Abstract.Tuning the alternating Schwarz method to the exterior problems is the subject of this paper. We present the original algorithm and we propose a modification of it, so that the solution of the subproblem involving the condition at infinity has an explicit integral representation formulas while the solution of the other subproblem, set in a bounded domain, is approximated by classical variational methods. We investigate many of the advantages of the new Schwarz approach: a geometrical convergence rate, an easy implementation, a substantial economy in computational costs and a satisfactory accuracy in the numerical results as well as their agreement with the theoretical statements.Mathematics Subject Classification. 35J20, 65N38, 65N55.

Abstract. The purpose of this paper is to comment a frequent observation by the engineers studying acoustic scattering. It is related to the convergence of the GMRES method when solving systems Ax = b with A = I − B. The paper includes a theorem which expresses the convergence rate when some eigenvalues of B have modulus larger than one; that rate depends on the rate measured when solving the system obtained by spectral projection onto the invariant subspace corresponding to the other eigenvalues. The conclusion of the theorem is illustrated on the Helmholtz equation.

In this work, we are interested in the modelling of the acoustic attenuation of exhaust mufflers including perforated ducts, and its numerical computation. The study is worked out in harmonic time regime, for the two-dimensional case. The hole diameter and the center-to-center distance between consecutive holes are supposed of same order, and small compared to the size of the muffler. The formulation is derived by using multiscale techniques and matching the asymptotic expansions. The numerical method couples finite elements in the muffler with modal decomposition in the inlet and the outlet of the duct.

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SUMMARYWe consider the Jacobi preconditioner of the GMRES method introduced by Liu and Jin for the scattering problem (IEEE Trans. Ante. Prop. 2002; 50:132-140). We explain why it is a particular form of the Schwarz' preconditioner with a complete overlap and specific transmission conditions. So far, a superlinear convergence has been predicted by the general theory without any additional indication on the convergence rates. Here, we establish error bounds that provide accurate convergence rates in two and three dimensions. Courant-Weyl's min-max principle applied to some kernel operators together with some polynomial approximation estimates are the milestones for the proofs.