Résolution rapide d'équations intégrales pour un problème d'antennes par des méthodes d'ondelettes

Cyril Safa
Abstract : Integral equations methods for solving PDEs, and in particular Maxwell's equations system, are now well known for about twenty years. After discretization by finite elements, a dense linear system arises, which is unfortunate for any numerical implementation. For positive order operators, some successfull works have been led to reduce the discrete linear system to a sparse one. Some difficulties remained for Maxwell's system: energy space(s), presence of a negative order operator, and thus, choice of wavelets. In this thesis, I give a method to bring Maxwell's scattering problem back to classical Sobolev spaces on a manifold using Hodge decompositions. I also give a way to compress the discrete problem provided wavelets satisfy some properties (moment conditions, stability). The compression works even with wavelets based on piecewise linear finite elements, despite the presence of a negative order operator, and does not affect optimal convergence rates. The analysis has been done on a closed simply connected regular manifold as well as on a part of such manifolds, with a polygonal boundary (open surface). Functional spaces and wavelet compression, much more complicated in the latter case, are studied in detail.
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Submitted on : Thursday, June 24, 2004 - 12:18:46 PM
Last modification on : Thursday, June 24, 2004 - 12:18:46 PM
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Cyril Safa. Résolution rapide d'équations intégrales pour un problème d'antennes par des méthodes d'ondelettes. Mathématiques [math]. Université Rennes 1, 2001. Français. ⟨tel-00006317⟩

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