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International audience; Fused Filament Fabrication (FFF) is a promising technology that is largely developed in small series, as this technology optimizes supply chains by reducing production time and costs. However, its shortcomings have slowed its adoption as a dominant production technology. Among its weaknesses, this work focuses on geometric and dimensional accuracy within tolerance range. There is a need for understanding the sources of geometrical inaccuracies and for methods of characterizing them, in order to modify the input parameters to eventually obtain the desired geometry. This work first focuses on the geometric and dimensional accuracy of parts printed by the FFF process by studying the influence of the inner radius of a cylindrical part, the type of material and the type of filling pattern. The levels with the greatest dimensional dispersion are the largest radius, the nylon material, and the hexagonal filling pattern. Secondly, a defect characterization method associated with a parametric mathematical model is developed. The 3D scanner enables the retrieval of the coordinates of the printed geometry; this allows to characterize the errors with respect to the theoretical 3D model and to modelize the printed part by a series of ellipses of which we obtain the analytical equations, as a first step of a correction process.

International audience; In this paper, we study in two and three space dimensions, the a posteriori error estimates for the Large Eddy Simulation applied to the Navier-Stokes system. We begin by introducing the Navier-Stokes and the corresponding Large Eddy Simulation (LES) equations. Then we introduce the corresponding discrete problem based on the finite element method. We establish an a posteriori error estimation with three types of error indicators related to the filter of the LES method, to the discretization and to the linearization. Finally, numerical investigations are shown and discussed.

International audience; This paper addresses an enriched nonconforming Multiscale Finite Element Method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media. In the work of [B. P. Muljadi, J. Narski, A. Lozinski, and P. Degond, Multiscale Modeling \& Simulation 2015 13:4, 1146-1172] and [G. Jankowiak and A. Lozinski, arXiv:1802.04389 [math.NA], 2018], a nonconforming MsFEM has been first developed for Stokes problems in such media. Based on these works, we propose an innovative enriched nonconforming MsFEM where the approximation space of both velocity and pressure are enriched by weighting functions which are defined by polynomials of higher-degree. Numerical experiments show that this enriched nonconforming MsFEM improves significantly the accuracy of the nonconforming MsFEMs. Theoretically, this method provides a general framework which allows to find a good compromise between the accuracy of the method and the computing costs, by varying the degrees of polynomials.

International audience; Classical finite volume schemes for the Euler system are not accurate at low Mach number and some fixes have to be used and were developed in a vast literature over the last two decades. The question we are interested in in this article is: What about if the porosity is no longer uniform? We first show that this problem may be understood on the linear wave equation taking into account porosity. We explain the influence of the cell geometry on the accuracy property at low Mach number. In the triangular case, the stationary space of the Godunov scheme approaches well enough the continuous space of constant pressure and divergence-free velocity, while this is not the case in the Cartesian case. On Cartesian meshes, a fix is proposed and accuracy at low Mach number is proved to be recovered. Based on the linear study, a numerical scheme and a low Mach fix for the non-linear system, with a non-conservative source term due to the porosity variations, is proposed and tested.

International audience; We propose and analyse a parallel method, both in the time and space directions, that couples the Parareal algorithm with the Optimized Schwarz waveform relaxation (OSWR) method, with only few OSWR iterations in the fine propagator and with a simple coarse propagator deduced from the Backward Euler method. The analysis of this coupled method is presented for a one-dimensional advection-reaction-diffusion equation. We prove a general convergence result for this method via energy estimates. Numerical results for two-dimensional advection-diffusion problems and for a diffusion equation with strong heterogeneities are presented, to illustrate the performance of the coupled Parareal-OSWR algorithm.

International audience; In this article, we study the time dependent convection-diffusion-reaction equation coupled with the Darcy equation. We propose and analyze two numerical schemes based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. An optimal a priori error estimate is then derived for each numerical scheme. Finally, we present some numerical experiments that confirm the theoretical accuracy of the discretization.

In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with a damping on the boundary. The well-posedness of the model is shown in a class of domains with d-set boundaries (N -- 1 $\le$ d < N). We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall exists in the following sense: We prove the existence of a Radon measure on this shape, greater than or equal to the usual Lebesgue measure, for which the corresponding solution of the Helmholtz problem realizes the infimum of the acoustic energy defined with the Lebesgue measure on the boundary. If this Radon measure coincides with the Lebesgue measure, the corresponding solution realizes the minimum of the energy. For a fixed porous material, considered as an acoustic absorbent, we derive the damping parameters of its boundary from the corresponding time-dependent problem described by the damped wave equation (damping in volume)., SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, In press

International audience; We present an application of the discrete duality finite volume method to the numerical approximation of the 2D Stokes or (unsteady) Navier–Stokes equations associated to Dirichlet boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical MAC scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. Different numerical examples over triangular, cartesian, quadrangular and locally refined meshes are led in order to illustrate the possibilities and weaknesses of the method.

