Quasi-optimal Schwarz methods for the conforming spectral element discretization

SIAM Journal on Numerical Analysis

Volumn 34, Issue 6, 1997, Pages 2482-2502

  Casarin, Mario A ^a  

^a UNIVERSITY OF CAMPINAS   (Brazil)

Author keywords

Domain decomposition; Iterative substructuring; Preconditioned conjugate gradients; Schwarz methods; Spectral element method

Indexed keywords

EID: 0000997330     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142995292281     Document Type: Article

References (38)

1. C. BERNARDI AND Y. MADAY, Approximations Spectrales de Problmès aux Limites Ellip- e tiques, Math. Appl., 10, Springer-Verlag, France, Paris, 1992.
2. C. BERNARDI AND Y. MADAY, Polynomial interpolation results in Sobolev spaces, J. Comput. Appl. Math., 43 (1992), pp. 53-80.
3. J. H. BRAMBLE, A second order finite difference analogue of the first biharmonic boundary value problem, Numer. Math., 9 (1966), pp. 236-249.
4. J. H. BRAMBLE, J. E. PASCIAK, AND A. H. SCHATZ, The construction of preconditioned for elliptic problems by substruc.turing, IV, Math. Comp., 53 (1989), pp. 1-24.
5. J. H. BRAMBLE, J. E. PASCIAK, J. WANG, AND J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp., 57 (1991), pp. 1-21.
6. 2 projection, Math. Comp., 56 (1991), pp. 463-476.
7. C. CANUTO, Stabilization of spectral methods by finite element bubble functions, Comput. Methods Appl. Mech. Erigrg, 116 (1994), pp. 13-26.
8. C. CANUTO AND D. FUNARO, The Schwarz algorithm for spectral methods, SIAM J. Numer. Anal., 25 (1988), pp. 24-40.
9. M. A. CASARIN, Schwarz Preconditioned for Spectral and Mortar Finite. Element Methods with Applications to Incompressible Fluids, Ph.D. thesis, Courant Institute of Mathematical Sciences, 1996; Tech. Report 717, Department of Computer Science, Courant Institute, New York.
10. J. W. DEMMEL, M. T. HEATH, AND H. A. VAN DER VORST, Parallel numerical linear algebra. in Acta Numerica 1993, Cambridge University Press. London, 1993, pp. 111-197.
11. M. O. DEVILLE AND E. H. MUND, Finite-element preconditioning for pseudospectral solutions of elliptic problems, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 311-342.
12. M. DRYJA, M. V. SARKIS, AND O. B. WIDLUND, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer. Math., 72 (1996), pp. 313-348.
13. 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), pp. 1662-1694.
14. M. DRYJA AND O. B. WIDLUND, Domain decomposition algorithms with small overlap, SIAM J. Sci. Comput., 15 (1994). pp. 604-620.
15. P. F. FISCHER AND E. RONQUIST, Spectral element methods for large scale parallel Navier- Stokes calculations, Comput. Methods Appl. Mech. Engrg, 116 (1994), pp. 69-76; in Proc. ICOSAHOM 92, Montpellier. France, June 22-26, 1992.
16. P. F. FISCHER AND E. RONQUIST, Spectral element methods for large scale parallel Navier- Stokes calculations, Comput. Methods Appl. Mech. Engrg, 116 (1994), pp. 69-76; in Proc. ICOSAHOM 92, Montpellier. France, June 22-26, 1992.
17. P. GERVASIO, Risoluzione di equazioni alle derivate parziali con metodi spettrali in regioni partizionate in sottodomini, Ph.D. thesis, Department of Mathematics, Politecnico di Milano, Italy, 1995.
18. P. GERVASIO, E. I. OVTCHINNIKOV, AND A. QUARTERONI, The Spectral Projection Decomposition Method for Elliptic Equations, SIAM J. Numer. Anal., 34 (1997), pp. 1616-1639.
19. V. GIRAULT AND P.-A. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, New York, 1986.
