Méthodes de de Rham discrètes sur domains à topologie non triviale

Discrete de Rham methods on domains with non-trivial topology

• Auteur / Autrice : Vito Patierno
• Direction : Daniele Di pietro, Jérôme Droniou
• Type : Projet de thèse
• Discipline(s) : Mathématiques et Modélisation
• Date : Inscription en doctorat le 30/09/2024
• Etablissement(s) : Université de Montpellier (2022-....)
• Ecole(s) doctorale(s) : École Doctorale Information, Structures, Systèmes
• Partenaire(s) de recherche : Laboratoire : IMAG - Institut Montpelliérain Alexander Grothendieck
• Partenaire(s) de recherche : Equipe de recherche : ACSIOM - Equipe d'Analyse, Calcul Scientifique Industriel et Optimisation de Montpellier

Mots clés

Mots clés libres

Analyse numérique

Topologie algébrique

Méthodes polyédriques

Eléments finis

Finite elements

Polyhedral methods

Numerical analysis

Algebraic topology

Résumé

Voir résumé en anglais

The discrete de Rham (DDR) method [1,2] provides a paradigm for the discretisation on general meshes of partial differential equations (PDE) problems that relate to the de Rham complex. To this day, DDR methods have been used to carry out computations on domains with simple topology. Recent results [3] point out a way to efficiently treat the case of domains with complex topology, where non-trivial spaces of harmonic forms may appear. The goal of this PhD thesis is precisely to develop the numerical tools required in this case. The first application will be to problems in magnetostatics. Depending on the outcome, more complicated problems involving advanced Hilbert complexes may also be treated. Other methodological developments and applications may also be considered at a later stage of this PhD thesis. The candidates for this PhD position will hold a MSc in Applied Mathematics or Engineering. Experience in programming will be a welcome addition. This PhD position will be funded by the ERC project NEMESIS.