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New formulation of iterative substructuring methods without Lagrange multipliers
 NeumannNeumann and FETI, Numer. Methods Partial Dierential Equations
"... This article is devoted to introduce a new approach to iterative substructuring methods that, without recourse to Lagrange multipliers, yields positive definite preconditioned formulations of the Neumann–Neumann and FETI types. To my knowledge, this is the first time that such formulations have been ..."
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This article is devoted to introduce a new approach to iterative substructuring methods that, without recourse to Lagrange multipliers, yields positive definite preconditioned formulations of the Neumann–Neumann and FETI types. To my knowledge, this is the first time that such formulations have been made without resource to Lagrange multipliers. A numerical advantage that is concomitant to such multipliersfree formulations is the reduction of the degrees of freedom associated with the Lagrange multipliers. Other attractive features are their generality, directness, and simplicity. The general framework of the new approach is rather simple and stems directly from the discretization procedures that are applied; in it, the differential operators act on discontinuous piecewisedefined functions. Then, the Lagrange multipliers are not required because in such an environment the functionsdiscontinuities are not an anomaly that need to be corrected. The resulting algorithms and equationssystems are also derived with considerable detail. © 2007 Wiley Periodicals, Inc.
Unified MultipliersFree Theory of DualPrimal Domain Decomposition Methods
, 2007
"... The concept of dualprimal methods can be formulated in a manner that incorporates, as a subclass, the non preconditioned case. Using such a generalized concept, in this article without recourse to “Lagrange multipliers, ” we introduce an allinclusive unified theory of nonoverlapping domain decompo ..."
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The concept of dualprimal methods can be formulated in a manner that incorporates, as a subclass, the non preconditioned case. Using such a generalized concept, in this article without recourse to “Lagrange multipliers, ” we introduce an allinclusive unified theory of nonoverlapping domain decomposition methods (DDMs). Onelevel methods, such as Schurcomplement and onelevel FETI, as well as twolevel methods, such as NeumannNeumann and preconditioned FETI, are incorporated in a unified manner. Different choices of the dual subspaces yield the different dualprimal preconditioners reported in the literature. In this unified theory, the procedures are carried out directly on the matrices, independently of the differential equations that originated them. This feature reduces considerably the codedevelopment effort required for their implementation and permit, for example, transforming 2D codes into 3D codes easily. Another source of this simplification is the introduction of two projectionmatrices, generalizations of the average and jump of a function, which possess superior computational properties. In particular, on the basis of numerical results reported there, we claim that our jump matrix is the optimal choice of the B operator of the FETI methods. A new formula for the SteklovPoincaré operator, at the discrete level, is also introduced. © 2008 Wiley