The study deals with collocated Godunov type finite volume schemes applied to the two-dimensional linear wave equation with Coriolis source term. The purpose is to explain the wrong behaviour of the classic scheme and to modify it in order to avoid accuracy issues around the geostrophic equilibrium and in geostrophic adjustment processes. To do so, a Hodge-like decomposition is introduced. Then three different well-balanced strategies are introduced. Some properties of the associated modified equations are proven and then extended to the semi-discrete case. Stability of fully discrete schemes under a suitable CFL condition is established thanks to a Von Neumann analysis. Some numerical results reinforce the purpose and exhibit the concrete improvements achieved by the application of these new techniques in both linear and nonlinear cases.

International audience; The shallow water equations can be used to model many phenomena in geophysical fluid mechanics. For large scales, the Coriolis force plays an important role and the geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis force is an important feature. In this communication , we investigate the stability condition and the behavior of the so-called Apparent Topography scheme which is capable of capturing a discrete version of the geostrophic equilibrium.

International audience

International audience

International audience; This article is composed of three self-consistent chapters that can be read independently of each other. In Chapter 1, we define and we analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Chapter 2, we apply the all Mach correction proposed in Chapter 1 to the case of the non-linear barotropic Euler system when the Godunov type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this non-linear case by showing how this construction is related with the linear analysis of Chapter 1. At last, we apply in Chapter 3 the all Mach correction proposed in Chapter 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Chapters 1, 2 and 3 justify the theoretical results and show that the obtained all Mach Godunov type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.

We introduce continuous tools to study the low Mach number behavior of the Godunov scheme applied to the linear wave equation with porosity on cartesian meshes. More precisely, we extend the Hodge decomposition to a weighted L2 space in the continuous case and we study the properties of the modified equation associated to this Godunov scheme. This allows to partly explain the inaccuracy of the Godunov scheme at low Mach number on cartesian meshes and to propose two corrections: a first one named low Mach and a second one named all Mach. These results are preliminary since it remains to prove them in the discrete case.

International audience; We derive an a posteriori error estimation for the discrete duality finite volume (DDFV) discretization of the stationary Stokes equations on very general twodimensional meshes, when a penalty term is added in the incompressibility equation to stabilize the variational formulation. Two different estimators are provided: one for the error on the velocity and one for the error on the pressure. They both include a contribution related to the error due to the stabilization of the scheme, and a contribution due to the discretization itself. The estimators are globally upper as well as locally lower bounds for the errors of the DDFV discretization. They are fully computable as soon as a lower bound for the inf-sup constant is available. Numerical experiments illustrate the theoretical results and we especially consider the influence of the penalty parameter on the error for a fixed mesh and also of the mesh size for a fixed value of the penalty parameter. A global error reducing strategy that mixes the decrease of the penalty parameter and adaptive mesh refinement is described.

International audience; Cell-centered and vertex-centered finite volume schemes for the Laplace equation with homogeneous Dirichlet boundary conditions are considered on a triangular mesh and on the Voronoi diagram associated to its vertices. A broken $P^1$ function is constructed from the solutions of both schemes. When the domain is two-dimensional polygonal convex, it is shown that this reconstruction converges with second-order accuracy towards the exact solution in the~$L^2$ norm, under the sufficient condition that the right-hand side of the Laplace equation belongs to~$H^1(\Omega)$.

