Development and comparison of finite element bases for solving the transport equation on hexagonal meshes ; Développement et comparaison de bases d'éléments finis pour la résolution de l'équation de transport sur des maillages hexagonaux

In the realm of applied mathematics, this Ph.D. proposal is related to the numerical solution of partial differential equations through the discretization of the associated dependent variables. In the frame of finite element analysis, it is focused on the construction of high-order schemes on polygonal meshes. The industrial application of such developments is the solution to the Boltzmann equation that describes the transport of neutrons within the core of a nuclear reactor. In this context, many modern codes use a finite-element-based discretization (more precisely, an upwind discontinuous Galerkin scheme) on triangular or quadrilateral meshes spanning the spatial domain. This efficient and versatile method is well suited for the development of adaptive mesh refinement techniques. The objective of this Ph.D. work is the analysis and the development of such a scheme optimised for honeycomb meshes (i.e composed of regular hexagons) that are typical of the assembly pattern in sodiumcooled fast reactors. From the application point of view, this optimisation is required in order to improve the time-consuming multiphysics simulations of accidental transient for such reactors. In order to reach this goal, the idea is to construct high-order finite element bases over the hexagon. In particular, the development of such a new scheme will encompass a p−adaptive mesh refinement procedure that requires an a posteriori error estimation in such a way that the scheme order can be locally increases in order to decrease the error in an optimal way with respect to the computational time. ; Cette proposition de thèse s'inscrit dans le cadre mathématique de la résolution numérique d'équations aux dérivées partielles par le biais d'une discrétisation des variables dépendantes. Elle s'intéresse, dans un formalisme d'éléments finis, à la construction de schémas d'ordre élevé (au sens de l'espace discrétisé) sur des maillages polygonaux. Le cadre industriel applicatif est la résolution de l'équation de Boltzmann appliquée au transport ...