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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 90
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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

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.

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.

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
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
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
I. Herrera, R. Yates, "The Multipliers-Free Dual-Primal Method", Numer Meth Part D E, To be submitted, 2009.

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