This document is a synthesis of a set of works concerning the development and the analysis of finite volume methods used for the numerical approximation of partial differential equations (PDEs) stemming from physics. In the first part, the document deals with colocalized Godunov type schemes for the Maxwell and wave equations, with a study on the loss of accuracy of this scheme at low Mach number. In the second part, discrete differential operators are built on fairly general, in particular very distorted or nonconforming, bidimensional meshes. These operators are used to approach the solutions of PDEs modelling diffusion, electro and magnetostatics and electromagnetism by the discrete duality finite volume method (DDFV) on staggered meshes. The third part presents the numerical analysis and some a priori as well as a posteriori error estimations for the discretization of the Laplace equation by the DDFV scheme. The last part is devoted to the order of convergence in the L^2 norm of the finite volume approximation of the solution of the Laplace equation in one dimension and on meshes with orthogonality properties in two dimensions. Necessary and sufficient conditions, relatively to the mesh geometry and to the regularity of the data, are provided that ensure the second-order convergence of the method.; Ce document synthétise un ensemble de travaux portant sur le développement et l'analyse de méthodes de volumes finis utilisées pour l'approximation numérique d'équations aux dérivées partielles issues de la physique. Le mémoire aborde dans sa première partie des schémas colocalisés de type Godunov d'une part pour les équations de l'électromagnétisme, et d'autre part pour l'équation des ondes acoustiques, avec une étude portant sur la perte de précision de ce schéma à bas nombre de Mach. La deuxième partie est consacrée à la construction d'opérateurs différentiels discrets sur des maillages bidimensionnels relativement quelconques, en particulier très déformés ou encore non-conformes, et à leur utilisation pour la discrétisation d'équations aux dérivées partielles modélisant des phénomènes de diffusion, d'électrostatique et de magnétostatique et d'électromagnétisme par des schémas de type volumes finis en dualité discrète (DDFV) sur maillages décalés. La troisième partie aborde ensuite l'analyse numérique et les estimations d'erreur a priori et a posteriori associées à la discrétisation par le schéma DDFV de l'équation de Laplace. La quatrième et dernière partie est consacrée à la question de l'ordre de convergence en norme L^2 de la solution numérique du problème de Laplace, issue d'une discrétisation volumes finis en dimension un et en dimension deux sur des maillages présentant des propriétés d'orthogonalité. L'étude met en évidence des conditions nécessaires et suffisantes relatives à la géométrie des maillages et à la régularité des données du problème afin d'obtenir la convergence à l'ordre deux de la méthode.

International audience; The cell-centered finite volume approximation of the Laplace equation in dimension one is considered. An exact expression of the error between the exact and numerical solutions is derived through the use of continuous and discrete Green functions. This allows to discuss convergence of the method in the L infinity and L2 norms with respect to the choice of the control points in the cells and with respect to the regularity of the data. Well-known second-order convergence results are recovered if those control points are properly chosen and if the data belongs to H1. Counterexamples are constructed to show that second-order may be lost if these conditions are not met.

International audience; We define discrete differential operators such as grad, div and curl, on general two-dimensional non-orthogonal meshes. These discrete operators verify discrete analogues of usual continuous theorems: discrete Green formulae, discrete Hodge decomposition of vector fields, vector curls have a vanishing divergence and gradients have a vanishing curl. We apply these ideas to discretize div-curl systems. We give error estimates based on the reformulation of these systems into equivalent equations for the potentials. Numerical results illustrate the use of the method on several types of meshes, among which degenerating triangular meshes and non-conforming locally refined meshes.

We study in this thesis space-time domain decomposition methods, in particular, the Parareal method, the Optimized Schwarz Waveform Relaxation (OSWR) method and their coupling, applied to the numerical simulation of parabolic equations and of the Stokes equations. We first propose and analyze a coupling of the Parareal method with the OSWR method. The obtained coupled Parareal-OSWR method is a parallel method, both in the time and space directions, with only few OSWR iterations in the fine propagator in order to reduce computational costs and with a simple coarse propagator deduced from the Backward Euler method. The analysis of this coupled method is presented for a one-dimensional advection-reaction-diffusion equation. For the coupling of Parareal with non-overlapping OSWR, we prove a general convergence result via energy estimates. Numerical results for two-dimensional advection-diffusion problems and for a diffusion equation with strong heterogeneities are presented, to illustrate the performance of the coupled Parareal-OSWR algorithm. We then present also an algorithm that couples Parareal with overlapping OSWR, and we analyze its convergence factor by using the linear convergence of overlapping OSWR that we obtain through a Fourier analysis.For the Stokes equations, we present a well-posed OSWR algorithm and an energy estimate for the convergence of the velocities. Then we show that, in general, the pressure does not converge and we propose a correction against this. A similar strategy based on Fourier transform is performed to get the formulation of the convergence factor. Numerical tests follow to illustrate the performance of the OSWR method with correction. In addition, these results are also extended to get similar ones on the Oseen equation. Finally, we propose the Parareal algorithm and a Parareal-OSWR coupling for the Stokes equations, and prove some of their basic properties, Nous étudions dans cette thèse les méthodes de décomposition de domaine spatio-temporelles, en particulier la méthode Pararéele, la méthode de Relaxation d’Ondes Optimisée (OSWR) et leur couplage, appliqués à la simulation numérique des équations paraboliques et des équations de Stokes. Nous proposons et analysons dans un premier temps un couplage de la méthode Pararéel avec la méthode OSWR. La méthode couplée Pararéel- OSWR obtenue est une méthode parallèle, à la fois en temps et en espace, avec seulement peu d’itérations OSWR dans le propagateur fin afin de réduire les coûts de calcul et avec un propagateur grossier classique déduit de la méthode d’Euler implicite. L’analyse de cette méthode couplée est présentée pour une équation d’advection-réaction-diffusion unidimensionnelle. Pour le couplage de Pararéel avec OSWR sans recouvrement, nous prouvons un résultat de convergence général via des estimations d’énergie. Des résultats numériques pour des problèmes d’advection-diffusion bidimensionnels et pour une équation de diffusion avec de fortes hétérogénéités sont présentés, pour illustrer les performances de l’algorithme couplé Pararéel-OSWR. Nous présentons ensuite également un algorithme qui couple Pararéel avec OSWR avec recouvrement, et nous analysons son facteur de convergence en utilisant la convergence linéaire de OSWR avec recouvrement que nous obtenons par une analyse de Fourier. Pour les équations de Stokes, nous présentons un algorithme OSWR bien posé et une estimation d’énergie pour la convergence des vitesses. Ensuite, nous montrons qu’en général, la pression ne converge pas et nous proposons une correction pour remédier à cela. Une stratégie similaire basée sur la transformée de Fourier est effectuée pour obtenir la formulation du facteur de convergence. Des tests numériques suivent pour illustrer les performances de la méthode OSWR avec correction. De plus, ces résultats sont également étendus pour obtenir des résultats similaires sur l’équation d’Oseen. Enfin, nous proposons l’algorithme Pararéel et un couplage Pararéel-OSWR pour les équations de Stokes, et démontrons certaines de leurs propriétés de base

