Theory and implementation of $\mathcal{H}$-matrix based iterative and direct solvers for oscillatory kernels

Stéphanie Chaillat 1 Luca Desiderio 1 Patrick Ciarlet 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, ENSTA ParisTech UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : In this work, we study the accuracy and efficiency of hierarchical matrix ($\mathcal{H}$-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green's tensors. It is well known in the literature that standard $\mathcal{H}$-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. $\mathcal{H}^2$-matrix and directional approaches have been proposed to overcome this problem. However the implementation of such methods is much more involved than the standard $\mathcal{H}$-matrix representation. The central questions we address are twofold. (i) What is the frequency-range in which the $\mathcal{H}$-matrix format is an efficient representation for 3D elastodynamic problems? (ii) What can be expected of such an approach to model problems in mechanical engineering? We show that even though the method is not optimal (in the sense that more involved representations can lead to faster algorithms) an efficient solver can be easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01543919
Contributeur : Stéphanie Chaillat <>
Soumis le : mercredi 21 juin 2017 - 11:33:13
Dernière modification le : mardi 18 juillet 2017 - 14:44:09

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  • HAL Id : hal-01543919, version 1
  • ARXIV : 1706.09384

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Stéphanie Chaillat, Luca Desiderio, Patrick Ciarlet. Theory and implementation of $\mathcal{H}$-matrix based iterative and direct solvers for oscillatory kernels. 2017. <hal-01543919>

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