Maio 2014 vol. 1 num. 1 - 10th World Congress on Computational Mechanics

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THE DERIVED-VECTOR-SPACE: A UNIFIED FRAMEWORK FOR NONOVERLAPPING DDM

Herrera, I. ;

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We present and discuss a framework called the ‘derived-vector-space (DVS) framework’. In part, the relevance of this framework is because in its realm four preconditioned massively parallelizable DVS-algorithms with constraints of general applicability (by this we mean: applicable to symmetric-definite, non-symmetric and indefinite matrices) have been developed. Such algorithms yield codes that are almost 100% in parallel; more precisely, they achieve what we call the ‘leitmotif of DDM research’: to obtain algorithms that yield the global solution by solving local problems exclusively. This has been possible because in the DVS-approach the original PDE, or system of such equations, is transformed into a problem formulated in the derived-vector-space, which is a Hilbert-space that is well-defined by itself independently of the particular PDE considered, and afterwards all the numerical work is done in that space. Thus, the applicability of the algorithms developed in this framework is essentially independent of the specific problem considered, and furthermore this allows the development of codes, which to a large extent are of universal applicability. Thus, the DVSalgorithms are very suitable for programming in an efficient manner the most powerful parallel computers available at present.

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Palavras-chave: Massively-parallel-algorithms, parallel-computers, non-overlapping-DDM, BDDC, FETI-DP.,

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DOI: 10.5151/meceng-wccm2012-18051

Referências bibliográficas
  • [1] Herrera, I. Andamp; Rosas-Medina A. “Four General Purpose Massively Parallel DDM Algorithms”, Available as Memoria #7, GMMC, Instituto de Geofísica, UNAM, 2012.
  • [2] Herrera, I., Carrillo-Ledesma A. Andamp; Rosas-Medina Alberto “A Brief Overview of Nonoverlapping Domain Decomposition Methods”, Geofisica Internacional, Vol. 50(4), pp 445-463, 2011.
  • [3] Herrera, I. Andamp; Yates R. A. The Multipliers-Free Dual Primal Domain Decomposition Methods for Nonsymmetric Matrices NUMER. METH. PART D. E. 27(5) pp. 1262-1289, 2011. DOI 10.1002/Num. 20581. (Published on line April 28, 2010).
  • [4] Herrera, I. Andamp; Yates R. A. The Multipliers-free Domain Decomposition Methods NUMER. METH. PART D. E. 26: 874-905 July 2010, DOI 10.1002/num. 20462. (Published on line Jan 28, 2009)
  • [5] Herrera I. and R. Yates “Unified Multipliers-Free Theory of Dual-Primal Domain Decomposition Methods. NUMER. METH. PART D. E. Eq. 25:552-581, May 2009, (Published on line May 13, 08) DOI 10.1002/num. 20359.
  • [6] Herrera, I. “New Formulation of Iterative Substructuring Methods without Lagrange Multipliers: Neumann-Neumann and FETI”, NUMER METH PART D E 24(3) pp 845-878, 2008 (Published on line Sep 17, 2007) DOI 10.1002 NO. 20293.
  • [7] Herrera, I. “Theory of Differential Equations in Discontinuous Piecewise-Defined- Functions”, NUMER METH PART D E, 23(3), pp 597-639, 2007 DOI 10.1002 NO. 20182.
  • [8] DDM Organization, Proceedings of 20 International Conferences on Domain Decomposition Methods. www.ddm.org, 1988-2009.
  • [9] Toselli A. and O. Widlund, Domain decomposition methods- Algorithms and Theory, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2005, 450p.
  • [10] Mandel J. Balancing domain decomposition, Commun. Numer. Methods Engrg. 1(1993) 233-241.
  • [11] Mandel J. and Brezina M. Balancing domain decomposition for problems with large jumps in coefficients. Math. Comput. 65, pp 1387-1401, 1996.
  • [12] Dohrmann C.R., A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25(1):246-258, 2003.
  • [13] Mandel J. and C. R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl., 10(7):639-659, 2003.
  • [14] Brenner S. and Sung L. BDDC and FETI-DP without matrices or vectors Comput. Methods Appl. Mech. Engrg. 196(8): 1429-1435. 2007.
  • [15] Farhat Ch., and Roux F. A method of finite element tearing and interconnecting and its parallel solution algorithm. Internat. J. Numer. Methods Engrg. 32:1205-1227, 1991.
  • [16] Mandel J. and Tezaur R. Convergence of a substructuring method with Lagrange multipliers. Numer. Math 73(4): 473-487, 1996.
  • [17] Farhat C., Lessoinne M. and Pierson K. A scalable dual-primal domain decomposition method, Numer. Linear Algebra Appl. 7, pp 687-714, 2000.
  • [18] Mandel J. and Tezaur R., On the convergence of a dual-primal substructuring method, SIAM J. Sci. Comput., 25, pp 246-258, 2001.
  • [19] Farhat C., Lessoinne M. LeTallec P., Pierson K. and Rixen D. FETI-DP a dual-primal unified FETI method, Part I: A faster alternative to the two-level FETI method. Int. J. Numer. Methods Engrg. 50, pp 1523-1544, 2001.
  • [20] Cai, X-C. Andamp; Widlund, O.B., Domain Decomposition Algorithms for Indefinite Elliptic Problems, SIAM J. Sci. Stat. Comput. 1992, Vol. 13 pp. 243-258
  • [21] Farhat C., and Li J. An iterative domain decomposition method for the solution of a class of indefinite problems in computational structural dynamics. ELSEVIER Science Direct Applied Numerical Math. 54 pp 150-166. 2005.
  • [22] Li J. and Tu X. Convergence analysis of a Balancing Domain Decomposition method for solving a class of indefinite linear systems. Numer. Linear Algebra Appl. 2009; 16:745–773
  • [23] Tu X. and Li J., A Balancing Domain Decomposition method by constraints for advection- diffusion problems. www.ddm.org/DD18/
Como citar:

Herrera, I.; "THE DERIVED-VECTOR-SPACE: A UNIFIED FRAMEWORK FOR NONOVERLAPPING DDM", p. 651-658 . In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 2358-0828, DOI 10.5151/meceng-wccm2012-18051

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