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open 2 no From the introduction to the monograph: Students of scientific disciplines usually meet vector fields during their undergraduate courses in connection with physics, when studying conservative forces and potentials. Then the vector fields reappear in the context of the differential geometry, in the manifold theory, and again in the theory of Lie groups and in the advanced theory of ordinary differential equations. The aim of this book is to provide the reader with a gentle path through the multifaceted theory of vector fields, starting from the definition and the basic properties of vector fields and flows, and ending with some of their countless applications. The building blocks of rich background material comprise the following topics: ∙ commutators and Lie derivatives; the semi-group property and the equation of variation; global vector fields, C^0,C^k,C^ω-dependence; ∙ relatedness, invariance and commutability; Hadamard-type formulas; ∙ composition of flows: Taylor approximation and exact formulas; ∙ the algebraic Campbell-Baker-Hausdorff-Dynkin Theorem; the relevant series and convergence; the Campbell-Baker-Hausdorff-Dynkin operation and its local associativity; Poincaré-type ordinary differential equations; ∙ iterated commutators and the Hörmander bracket-generating condition; connectivity; sub-unit curves and the control distance. Once the background material is established in the first part of the monograph, the application mainly deals, according to the choice of the authors, with the following three settings: ∙ ordinary differential equations theory; ∙ weak and strong maximum principles, propagation principles; ∙ Lie groups, with an emphasis on the construction of Lie groups. This book also provides an introduction to the basic theory of geometrical analysis with a new foundational presentation based on ordinary differential equations techniques, in a unitary and self-contained way. open S Biagi; A Bonfiglioli S Biagi; A Bonfiglioli

We propose in this thesis a methodology to compute the Helmholtz-Hodge decomposition on discrete polyhedral meshes. The challenge of this work isto preserve the properties of the decomposition at the discrete level. In our literature survey, we have identified the need of mimetic schemes to achieve our goal. The description and validation of our implementation of these schemes are presented inthis document. We revisit and improve the methods of decomposition we then study through numerical experiments. In particular, we detail our choice of linear solvers and the convergence of extracted quantities on various series of polyhedral meshes and boundary conditions. Finally, we apply the Helmholtz-Hodge decomposition to the study of two turbulent flows: a turbulent channel flow and a homogeneous isotropic turbulent flow., Nous proposons dans ce mémoire de thèse une méthodologie permettant la résolution du problème de la décomposition de Hodge-Helmholtz discrète sur maillages polyédriques. Le défi de ce travail consiste à respecter les propriétés de la décomposition au niveau discret. Pour répondre à cet objectif, nous menons une étude bibliographique nous permettant d'identifier la nécessité de la mise en oeuvre de schémas numériques mimétiques. La description ainsi que la validation de la mise en oeuvre de ces schémas sont présentées dans ce mémoire. Nous revisitons et améliorons les méthodes de décomposition que nous étudions ensuite au travers d'expériences numériques. En particulier, nous détaillons le choix d'un solveur linéaire ainsi que la convergence des quantités extraites sur un ensemble varié de maillages polyédriques et de conditions aux limites. Nous appliquons finalement la décomposition de Hodge-Helmholtz à l'étude de deux écoulements turbulents : un écoulement en canal plan et un écoulement turbulent homogène isotrope.