TR2004-855 DUAL-PRIMAL FETI METHODS FOR LINEAR ELASTICITY AXEL KLAWONN * AND OLOF B. WIDLUND † September 14, 2004 Abstract. Dual-Primal FETI methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold through- out the iterations, as in primal iterative substructuring methods, while most of the constraints are enforced by Lagrange multipliers, as in one-level FETI methods. 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, which are independent of even large changes in the stiffnesses of the subdomains across the interface between them. A theoretical analysis is provided and condition number bounds are established which are uniform with respect to arbitrarily large jumps in the Young’s modulus of the material and otherwise only depend polylogarithmically on the number of unknowns of a single subdomain. Key words. domain decomposition, Lagrange multipliers, FETI, preconditioners, elliptic sys- tems, elasticity, finite elements. AMS subject classifications. 65F10,65N30,65N55 1. Introduction. We will consider iterative substructuring methods with La- grange multipliers for the elliptic system of linear elasticity. The algorithms be- long to the family of dual-primal FETI (finite element tearing and interconnection) methods which was introduced for linear elasticity problems in the plane in [8] and then extended to three dimensional elasticity problems in [9]. In dual-primal FETI (FETI-DP) methods, some continuity constraints on primal displacement variables are required to hold throughout the iterations, as in primal iterative substructuring methods, while most of the constraints are enforced by the use of dual Lagrange mul- tipliers, as in the older one-level FETI algorithms. The primal constraints should be chosen so that the local problems become invertible. They also provide a coarse problem and they should be selected so that the iterative method converges rapidly. We also wish to use relatively few, and effective, primal constraints since the they represent a global part of the preconditioner which is relatively difficult to parallelize. More recently, the family of algorithms for scalar elliptic problems in three dimen- sions was extended and a theory was provided in [15, 16]; see also [24, Section 6.4]. It was shown that the condition number of the dual-primal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subre- gion problems and that the bounds can otherwise be made independent of the number of subdomains, the mesh size, and jumps in the coefficients. In the case of the elliptic system of partial differential equations arising from linear elasticity, essential changes in the selection of the primal constraints have to be made in order to obtain the same quality bounds for elasticity problems as in the scalar case. Special emphasis will be given to developing robust condition number estimates with bounds which are inde- pendent of arbitrarily large jumps of the material coefficients. For benign coefficients, without large jumps, it is sufficient to select an appropriate set of edge averages as * Fachbereich Mathematik, Universit¨at Duisburg-Essen, Campus Essen, Universit¨atsstraße 3, D- 45117 Essen, Germany. E-mail: klawonn@math.uni-essen.de, URL: http://www.uni-essen.de/numa. † Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA. E-mail: widlund@cs.nyu.edu, URL: http://www.cs.nyu.edu/cs/faculty/widlund. This work was supported in part by the US Department of Energy under Contracts DE-FG02-92ER25127 and DE-FC02-01ER25482. 1