Convergence of a Balancing Domain Decomposition by Constraints and Energy Minimization Jan Mandel ∗ Clark R. Dohrmann † November 29, 2002 Dedicated to Professor Ivo Marek on the occasion of his 70th birthday. Abstract A convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts. The preconditioner is formulated as an Additive Schwarz method and analyzed by building on existing results for Balancing Domain Decomposition. The main result is a bound on the condition number based on inequalities involving the matrices of the preconditioner. Estimates of the usual form C(1 + log 2 (H/h)) are obtained under the standard assumptions of substructuring theory. Computational results demonstrating the performance of method are included. Keywords. Iterative substructuring, FETI, Balancing Domain Decomposition, Neumann- Neumann, Additive Schwarz methods, non-overlapping domain decomposition methods AMS Subject Classification. 65N55, 65F10, 65N30 1 Introduction The purpose of this paper is to restate the method of Dohrmann [6] in an abstract manner, and to provide a convergence theory. The first step to this end is to define the method as an Additive Schwarz preconditioner. The main theoretical result can then be obtained by understanding the preconditioner as a Balancing Domain Decomposition type method and applying similar analysis techniques. The present analysis is focused on a two-level method for positive definite systems of equations, but we hope it will provide a solid foundation for multilevel extensions and variants for other problem types. The Balancing Domain Decomposition (BDD) method was created by Mandel [16] by adding a coarse problem to an earlier method of Neumann-Neumann type by DeRoeck and Le Tallec [20]. The original BDD method used a multiplicative coarse correction to make sure that the residuals, which become right-hand sides of the local problems, are in the range of the substructure matrices, that is, “balanced”. The method was extended to problems with jumps in coefficients between substructures by Dryja and Widlund [8] and Mandel and Brezina [18]. Le Tallec, Mandel, and * Department of Mathematics, University of Colorado at Denver, and Department of Aerospace Engineering Sciences, University of Colorado at Boulder. Supported by the National Science Foundation under Grant DMS-0074278. † Structural Dynamics Research Department, Sandia National Laboratories, Albuquerque, New Mexico. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000. 1