Combining Analytic Preconditioner and Fast Multipole Method for the 3-D Helmholtz Equation

Marion Darbas (1) , Eric Darrigrand (2) , Yvon Lafranche (2)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 IRMAR - Institut de Recherche Mathématique de Rennes

The paper presents a detailed numerical study of an iterative solution to 3-D sound-hard acoustic scattering problems at high frequency considering the Combined Field Integral Equation (CFIE). We propose a combination of an OSRC preconditioning technique and a Fast Multipole Method which leads to a fast and efficient algorithm independently of both a frequency increase and a mesh refinement. The OSRC-preconditioned CFIE exhibits very interesting spectral properties even for trapping domains. Moreover, this analytic preconditioner shows highly-desirable advantages: sparse structure, ease of implementation and low additional computational cost. We first investigate the numerical behavior of the eigenvalues of the related integral operators, CFIE and OSRC-preconditioned CFIE, in order to illustrate the influence of the proposed preconditioner. We then apply the resolution algorithm to various and significant test-cases using a GMRES solver. The OSRC-preconditioning technique is combined to a Fast Multipole Method in order to deal with high-frequency 3-D cases. This variety of tests validates the effectiveness of the method and fully justifies the interest of such a combination.

Mots clés: integral equation methods; analytic preconditioner; FMM; high-frequency scattering; Helmholtz

Domaines: Analyse numérique [math.NA]

Marion Darbas, Eric Darrigrand, Yvon Lafranche. Combining Analytic Preconditioner and Fast Multipole Method for the 3-D Helmholtz Equation. Journal of Computational Physics, 2013, 236, pp.289-316. ⟨10.1016/j.jcp.2012.10.059⟩. ⟨hal-00749822⟩