Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. SCI. COMPUT. c  2010 Society for Industrial and Applied Mathematics Vol. 32, No. 6, pp. 3604–3626 BDDC PRECONDITIONERS FOR SPECTRAL ELEMENT DISCRETIZATIONS OF ALMOST INCOMPRESSIBLE ELASTICITY IN THREE DIMENSIONS ∗ LUCA F. PAVARINO † , OLOF B. WIDLUND ‡ , AND STEFANO ZAMPINI § Abstract. Balancing domain decomposition by constraints (BDDC) algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with Gauss–Lobatto– Legendre spectral elements in three dimensions. Initially mixed spectral elements are employed to discretize the almost incompressible elasticity system, but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped. Appropriate sets of primal constraints can be associated with the subdomain vertices, edges, and faces so that the resulting BDDC methods have a fast convergence rate independent of the almost incompressibility of the material. In particular, the condition number of the BDDC preconditioned operator is shown to depend only weakly on the polynomial degree n, the ratio H/h of subdomain and element diameters, and the inverse of the inf-sup constants of the subdomains and the underlying mixed formulation, while being scalable, i.e., independent of the number of subdomains and robust, i.e., independent of the Poisson ratio and Young’s modulus of the material considered. These results also apply to the related dual-primal finite element tearing and interconnect (FETI-DP) algorithms defined by the same set of primal constraints. Numerical experiments, carried out on parallel computing systems, confirm these results. Key words. domain decomposition, balancing domain decomposition by constraints precondi- tioners, almost incompressible elasticity, mixed spectral elements AMS subject classifications. 65F08, 65N30, 65N35, 65N55 DOI. 10.1137/100791701 1. Introduction. The purpose of this paper is to construct and analyze bal- ancing domain decomposition by constraints (BDDC) algorithms, (see [13, 41]) for the system of almost incompressible elasticity in three dimensions, discretized with Gauss–Lobatto–Legendre (GLL) spectral elements. As the material becomes almost incompressible, the resulting discrete system becomes extremely ill conditioned, par- ticularly so when increasing the polynomial degree of the spectral elements. Therefore, it is quite a challenge to devise domain decomposition preconditioners that maintain scalability and fast convergence rates also for almost incompressible materials. Our algorithm builds on earlier work by Li and Widlund [36] for the Stokes system, but here we can work with a positive definite reformulation of an underlying mixed formu- lation of the elasticity system, obtained by eliminating all displacement and pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped. Our overall strategy assumes that the set of primal constraints works well in the compressible case and, in addition, that a no net flux condition is satisfied across the boundary of each subdomain. We show that appropriate sets of primal ∗ Received by the editors April 9, 2010; accepted for publication (in revised form) September 21, 2010; published electronically December 21, 2010. http://www.siam.org/journals/sisc/32-6/79170.html † Dipartimento di Matematica, Universit`a di Milano, Via Saldini 50, 20133 Milano, Italy (luca. pavarino@unimi.it). This work was supported by grants of M.I.U.R. (PRIN 200774A7LH 003). ‡ Courant Institute of Mathematical Sciences, New York, NY 10012 (widlund@cims.nyu.edu). This work was supported in part by the U.S. Department of Energy under contract DE-FG02-06ER25718 and in part by the National Science Foundation grant DMS-0914954. § Dipartimento di Matematica, Universit`a di Milano, Via Saldini 50, 20133 Milano, Italy (stefano. zampini@unimi.it). This work was supported by grants of M.I.U.R. (PRIN 200774A7LH 003). 3604 Downloaded 07/31/13 to 216.165.95.77. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php