Unified Multipliers-Free Theory of Dual-Primal Domain Decomposition Methods Ismael Herrera, Robert A. Yates * Instituto de Geofísica, Universidad Nacional Autónoma de México (UNAM), 14000 Mexico, D.F., Mexico Received 27 November 2007; accepted 29 February 2008 Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.20359 The concept of dual-primal 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 all-inclusive unified theory of nonoverlapping domain decomposition methods (DDMs). One-level methods, such as Schur-complement and one-level FETI, as well as two-level meth- ods, such as Neumann-Neumann and preconditioned FETI, are incorporated in a unified manner. Different choices of the dual subspaces yield the different dual-primal 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 code-development 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 projection-matrices, 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 Steklov-Poincaré operator, at the discrete level, is also introduced. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 000: 000–000, 2008 Keywords: domain decomposition methods; dual-primal; Lagrange multipliers; preconditioners; discon- tinuous Galerkin; FETI; Neumann-Neumann I. INTRODUCTION Mathematical models of many systems of interest, including very important continuous systems of Engineering and Science, lead to a great variety of partial differential equations whose solution methods are based on the computational processing of large-scale algebraic systems. Furthermore, the incredible growth experienced by the existing computational hardware and software has made amenable to effective treatment an ever increasing diversity and complexity of problems, posed by engineering and scientific applications. Parallel computing is outstanding among the new computational tools, especially at present when further increases in hardware speed apparently have reached insurmountable barriers [1]. Correspondence to: Ismael Herrera, Instituto de Geofísica, Universidad Nacional Autónoma de México (UNAM), Apdo. Postal 22-582, 14000 Mexico, D.F., Mexico (e-mail: iherrera@servidor.unam.mx) ∗ Present address: Alternativas en Computación, SA de CV, Shakespeare #15, 11590 México, D.F., Mexico © 2008 Wiley Periodicals, Inc.