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Non-local Impedance Operator for Non-overlapping DDM for the Helmholtz Equation

Francis Collino 1 Patrick Joly 1 Emile Parolin 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : In the context of time harmonic wave equations, the pioneering work of B. Després [4] has shown that it is mandatory to use impedance type transmission conditions in the coupling of sub-domains in order to obtain convergence of nonoverlapping domain decomposition methods (DDM). In later works [2, 3], it was observed that using non-local impedance operators leads to geometric convergence, a property which is unattainable with local operators. This result was recently extended to arbitrary geometric partitions, including configurations with cross-points, with provably uniform stability with respect to the discretization parameter [1]. We present a novel strategy to construct suitable non-local impedance operators that satisfy the theoretical requirements of [1] or [2, 3]. It is based on the solution of elliptic auxiliary problems posed in the vicinity of the transmission interfaces. The definition of the operators is generic, with simple adaptations to the acoustic or electromagnetic settings, even in the case of heterogeneous media. Besides, no complicated tuning of parameters is required to get efficiency. The implementation in practice is straightforward and applicable to sub-domains of arbitrary geometry, including ones with rough boundaries generated by automatic graph partitioners. We first provide in Section 1 a general definition of this novel transmission operator in a two-domain configuration. In Section 2 we then study more quantitatively the convergence in the geometric configuration of a closed wave-guide. Section 3 illustrates the results using actual finite element computations.
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Contributor : Emile Parolin <>
Submitted on : Wednesday, May 12, 2021 - 4:51:36 PM
Last modification on : Thursday, May 20, 2021 - 12:20:18 PM


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  • HAL Id : hal-03225581, version 1


Francis Collino, Patrick Joly, Emile Parolin. Non-local Impedance Operator for Non-overlapping DDM for the Helmholtz Equation. 2021. ⟨hal-03225581⟩



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