Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems Charbel Farhat a, * , Antonini Macedo a , Michel Lesoinne a , Francois-Xavier Roux b , Fr  ed eric Magoul  es b , Armel de La Bourdonnaie c a Department of Aerospace Engineering Sciences and Center for Aerospace Structures, University of Colorado at Boulder, Boulder, CO 80309-0429, USA b ONERA, Direction de l'Informatique, 29 Av. de la Division Leclerc, BP72 92322 Chatillon Cedex, France c CERMICS/CAIMAN, INRIA BP.93 - F06902, Sophia Antipolis Cedex, France Abstract We present two dierent but related Lagrange multiplier based domain decomposition (DD) methods for solving iteratively large- scale systems of equations arising from the ®nite element discretization of high-frequency exterior Helmholtz problems. The proposed methods are essentially two distinct extensions of the regularized ®nite element tearing and interconnecting (FETI) method to inde®nite or complex problems. The ®rst method employs a single Lagrange multiplier ®eld to glue the local solutions at the subdomain interface boundaries. The second method employs two Lagrange multiplier ®elds for that purpose. The key ingredients of both of these FETI methods are the regularization of each subdomain matrix by a complex lumped mass matrix de®ned on the subdomain interface boundary, and the preconditioning of the global interface problem by a coarse second-level problem constructed with planar waves. We show numerically that both methods are scalable with respect to the mesh size, the subdomain size, and the wavenumber, but that the FETI method with a single Lagrange multiplier ®eld ± labeled FETI-H (H for Helmholtz) in this paper ± delivers superior computational performances. We apply the FETI-H method to the parallel solution on a 24-processor Origin 2000 of an acoustic scattering problem with a submarine shaped obstacle, and report performance results that highlight the unique eciency of this DD method for the solution of high frequency acoustic scattering problems. Ó 2000 Elsevier Science S.A. All rights reserved. MSC: 57; 49; 18; 20 1. Introduction The d-dimensional scattering of time-harmonic acoustic waves by an impenetrable obstacle embedded in a homogeneous medium X  R d can be formulated as the following exterior Helmholtz boundary value problem r 2 u  k 2 u  f in X; u  g 1 on C D ; ru  m  g 2 on C N ; 1 lim r!1 r d 1=2 ou or   iku   0; www.elsevier.com/locate/cma Comput. Methods Appl. Mech. Engrg. 184 (2000) 213±239 * Corresponding author. Tel.: +1-303-492-3992; fax: +1-303-492-4990. E-mail address: charbel@alexandra.colorado.edu (C. Farhat). 0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 2 2 9 - 7