Preconditioning on element interfaces for the p-version finite element method and spectral element method

SIAM Journal of Scientific Computing

Volumn 21, Issue 2, 1999, Pages 522-551

  Cao, Weiming ^a   Guo, Benqi ^b  

^a SIMON FRASER UNIVERSITY   (Canada)

^b UNIVERSITY OF MANITOBA   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; ITERATIVE METHODS; LINEAR SYSTEMS; MATRIX ALGEBRA; PROBLEM SOLVING;

ELEMENT INTERFACES; PRECONDITIONING; SPECTRAL ELEMENT METHOD;

FINITE ELEMENT METHOD;

EID: 0033294861     PISSN: 10648275     EISSN: None     Source Type: None    
DOI: 10.1137/S1064827596306951     Document Type: Article

References (26)

1. M. AINSWORTH, A preconditioner based on domain decomposition for h-p finite-element approximation on quasi-uniform meshes, SIAM J. Numer. Anal., 33 (1996), pp. 1358-1376.
2. I. BABUŠKA, A. CRAIG, J. MANDEL, AND J. PITKÄRANTA, Efficient preconditioning for the p-version finite element method in two dimensions, SIAM J. Numer. Anal., 28 (1991), pp. 624-661.
3. I. BABUŠKA, M. GRIEBEL, AND J. PITKÄRANTA, The problem of selecting the shape functions for a p-type finite element, J. Comput. Appl. Math., 28 (1989), pp. 157-187.
4. P. E. BJØRSTAD AND O. B. WIDLUND, Iterative methods for the solution of elliptic problems on regions partitioned into substructures, SIAM J. Numer. Anal., 23 (1986), pp. 1097-1120.
5. J. BRAMBLE, J. PASCIAK, AND A. SCHATZ, The construction of preconditioners for elliptic problems by substructuring, I, Math. Comp., 175 (1986), pp. 103-134.
6. J. BRAMBLE, J. PASCIAK, AND A. SCHATZ, The construction of preconditioners for elliptic problems by substructuring, IV, Math. Comp., 53 (1989), pp. 1-24.
7. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1987.
8. C. CANUTO AND A. QUARTERONI, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38 (1982), pp. 67-86.
9. M. CASARIN, Diagonal edge preconditioners in p-version and spectral element methods, SIAM J. Sci. Comput., 18 (1997), pp. 610-620.
10. T. F. CHAN AND D. C. RESASCO, Analysis of Domain Decomposition Preconditioners on Irregular Regions, CAM Report 87-05, UCLA, Los Angeles, CA, 1987.
11. W. S. DON AND D. GOTTLIEB, The Chebyshev-Legendre method: Implementing Legendre methods on Chebyshev points, SIAM J. Numer. Anal., 31 (1994), pp. 1519-1534.
12. M. DRYJA AND O. B. WIDLUND, Some domain decomposition algorithms for elliptic problems, in Iterative Methods for Large Scale Linear Systems, L. Hayes and D. Kincaid, eds., Academic Press, San Diego, CA, 1989, pp. 273-291.
13. M. DRYJA, B. 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. B. Q. GUO AND W. M. CAO, A preconditioner for the h-p version of the finite element method in two dimensions, Numer. Math., 75 (1996), pp. 59-77.
15. B. Q. GUO AND W. M. CAO, An additive Schwarz method for the h-p version of the finite element method in three dimensions, SIAM J. Numer. Anal., 35 (1998), pp. 632-654.
16. J. L. LIONS AND E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, Heidelberg, 1972.
17. Y. MADAY AND A. T. PATERA, Spectral element method for the Navier-Stokes equations, in State of the Art Survey in Computational Mechanics, A. Noor and J.T. Oden, eds., New York, 1989.
18. J. MANDEL, Iterative solvers by substructuring for the p-version finite element method, Comput. Methods Appl. Mech. Engrg., 80 (1990), pp. 117-128.
19. J. MANDEL, Two-level domain decomposition preconditioning for the p-version finite element method in three dimensions, Internat. J. Numer. Methods Engrg., 29 (1990), pp. 1095-1108.
20. 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), pp. 1303-1335.
21. 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.
22. J. SHEN, Efficient spectral-Galerkin method II. Direct solvers of second-and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comput., 16 (1995), pp. 74-87.
23. B. SZABÓ AND I. BABUŠKA, Finite Element Analysis, Wiley, New York, 1990.
24. O. B. WIDLUND, Iterative substructuring methods: Algorithms and theory for elliptic problems in the plane, in Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, and J. Périaux, eds., SIAM, Philadelphia, PA, 1988.
25. O. B. WIDLUND, private communications, 1996.
26. J. XU, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), pp. 581-613.