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A Quasi-Optimal Non-Overlapping Domain Decomposition Algorithm for the Helmholtz Equation

Summary: A Quasi-Optimal Non-Overlapping Domain Decomposition
Algorithm for the Helmholtz Equation
Y. Boubendir
, X. Antoine
, C. Geuzaine
February 23, 2011
This paper presents a new non-overlapping domain decomposition method for the Helmholtz
equation, whose effective convergence is quasi-optimal. These improved properties result from a
combination of an appropriate choice of transmission conditions and a suitable approximation of the
Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical
results validating the new approach are presented in both two and three dimensions.
1 Introduction
In this paper, we are interested in non-overlapping Domain Decomposition Methods (DDMs) for the
Helmholtz equation. Such methods were introduced by Lions [36] for the Laplace equation and extended
to the Helmholtz equation by Despr´es [19, 20, 21]. Essentially, the method consists in combining the
continuity conditions (of the field and its normal derivative) on the artificial interfaces between subdo-
mains, in order to obtain Robin boundary conditions and to solve the overall problem by iterating over
the subdomains [39, 47, 45]. Robin conditions (also called absorbing or impedance boundary conditions)
are chosen to couple the subdomains because using the natural conditions leads to divergent iterative


Source: Antoine, Xavier - Institut de Mathématiques Élie Cartan, Université Henri Poincaré - Nancy 1


Collections: Mathematics