The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Dual-primal FETI methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations, as in primal iterative substructuring methods, while most of the constraints are enforced by Lagrange multipliers, as in one-level FETI methods. These methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems. The purpose of this article is to develop strategies for selecting these constraints, which are enforced throughout the iterations, such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them. The algorithms are described in terms of a change of basis that has proven to be quite robust in practice. A theoretical analysis is provided for the case of linear elasticity, and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the Young's modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain. The strategies have already proven quite successful in large-scale implementations of these iterative methods.

No abstract

Finite element problems can often naturally be divided into subproblems which correspond to subregions into which the region has been partitioned or from which it was originally assembled. A class of iterative methods is discussed in which these subproblems are solved by direct methods, while the interaction across the curves or surfaces which divide the region is handled by a conjugate gradient method.A mathematical framework for this work is provided by regularity theory for elliptic finite element problems and by block Gaussian elimination. A full development of the theory, which shows that certain of these methods are optimal, is given for Lagrangian finite element approximations of second order linear elliptic problems in the plane. Results from numerical experiments are also reported.

Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate solvers for the same partial differential equation restricted to a set of subregions into which the given region has been divided. In addition, the preconditioner is often augmented by a coarse, second-level approximation that provides additional, global exchange of information that can enhance the rate of convergence considerably. The iterative substructuring methods, based on decompositions of the region into nonoverlapping subregions, form one of the main families of such algorithms.Many domain decomposition algorithms can conveniently be described and analyzed as Schwarz methods. These algorithms are fully defined in terms of a set of subspaces and auxiliary bilinear forms.A general theoretical framework has previously been developed. In this paper, these techniques are used in an analysis of iterative substructuring methods for elliptic problems in three dimensions. A special emphasis is placed on the difficult problem of designing good coarse models and obtaining robust methods for which the rate of convergence is insensitive to large variations in the coefficients of the differential equation.Domain decomposition algorithms can conveniently be built from modules that represent local and global components of the preconditioner. In this paper, a number of such possibilities are explored, and it is demonstrated how a great variety of fast algorithms can be designed and analyzed.