Unified multipliers‐free theory of dual‐primal domain decomposition methods

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 methods, 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, 2009