in the case of compressible elasticity in pure displacement form, or saddle point problems, as in the case of almost incompressible elasticity in mixed form and Stokes equations. Iterative substructuring algorithms are developed for both cases. They are domain decomposition preconditioners constructed from local solvers for the interior of each element and for each face of the elements and a coarse, global solver related to the wire basket of the elements. In the positive definite case, the condition number of the resulting preconditioned operator is independent of the number of spectral elements and grows at most in proportion to the square of the logarithm of the spectral degree. For saddle point problems, there is an additional factor in the estimate of the condition number, namely, the inverse of the discrete inf-sup constant of the problem.

This work was supported by I.A.N-CNR, Pavia and by the National Science Foundation under Grant NSF-CCR-9503408. Work on this project began when both of the authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001 and were supported by NASA Contract No. NAS1-19480.

This work was supported in part by the National Science Foundation under Grants NSF-CCR-9503408 and in part by the U.S. Department of Energy under contract DEFG02-92ER25127.

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References

  1. I. BABUŠKA AND M. SURI, Locking effects in the finite element approximation of elasticity problems, Numer. Math., 62 (1992), 439–463.

    Article  MathSciNet  MATH  Google Scholar 

  2. K.-J. BATHE AND D. CHAPELLE, The inf-sup test, Computers and Structures, 47 (1993), 537–545.

    Article  MathSciNet  MATH  Google Scholar 

  3. C. BERNARDI AND Y. MADAY, Approximations Spectrales de Problèmes aux Limites Elliptiques, Springer-Verlag France, Paris, 1992.

    MATH  Google Scholar 

  4. Spectral Methods, in Handbook of Numerical Analysis, Volume V: Techniques of Scientific Computing (Part 2), North-Holland, 1997, 209–485.

    Google Scholar 

  5. J. BRAMBLE AND J. PASCIAK, A domain decomposition technique for Stokes problems, Appl. Numer. Math., 6 (1989/90), 251–261.

    Article  MathSciNet  Google Scholar 

  6. S. BRENNER, Multigrid methods for parameter dependent problems, RAIRO M 2 AN, 30 (1996), 265–297.

    MATH  Google Scholar 

  7. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods,Springer-Verlag, Berlin, 1991.

    Book  MATH  Google Scholar 

  8. C. CANUTO, Stabilization of spectral methods by finite element bubble functions, Comp. Meths. Appl. Mech. Eng., 116 (1994), 13–26.

    Article  MathSciNet  MATH  Google Scholar 

  9. C. CANUTO AND V.V. KEMENADE, Bubble-stabilized spectral methods for the incompressible Navier-Stokes equations,Comp. Meths. Appl. Mech. Eng., 135 (1996), 35–61.

    Article  MATH  Google Scholar 

  10. M.A. CASARIN, Schwarz Preconditioners for Spectral and Mortar Finite Element Methods with Applications to Incompressible Fluids,PhD thesis, Dept. of Mathematics, Courant Institute of Mathematical Sciences, New York University, March 1996.

    Google Scholar 

  11. P.G. CIARLET, Mathematical Elasticity, North-Holland, Amsterdam, 1988.

    MATH  Google Scholar 

  12. M. DRYJA, B.F. SMITH, AND O.B. WIDLUND, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal, 31 (1994), 1662–1694.

    Article  MathSciNet  MATH  Google Scholar 

  13. H.C. ELMAN, Multigrid and Krylov subspace methods for the discrete Stokes equations, Int. J. Numer. Meth. Fluids, 227 (1996), 755–770.

    Article  Google Scholar 

  14. H.C. ELMAN, Preconditioning for the steady-state Navier—Stokes equations with low viscosity, SIAM J. Sci. Comput., 20(4) (1999), 1299–1316.

    Article  MathSciNet  MATH  Google Scholar 

  15. H.C. ELMAN AND D. SILVESTER, Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comp., 17 (1996), 33–46.

    Article  MathSciNet  MATH  Google Scholar 

  16. C. FARHAT AND F.-X. ROUX, Implicit parallel processing in structural mechanics,in Computational Mechanics Advances, J.T. Oden, ed., Vol. 2 (1), North-Holland, 1994, 1–124.

    MathSciNet  MATH  Google Scholar 

  17. P. FISCHER AND E.RØONQUIST, Spectral element methods for large scale parallel Navier-Stokes calculations, Comp. Meths. Appl. Mech. Eng., 116 (1994), 69–76.

    Article  MATH  Google Scholar 

  18. V. GIRAULT AND P.-A. RAVIART, Finite Element Methods for Navier-Stokes equations,Springer-Verlag, Berlin, 1986.

    Book  MATH  Google Scholar 

  19. A. KLAWONN, Block-triangular preconditioners for saddle point problems with a penalty term, SIAM J. Sci. Comput., 19(1) (1998), 172–184.

    Article  MathSciNet  MATH  Google Scholar 

  20. Preconditioners for Indefinite Problems, PhD thesis, Westfälische Wilhelms-Universität Münster, Angewandte Mathematik und Informatik, 1996. Tech. Rep. 8/96-N.

