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Keywords: domain decomposition methods, dual-primal, Lagrange multipliers, preconditioners, discontinuous Galerkin, FETI, Neumann-Neumann.

Summary

Nowadays parallel computing is the most effective means for increasing computational speed. In turn, the domain decomposition methods (DDM) are most efficient for applying parallel-computing to the solution of partial differential equations. The non-overlapping class of such methods, which are especially effective, is constituted mainly by the Schur complement and the non-preconditioned FETI (or Neumann) methods, here grouped generically as the one-way methods, together with the Neumann-Neumann and the preconditioned FETI methods, here grouped generically as the round-trip methods. More recently, such methods have been improved by the introduction of the dual-primal methods, in which a relatively small number of continuity constraints across the interfaces are enforced. However, the treatment of round-trip algorithms up to now has been done with recourse to Lagrange multipliers exclusively. Recently, however, Herrera and his collaborators [1,2,3,4] have introduced a more direct treatment, the multipliers-free formulation, in which the differential operators are applied to discontinuous functions, and the matrices are applied to discontinuous vectors. Among some other advantages of such a direct treatment, it allows the development of more explicit and general expressions of the algorithm matrices, which are here reported. Such matrix-expressions in turn allow the development of more robust and simple computational codes.

References

1: I. Herrera, "Theory of Differential Equations in Discontinuous Piecewise-Defined-Functions", Numer Meth Part D E, 23(3), 597-639, 2007. doi:10.1002/num.20182
2: I. Herrera, "New Formulation of Iterative Substructuring Methods without Lagrange Multipliers: Neumann-Neumann and FETI", Numer Meth Part D E, 24(3), 845-878, 2008. doi:10.1002/num.20293
3: I. Herrera, R. Yates, "Unified Multipliers-Free Theory of Dual Primal Domain Decomposition Methods", Numer Meth Part D E, To appear in 2009. doi:10.1002/num.20359
4: I. Herrera, R. Yates, "The Multipliers-Free Dual-Primal Method", Numer Meth Part D E, To be submitted, 2009.

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 90 PROCEEDINGS OF THE FIRST INTERNATIONAL CONFERENCE ON PARALLEL, DISTRIBUTED AND GRID COMPUTING FOR ENGINEERING Edited by: Paper 5 New Matrix General Formulas of Dual-Primal Domain Decomposition Methods without recourse to Lagrange Multipliers I. Herrera¹ and R.A. Yates² ¹Institute of Geophysics, National Autonomous University of Mexico (UNAM), Mexico ²Alternativas en Computación, S.A. de C.V., Mexico doi:10.4203/ccp.90.5 purchase the full-text of this paper Full Bibliographic Reference for this paper I. Herrera, R.A. Yates, "New Matrix General Formulas of Dual-Primal Domain Decomposition Methods without recourse to Lagrange Multipliers", in , (Editors), "Proceedings of the First International Conference on Parallel, Distributed and Grid Computing for Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 5, 2009. doi:10.4203/ccp.90.5 Keywords: domain decomposition methods, dual-primal, Lagrange multipliers, preconditioners, discontinuous Galerkin, FETI, Neumann-Neumann. Summary Nowadays parallel computing is the most effective means for increasing computational speed. In turn, the domain decomposition methods (DDM) are most efficient for applying parallel-computing to the solution of partial differential equations. The non-overlapping class of such methods, which are especially effective, is constituted mainly by the Schur complement and the non-preconditioned FETI (or Neumann) methods, here grouped generically as the one-way methods, together with the Neumann-Neumann and the preconditioned FETI methods, here grouped generically as the round-trip methods. More recently, such methods have been improved by the introduction of the dual-primal methods, in which a relatively small number of continuity constraints across the interfaces are enforced. However, the treatment of round-trip algorithms up to now has been done with recourse to Lagrange multipliers exclusively. Recently, however, Herrera and his collaborators [1,2,3,4] have introduced a more direct treatment, the multipliers-free formulation, in which the differential operators are applied to discontinuous functions, and the matrices are applied to discontinuous vectors. Among some other advantages of such a direct treatment, it allows the development of more explicit and general expressions of the algorithm matrices, which are here reported. Such matrix-expressions in turn allow the development of more robust and simple computational codes. References 1 I. Herrera, "Theory of Differential Equations in Discontinuous Piecewise-Defined-Functions", Numer Meth Part D E, 23(3), 597-639, 2007. doi:10.1002/num.20182 2 I. Herrera, "New Formulation of Iterative Substructuring Methods without Lagrange Multipliers: Neumann-Neumann and FETI", Numer Meth Part D E, 24(3), 845-878, 2008. doi:10.1002/num.20293 3 I. Herrera, R. Yates, "Unified Multipliers-Free Theory of Dual Primal Domain Decomposition Methods", Numer Meth Part D E, To appear in 2009. doi:10.1002/num.20359 4 I. Herrera, R. Yates, "The Multipliers-Free Dual-Primal Method", Numer Meth Part D E, To be submitted, 2009. purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £72 +P&P)
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