Titre : An efficient domain decomposition method with cross-point treatment for Helmholtz problems
Année : 2018
Type : conf_acti
Auteurs : A. Modave, X. Antoine, C. Geuzaine
Résumé : Solving high-frequency time-harmonic scattering problems using finite element techniques is challenging, as such problems lead to very large, complex and indefinite linear systems. Optimized Schwarz domain decomposition methods (DDMs) are currently a very promising approach, where subproblems of smaller sizes are solved in parallel using direct solvers, and are combined in an iterative procedure.It is well-known that the convergence rate of these methods strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local transmission conditions based on high-order absorbing boundary conditions (HABCs) have proved well suited [Boubendir et al, 2012; Gander et al, 2002]. They represent a good compromise between basic impedance conditions (which lead to suboptimal convergence) and the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain (which is expensive to compute). However, a direct application of this approach for domain decomposition configurations with cross-points, where more than two subdomains meet, does not provide satisfactory results.We present an improved DDM that efficiently addresses configurations with cross points. Noting that these points actually are corners for the subdomains, our strategy consists in incorporating a corner treatment developed for HABCs into the DDM procedure. After a presentation of the key aspects of the methods, the effectiveness of our approach is discussed with two-dimensional finite element results.
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