GANDER , Martin

Multitrace formulations (MTF) for boundary integral equations (BIE) were developed over the last few years in [4] and [1, 2] for the simulation of electromagnetic problems in piecewise constant media, see also [3] for associated boundary integral methods. The MTFs are naturally adapted to the developments of new block preconditioners, as...

Schwarz domain decomposition methods can be analyzed both at the continuous and discrete level. For consistent discretizations, one would naturally expect that the discretized method performs as predicted by the continuous analysis. We show in this short note for two model problems that this is not always the case, and that the discretization...

Overlap is essential for the classical Schwarz method to be convergent when solving elliptic problems. Over the last decade, it was however observed that when solving systems of hyperbolic partial differential equations, the classical Schwarz method can be convergent even without overlap. We show that the classical Schwarz method without...

Multitrace formulations (MTF) for boundary integral equations (BIE) were developed over the last few years in [4] and [1, 2] for the simulation of electromagnetic problems in piecewise constant media, see also [3] for associated boundary integral methods. The MTFs are naturally adapted to the developments of new block preconditioners, as...

After the development of optimized Schwarz methods for the Helmholtz equation [2, 3, 4, 12, 14], extensions to the more difficult case of Maxwell's equations were developed: for curl-curl formulations, see [1]. For first order formulations without conductivity, see [7], and with conductivity, see [5, 11]. For DG discretizations of Maxwell's...

Over the last five years, classical and optimized Schwarz methods have been developed for anisotropic elliptic problems discretized with Discrete Duality Finite Volume (DDFV) schemes. Like for Discontinuous Galerkin methods (DG), it is not a priori clear how to appropriately discretize transmission conditions with DDFV, and numerical...

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This paper deals with the derivation and analysis of reduced order elliptic PDE models on fractured domains. We use a Fourier analysis to obtain coupling conditions between subdomains, when the fracture is represented as a hypersurface embedded in the surrounded rock matrix. We compare our results to prominent examples from the literature, for...

Schwarz waveform relaxation methods have been studied for a wide range of scalar linear partial differential equations (PDEs) of parabolic and hyperbolic type. They are based on a space-time decomposition of the computational domain and the subdomain iteration uses an overlapping decomposition in space. There are only few convergence studies...

Classical Schwarz methods need in general overlap to converge, but in the case of hyperbolic problems, they can also be convergent without overlap, see [7]. For the first order formulation of Maxwell equations, we have proved however in [18] that the classical Schwarz method without overlap does not converge in most cases in the presence of...

Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, and it was observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for...

Discrete Duality Finite Volume (DDFV) methods are very well suited to discretize anisotropic diffusion problems, even on meshes with low mesh quality. Their performance stems from an accurate reconstruction of the gradients between mesh cell boundaries, which comes however at the cost of using both a primal (cell centered) and a dual (vertex...

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We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the...

Over the last five years, classical and optimized Schwarz methods with Robin transmission conditions have been developed for anisotropic elliptic problems discretized by Discrete Duality Finite Volume (DDFV) schemes. We present here the case of higher order transmission conditions in the framework of DDFV. We prove convergence of the...

In [14, 15], it was discovered that heterogeneous media can actually im- prove the convergence of optimized Schwarz methods, provided that the coefficient jumps are aligned with the interfaces, and the jumps are taken into account in an appropriate way in the transmission conditions. Similar results were found for Maxwell's equations in [9] and...

We consider a Stokes flow along a thin fracture coupled to a Darcy flow in the surrounding matrix domain. In order to derive a dimensionally reduced model representing the fracture as an interface coupled to the surrounding matrix, we extend the methodology based on Fourier analysis developed in [1] for a Darcy-Darcy coupling. We show that this...

Local Multi-Trace Formulations (local MTF) are block-sparse boundary integral equations adapted to elliptic PDEs with piece-wise constant coefficients (typically multi-subdomain scattering problems) only recently introduced in [Hiptmair & Jerez-Hanckes, 2012]. In these formulations, transmission conditions are enforced by means of local...

Transmission conditions between subdomains have a substantial influence on the convergence of iterative domain decomposition algorithms. For Maxwell's equations, transmission conditions which lead to rapidly converging algorithms have been developed both for the curl-curl formulation of Maxwell's equation, see [2, 3, 1], and also for first...

The first ideas for optimized Schwarz methods for Maxwell's equations came from the analysis of optimized Schwarz methods for the Helmholtz equation, see [3, 4, 2, 11]. For the case of the rot-rot formulation of the Maxwell equations, optimized Schwarz methods were developed in [1]. The systematic study of Schwarz methods for Maxwell's...

In a previous paper, two of the authors have proposed and analyzed an entire hierarchy of optimized Schwarz methods for Maxwell's equations both in the time-harmonic and time-domain case. The optimization process has been perfomed in a particular situation where the electric conductivity was neglected. Here, we take into account this physical...

Like the Helmholtz equation, the high frequency time-harmonic Maxwell's equa- tions are difficult to solve by classical iterative methods. Domain decomposition methods are currently most promising: following the first provably convergent method in [4], various optimized Schwarz methods were developed over the last decade [2, 3, 10, 11, 1, 6,...

We are interested here in the numerical modeling of time-harmonic electromagnetic wave propagation problems in irregularly shaped domains and heterogeneous media. In this context, we are naturally led to consider volume discretization methods (i.e. finite element method) as opposed to surface discretization methods (i.e. boundary element...

The time-harmonic Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental for the simulation of many modern devices we have become used to in everyday life. The numerical solution of these equations is hampered by two fundamental problems: first, in the high frequency regime, very fine meshes need to be...