20. W. D. GROPP AND B. F. SMITH, Experiences with domain decomposition in three dimensions: Overlapping Schwarz methods, in Domain Decomposition Methods in Science and Engineering, 6th International Conference on Domain Decomposition, A. Quarteroni, Y. A. Kuznetsov, J. Périaux, and O. B. Widlund, eds., Contemp. Math., 157, AMS, 1994, pp. 323-334.
21. D. E. KEYES, T. F. CHAN, G. A. MEURANT, J. S. SCROGGS, AND R. G. VOIGT, EDS., 5th International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1992.
22. D. E. KEYES AND J. XU, EDS., Domain Decomposition Methods in Science and Engineering, Contemp. Math., 180, AMS, Providence, RI, 1994.
23. Y. MADAY, D. MEIRON, A. T. PATERA, AND E. M. RØNQUIST, Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations, SIAM J. Sci. Comp., 14 (1993), pp. 310-337.
24. Y. MADAY AND A. T. PATERA, Spectral element methods for the Navier-Stokes equations, in State of the Art Surveys in Computational Mechanics, A. Noor and J. Oden, eds., American Society of Mathematical Engineers, Now York, 1989.
25. J. MANDEL AND M. BREZINA, Balancing Domain Decomposition: Theory and Computations in Two and Three. Dimensions, Tech. Report UCD/CCM 2, Center for Computational Mathematics, University of Colorado, Denver, CO, 1993.
26. S. A. ORZAG, Spectral methods for problems in complex geometries, J. Comput. Phys., 37 (1980), pp. 70-92.
27. S. S. PAUL, Schwarz Type Domain Decomposition Methods for Spectral Element Discretizations, Master's thesis, Department of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg. South Africa, 1993.
28. S. V. PARTER AND E. E. ROTHMAN, Preconditioning Legendre spectral collocation approximation to elliptic: problems, SIAM J. Numer. Anal., 32 (1995). pp. 333-385.
29. L. F. PAVARINO, Domain Decomposition Algorithms for the p-Version Finite Element Method for Elliptic Problems. Ph.D. thesis, Courant Institute, New York University, New York, 1992.
30. L. F. PAVARINO, Additive Schwarz methods for the p-version finite element method, Numer. Math., 66 (1994), pp. 493-515.
31. L. F. PAVARINO, Schwarz methods with local refinement for the p-version finite element method. Numer. Math., 69 (1994), pp. 185-211.
32. L. F. PAVARINO, Neumann-Neumann algorithms for spectral elements in three dimensions, Tech. Report 979, Istituto dii Analisi Numerica-Consiglio Nazionale delle Richerche, Pavia, Italy, 1995.
33. L. F. PAVARINO AND O. B. WIDLUND, Iterative substructuring methods for spectral elements: Problems in three dimensions based on numerical quadrature, Comput. Math. Appl., 33 (1997), pp. 193-209.
34. 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 (1997), pp. 1303-1335.
35. A. QUARTERONI, Y. A. KUZNETSOV, J. PÉRIAUX, AND O. B. WIDLUND, EDS., Domain Decomposition Methods in Science and Engineering, 6th International Conference on Domain Decomposition, Contemp. Math., 157, AMS, Providence, RI, 1994.
36. A. QUARTERONI AND E. ZAMPIERI, Finite element preconditioning for legendre spectral collocation approximations to elliptic equations and systems, SIAM J. Numer. Anal., 29 (1992), pp. 917-936.
37. B. F. SMITH, A domain decomposition algorithm for elliptic problems in three dimensions, Numer. Math., 60 (1991), pp. 219-234.
38. O. WIDLUND, Domain decomposition methods for mortar and higher order finite elements, 9th International Conference on Domain Decomposition Methods, Ullensvang, Norway, June 1996, University of Bergen, Norway.