Reference : A posteriori error estimation for finite element approximations of fractional Laplaci...
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A posteriori error estimation for finite element approximations of fractional Laplacian problems and applications to poro–elasticity
[fr] Estimation d’erreur a posteriori pour l’approximation de problèmes Laplaciens fractionnaires et applications en poro-élasticité
Bulle, Raphaël mailto [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Engineering (DoE) >]
University of Luxembourg, ​Esch-sur-Alzette, ​​Luxembourg
Université Bourgogne Franche-Comté ​​France, ​Besançon, ​​France
Docteur en Sciences de l'Ingénieur
Bordas, Stéphane mailto
Chouly, Franz
Hale, Jack mailto
Lozinski, Alexei
Francis, Olivier mailto
Bonito, Andrea
Vohralík, Martin
Becker, Roland
[en] This manuscript is concerned with a posteriori error estimation for the finite element discretization of standard and fractional partial differential equations as well as an application of fractional calculus to the modeling of the human meniscus by poro-elasticity equations. In the introduction, we give an overview of the literature of a posteriori error estimation in finite element methods and of adaptive refine- ment methods. We emphasize the state–of–the–art of the Bank–Weiser a posteriori error estimation method and of the adaptive refinement methods convergence results. Then, we move to fractional partial differential equations. We give some of the most common discretization methods of fractional Laplacian operator based equations. We review some results of a priori error estimation for the finite element discretization of these equations and give the state–of–the–art of a posteriori error estimation. Finally, we review the literature on the use of the Caputo’s fractional derivative in applications, focusing on anomalous diffusion and poro-elasticity applications. The rest of the manuscript is organized as follow. Chapter 1 is concerned with a proof of the reliability of the Bank–Weiser estimator for three–dimensional problems, extending a result from the literature. In Chapter 2 we present a numerical study of the Bank–Weiser estimator, provide a novel implementation of the estimator in the FEniCS finite element software and apply it to a variety of elliptic equations as well as goal-oriented error estimation. In Chapter 3 we derive a novel a posteriori estimator for the L2 error induced by the finite element discretization of fractional Laplacian operator based equations. In Chapter 4 we present new theoretical results on the convergence of a rational approximation method with consequences on the approximation of fractional norms as well as a priori error estimation results for the finite element discretization of fractional equations. Finally, in Chapter 5 we provide an application of fractional calculus to the study of the human meniscus via poro-elasticity equations.
[fr] Ce manuscrit traite d’estimation d’erreur a posteriori pour la discrétisation d’équations aux dérivées partielles standard et fractionnaires par les méthodes éléments finis ainsi que de l’application de l’analyse fractionnaire à la modélisation du ménisque humain par les équations de poro-élasticité. Dans l’introduction, nous donnons un aperçu de la littérature sur l’estimation d’erreur a posteriori pour les méth- odes éléments finis et des méthodes de raffinement adaptatif. Nous insistons particulièrement sur l’état de l’art de la méthode d’estimation d’erreur a posteriori de
Bank-Weiser et sur les résultats de convergence des méthodes adaptatives. Ensuite, nous nous intéressons aux équations aux dérivées partielles fractionnaires. Nous présentons certaines méthodes de discrétisation d’équations basées sur l’opérateur Laplacien fractionnaire et donnons l’état de l’art sur l’estimation d’erreur a posteriori. Finalement, nous donnons un aperçu de la littérature concernant les applications de la dérivée fractionnaire au sens de Caputo en nous concentrant sur le phénomène de diffusion anormale et les applications en poro-élasticité.
University of Luxembourg - UL
This thesis is made available under the Creative Commons Attribution Non-Commercial No Derivatives 4.0 licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work.
This thesis was also submitted to obtain the degree of Docteur de l’Université de Bourgogne Franche–Comté en Mathématiques.

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