Abstract
In this paper, we develop in a general framework a non overlapping Domain Decomposition Method that is proven to be well-posed and converges exponentially fast, provided that specific transmission operators are used. These operators are necessarily non local and we provide a class of such operators in the form of integral operators. To reduce the numerical cost of these integral operators, we show that a truncation process can be applied that preserves all the properties leading to an exponentially fast convergent method. A modal analysis is performed on a separable geometry to illustrate the theoretical properties of the method and we exhibit an optimization process to further reduce the convergence rate of the algorithm.
Highlights
Our goal is to provide a class of transmission operators and a corresponding iterative process, which is a relaxed version of the Jacobi algorithm (1.10), in order to achieve the following requirements (P1) The method is robust in the sense that the convergence is guaranteed, (P2) The method is fast in the sense that the convergence is exponential, these two properties being valid in the most general case: arbitrary variable coefficients and general geometry of the interface
We show that some strategies are compatible with a truncation procedure to achieve a quasi localization of the integral operators
We propose several solutions that all rely on integral operators with a singular kernel, the key point being that the singularity is imposed by condition (2.15) : roughly speaking, TR must be a pseudo-differential operator of order 1
Summary
The present work is motivated by the numerical simulation of electromagnetic scattering by multilayered coated obstacles. Ω1 represents the inner-most layer and ΩJ the outer domain Note that this onion-skin structure prevents that more than two subdomains touch each other (see Fig. 1). The internal structure of each layer can be highly heterogeneous This provides a natural geometrical splitting for domain decomposition. Our aim in this paper is to propose a novel iterative domain decomposition method (DDM) that behaves well on this type of geometry.
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