A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems

Abstract

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.

Scattered Field Computation With an Extended Feti-Dpem2 Method

“…After having constructed an augmented Lagrangian functional and found its saddle-point thanks to the Karush-Kuhn-Tucker conditions [21,24], we must solve the following equation, for each ith subdomain,…”

“…Finally the solution of the interface problem serves as the right-handside of each local problem. This method has been applied in many domains like mechanics [19,20], acoustic wave propagation [21][22][23], and in electromagnetism [24][25][26][27][28][29][30][31][32]. For example, related DDM methods have been developed for simulating the interactions of photonic crystals with electromagnetic waves [33,34].…”

Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem

“…[3]. A great variety of techniques based on local transmission conditions have thus been proposed over the years: these include the class of FETI-H methods [31,32,10,33], the optimized Schwarz approach [6], the evanescent modes damping algorithm [34,35,36] and the Padé-localized square-root operator [7]. All these local transmission conditions can be seen as approximations of the exact DtN operator; the better the related impedance operators approximate the exact DtN operator on all the modes of the solution, the better the convergence properties of the resulting DDM.…”

Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

“…In order to adapt this class of methods to Helmholtz problems, the first variant was the FETI-H method (for FETI-Helmholtz), see [26]. Instead of using Neumann transmission conditions in the dual Schur complement formulation, Robin conditions ∂ n − ik are used, but then still Dirichlet traces are matched in order to obtain a substructured formulation.…”