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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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Lower bounds for nonoverlapping domain decomposition preconditioners in two dimensions
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by Susanne C. Brenner and Li-Yeng Sung;
Math. Comp. 69 (2000), 1319-1339
DOI: https://doi.org/10.1090/S0025-5718-00-01236-9
Published electronically: April 12, 2000

Abstract:

Lower bounds for the condition numbers of the preconditioned systems are obtained for the Bramble-Pasciak-Schatz substructuring preconditioner and the Neumann-Neumann preconditioner in two dimensions. They show that the known upper bounds are sharp.
References
  • Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 482275
  • Petter E. Bjørstad and Jan Mandel, On the spectra of sums of orthogonal projections with applications to parallel computing, BIT 31 (1991), no. 1, 76–88. MR 1097483, DOI 10.1007/BF01952785
  • J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp. 47 (1986), no. 175, 103–134. MR 842125, DOI 10.1090/S0025-5718-1986-0842125-3
  • Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
  • Tony F. Chan and Tarek P. Mathew, Domain decomposition algorithms, Acta numerica, 1994, Acta Numer., Cambridge Univ. Press, Cambridge, 1994, pp. 61–143. MR 1288096, DOI 10.1017/S0962492900002427
  • Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
  • M. Dryja, A method of domain decomposition for three-dimensional finite element elliptic problems, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 43–61. MR 972511
  • M. Dryja and O.B. Widlund, Some domain decomposition algorithms for elliptic problems, Iterative Methods for Large Linear Systems (L. Hayes and D. Kincaid, eds.), Academic Press, New York, 1989, pp. 273–291.
  • Maksymilian Dryja and Olof B. Widlund, Towards a unified theory of domain decomposition algorithms for elliptic problems, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) SIAM, Philadelphia, PA, 1990, pp. 3–21. MR 1064335
  • Maksymilian Dryja and Olof B. Widlund, Additive Schwarz methods for elliptic finite element problems in three dimensions, Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk, VA, 1991) SIAM, Philadelphia, PA, 1992, pp. 3–18. MR 1189559
  • Maksymilian Dryja and Olof B. Widlund, Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems, Comm. Pure Appl. Math. 48 (1995), no. 2, 121–155. MR 1319698, DOI 10.1002/cpa.3160480203
  • M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math. 70 (1995), no. 2, 163–180. MR 1324736, DOI 10.1007/s002110050115
  • F. Kickinger, S.V. Nepomnyaschikh, R. Pfau and J. Schöberl, Numerical Estimates of Inequalities in $H^{1/2}$, Technical Report No 97-3, Institut für Mathematik, Johannes Kepler Universität, Linz, 1997.
  • S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, RI, 1982. Translated from the Russian by J. Szűcs. MR 649411
  • Y. Kuznetsov, P. Manninen and Y. Vassilevski, On numerical experiments with Neumann-Neumann and Neumann-Dirichlet domain decomposition preconditioners, Technical Report, University of Jyväkylä, 1993.
  • Patrick Le Tallec, Domain decomposition methods in computational mechanics, Comput. Mech. Adv. 1 (1994), no. 2, 121–220. MR 1263805
  • P.-L. Lions, On the Schwarz alternating method. I, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 1–42. MR 972510
  • Radosław Biernacki, An application of the cut-set method in the synthesis of block graphs, Arch. Elektrotech. (Warsaw) 25 (1976/77), no. 3, 659–676 (Polish, with English and Russian summaries). MR 444312
  • A.M. Matsokin and S.V. Nepomnyaschikh, Schwarz alternating method in subspaces, Soviet Mathematics 29 (1985), 78–84.
  • S.V. Nepomnyaschikh, On the application of the bordering method to the mixed boundary value problem for elliptic equations and on mesh norms in $W^{1/2}(S)$, Soviet J. Numer. Anal. Math. Modelling 4 (1989), 493–506.
  • —, Fictitious components and subdomain alternating methods, Soviet J. Numer. Anal. Math. Modelling 5 (1990), 53–68.
  • Barry F. Smith, Petter E. Bjørstad, and William D. Gropp, Domain decomposition, Cambridge University Press, Cambridge, 1996. Parallel multilevel methods for elliptic partial differential equations. MR 1410757
  • H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580
  • X. Zhang, Studies in Domain Decomposition: Multi-level Methods and the Biharmonic Dirichlet Problem, Dissertation, Courant Institute, 1991.
  • Xuejun Zhang, Multilevel Schwarz methods, Numer. Math. 63 (1992), no. 4, 521–539. MR 1189535, DOI 10.1007/BF01385873
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Bibliographic Information
  • Susanne C. Brenner
  • Affiliation: Department of Mathematics, University of South Carolina, Columbia, SC 29208
  • Email: brenner@math.sc.edu
  • Li-Yeng Sung
  • Affiliation: Department of Mathematics, University of South Carolina, Columbia, SC 29208
  • Email: sung@math.sc.edu
  • Received by editor(s): May 22, 1998
  • Published electronically: April 12, 2000
  • Additional Notes: The work of the first author was supported in part by the National Science Foundation under Grant No. DMS-96-00133.
  • © Copyright 2000 American Mathematical Society
  • Journal: Math. Comp. 69 (2000), 1319-1339
  • MSC (1991): Primary 65N55, 65N30
  • DOI: https://doi.org/10.1090/S0025-5718-00-01236-9
  • MathSciNet review: 1710656