    Google Scholar 

  21. A. KLAWONN, An optimal preconditioner for a class of saddle point problems with a penalty term, SIAM J. Sci. Comput., 19(2) (1998), 540–552.

    Article  MathSciNet  MATH  Google Scholar 

  22. P.LE TALLEC, Domain decomposition methods in computational mechanics,in Computational Mechanics Advances, J.T. Oden, ed., Vol. 1 (2), North-Holland, 1994, 121–220.

    Google Scholar 

  23. P.LE TALLEC and A. PATRA, Non-overlapping domain decomposition methods for adaptive hp approximations of the Stokes problem with discontinuous pressure fields,Comp. Meths. Appl. Mech. Eng., 145 (1997), 361–379.

    Article  MATH  Google Scholar 

  24. Y. MADAY, D. MEIRON, A. PATERA, AND E. RØNQUIST, Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations, SIAM J. Sci. Comp., 14 (1993), 310–337.

    Article  MATH  Google Scholar 

  25. Y. MADAY, A. PATERA, AND E. RØNQUIST, The P N x P N-2 method for the approximation of the Stokes problem, Tech. Rep. 92009, Dept. of Mech. Engr., M.I T., 1992.

    Google Scholar 

  26. J. MANDEL, Balancing domain decomposition, Comm. Numer. Meth. Engrg., 9 (1993), 233–241.

    Article  MathSciNet  MATH  Google Scholar 

  27. Iterative solvers for p-version finite element method in three dimensions, Comp. Meths. Appl. Mech. Eng., 116 (1994), 175–183. ICOSAHOM 92, Montpellier, France, June 1992.

    Google Scholar 

  28. L.F. PAVARINO, Preconditioned conjugate residual methods for mixed spectral discretizations of elasticity and Stokes problems, Comp. Meths. Appl. Mech. Eng., 146 (1997), 19–30.

    Article  MathSciNet  MATH  Google Scholar 

  29. Preconditioned mixed spectral element methods for elasticity and Stokes problems, SIAM J. Sci. Comput., 19(6) (1998), 1941–1957.

    Google Scholar 

  30. L.F. PAVARINO AND O.B. WIDLUND, A polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions, SIAM J. Numer. Anal., 33 (1996), 1303–1335.

    Article  MathSciNet  MATH  Google Scholar 

  31. Iterative substructuring methods for spectral element discretizations of elliptic systems. I: Compressible linear elasticity, To appear in SIAM J. Numer. Anal. (1999).

    Google Scholar 

  32. Iterative substructuring methods for spectral element discretizations of elliptic systems. II: Mixed methods for linear elasticity and Stokes flow, To appear in SIAM J. Numer. Anal. (1999).

    Google Scholar 

  33. Iterative substructuring methods for spectral elements: Problems in three dimensions based on numerical quadrature, Comp. Math. Appl., 33 (1997), 193–209.

    Google Scholar 

  34. A. QUARTERONI, Domain decomposition algorithms for the Stokes equations,in Domain Decomposition Methods, T. Chan, R. Glowinski, J. Périaux, and O. Widlund, eds., Philadelphia, 1989, SIAM.

    Google Scholar 

  35. E. RØONQUIST, A domain decomposition solver for the steady Navier-Stokes equations, in Proc. of ICOSAHOM ’95, A. Ilin and L. Scott, eds., 1996.

    Google Scholar 

  36. T. RUSTEN AND R. WINTHER, A preconditioned iterative method for saddle point problems, SIAM J. Matr. Anal. Appl., 13 (1992), 887–904.

    Article  MathSciNet  MATH  Google Scholar 

  37. D. SILVESTER AND A. WATHEN, Fast iterative solution of stabilised Stokes systems. Part II: Using general block preconditioners, SIAM J. Numer. Anal., 31 (1994), 1352–1367.

    Article  MathSciNet  MATH  Google Scholar 

  38. B.F. SMITH, P. BJORSTAD, AND W.D. GROPP, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, 1996.

    MATH  Google Scholar 

  39. M. SURI AND R. STENBERG, Mixed hp finite element methods for problems in elasticity and Stokes flow, Numer. Math., 72 (1996), 367–390

    Article  MathSciNet  MATH  Google Scholar 

  40. D. YANG, Stabilized schemes for mixed finite element methods with applications to elasticity and compressible flow problems, Tech. Rep. 1472, IMA Preprint, 1997.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Milano, Via Saldini 50, 20133, Milano, Italy

    Luca F. Pavarino

  2. Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y., 10012, USA

    Olof B. Widlund

Authors
  1. Luca F. Pavarino
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  2. Olof B. Widlund
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Editor information

Editors and Affiliations

  1. Institutt for Informatikk Hoyteknologisenteret, University of Bergen, Bergen, N-5020, Norway

    Petter Bjørstad

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Mitchell Luskin

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Pavarino, L.F., Widlund, O.B. (2000). Iterative Substructuring Methods for Spectral Element Discretizations of Elliptic Systems in Three Dimensions. In: Bjørstad, P., Luskin, M. (eds) Parallel Solution of Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 120. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1176-1_1

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  • DOI: https://doi.org/10.1007/978-1-4612-1176-1_1

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7034-8

  • Online ISBN: 978-1-4612-1176-1

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