We present a Lagrange multiplier based two-level domain decomposition method for solving iteratively large-scale systems of equations arising from the finite element discretization of high-frequency exterior Helmholtz problems. The proposed method is essentially an extension of the regularized FETI (Finite Element Tearing and Interconnecting) method to indefinite problems. Its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix, and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves. We show numerically that the proposed method is scalable with respect to the mesh size, the subdomain size, and the wavenumber. We report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements. Mathematics Subject Classification (1991): 65N55Correspondence to: C. Farhat 284 C. Farhat et al.

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The Galerkin method enriched with residual-free bubbles is considered for approximating the solution of the Helmholtz equation. Two-dimensional tests demonstrate the improvement over the standard Galerkin method and the Galerkin-least-squares method using piecewise bilinear interpolations.

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SUMMARYWe report on a generalization of the Bayliss-Gunzburger-Turkel non-re ecting boundary conditions to arbitrarily shaped convex artiÿcial boundaries. For elongated scatterers such as submarines, we show that this generalization can improve signiÿcantly the computational e ciency of ÿnite element methods applied to the solution of three-dimensional acoustic scattering problems.