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A Priori Error Analysis For A Penalty Finite Element MethodZerbinati, Umberto 04 April 2022 (has links)
Partial differential equations on domains presenting point singularities have always been of interest for applied mathematicians; this interest stems from the difficulty to prove regularity results for non-smooth domains, which has important consequences in the numerical solution of partial differential equations. In my thesis I address those consequences in the case of conforming and penalty finite element methods. The main results here contained concerns a priori error estimates for conforming and penalty finite element methods with respect to the energy norm, the $\mathcal{L}^2(\Omega)$ norm in both the standard and weighted setting.
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Bending of an orthotropic cusped plateJaiani, George V. January 1998 (has links)
The bending of an orthotropic cusped plate in energetic and weighted Sobolev spaces has been considered. The existence and uniqueness of generalized and weak solutions of admissible boundary value problems (BVPs) have been investigated.
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Contribution à l'analyse mathématique et à la résolution numérique d'un problème inverse de scattering élasto-acoustique / Contribution to the mathematical analysis and to the numerical solution of an inverse elasto-acoustic scattering problemEstecahandy, Elodie 19 September 2013 (has links)
La détermination de la forme d'un obstacle élastique immergé dans un milieu fluide à partir de mesures du champ d'onde diffracté est un problème d'un vif intérêt dans de nombreux domaines tels que le sonar, l'exploration géophysique et l'imagerie médicale. A cause de son caractère non-linéaire et mal posé, ce problème inverse de l'obstacle (IOP) est très difficile à résoudre, particulièrement d'un point de vue numérique. De plus, son étude requiert la compréhension de la théorie du problème de diffraction direct (DP) associé, et la maîtrise des méthodes de résolution correspondantes. Le travail accompli ici se rapporte à l'analyse mathématique et numérique du DP élasto-acoustique et de l'IOP. En particulier, nous avons développé un code de simulation numérique performant pour la propagation des ondes associée à ce type de milieux, basé sur une méthode de type DG qui emploie des éléments finis d'ordre supérieur et des éléments courbes à l'interface afin de mieux représenter l'interaction fluide-structure, et nous l'appliquons à la reconstruction d'objets par la mise en oeuvre d'une méthode de Newton régularisée. / The determination of the shape of an elastic obstacle immersed in water from some measurements of the scattered field is an important problem in many technologies such as sonar, geophysical exploration, and medical imaging. This inverse obstacle problem (IOP) is very difficult to solve, especially from a numerical viewpoint, because of its nonlinear and ill-posed character. Moreover, its investigation requires the understanding of the theory for the associated direct scattering problem (DP), and the mastery of the corresponding numerical solution methods. The work accomplished here pertains to the mathematical and numerical analysis of the elasto-acoustic DP and of the IOP. More specifically, we have developed an efficient numerical simulation code for wave propagation associated to this type of media, based on a DG-type method using higher-order finite elements and curved edges at the interface to better represent the fluid-structure interaction, and we apply it to the reconstruction of objects with the implementation of a regularized Newton method.
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