The direct numerical simulation (DNS) at high Reynolds number of fluid behavior described by the Navier-Stokes equations is particularly costly, if not impossible, since the mesh cells and time step sizes must be adapted to the smallest scales of fluctuations in the velocity and pressure fields that have an impact on the solution. For this reason, techniques such as the large eddy simulation method (LES)are used where the entire scale range is not solved, but the effect of the smallest scales on the resolved scales are modeled. In this thesis, we are interested in the Smagorinksy model, one of the simplest and most widely used among the LES models. This model expresses the effect of the small scales through an additional diffusion term, whose turbulent viscosity coefficient is a function of the resolved scales. We consider this model for the time dependent Navier-Stokes problem in dimension two and for the stationary Navier-Stokes problem in dimensions two and three.These problems are analyzed by introducing the equivalent variational formulations. Then, the corresponding discrete problems based on the finite element method for space discretization and on Euler’s scheme for time discretization are introduced. An a posteriori estimate that measures the error between the solution of the original Navier-Stokes system and the discrete computed solution is established. This estimate only depends on the calculated solution, the geometry of the mesh and the data of the problem ;three types of error indicators are involved : the first is related to the space discretization, the second to the filtering of the LES method and the third to time discretization in the time dependent case and to linearization in the stationary case. Finally, numerical investigations are shown where the whole process is implemented with the FreeFem++ software.; La simulation numérique directe (DNS) à nombre de Reynolds élevé du comportement d’un fluide décrit par les équations de Navier-Stokes est particulièrement coûteuse, voire impossible, puisque les tailles de maille et de pas de temps doivent être adaptées aux plus petites échelles des fluctuations des champs de vitesse et de pression ayant un impact sur la solution. Pour cette raison, on utilise des techniques comme la méthode de simulation des grandes échelles (LES) où l’on n’a pas besoin de résoudre l’intégralité de toutes les échelles, mais où l’effet des plus petites échelles sur les échelles résolues sera modélisé.Dans cette thèse, on s’intéresse au modèle de Smagorinksy, un des modèles les plus simples de LES et parmi les plus utilisés dans les codes de calcul. Il exprime l’effet des petites échelles par un terme de diffusion supplémentaire dont le coefficient de viscosité turbulente est une fonction des échelles résolues.Nous considérons ce modèle pour les équations instationnaires en dimension deux et stationnaires en dimensions deux et trois.On analyse ces problèmes en introduisant les formulations variationnelles équivalentes. Ensuite on introduit les problèmes discrets correspondants en se basant sur la méthode des éléments finis pour la discrétisation en espace et sur le schéma d’Euler pour la discrétisation en temps. On établit une estimation d’erreur a posteriori entre la solution des équations de Navier-Stokes originelles et la solution discrète calculée. Cette estimation ne dépend que de la solution discrète calculée, de la géométrie du maillage et des données du problème ; elle fait apparaître trois types d’indicateurs d’erreur : de discrétisation en espace, de filtrage dû à la méthode LES et de discrétisation en temps dans la cas instationnaire ou de linéarisation dans le cas stationnaire. Enfin, on montre des résultats numériques de validation où l’ensemble est implémenté à l’aide du logiciel FreeFem++.

The direct numerical simulation (DNS) at high Reynolds number of fluid behavior described by the Navier-Stokes equations is particularly costly, if not impossible, since the mesh cells and time step sizes must be adapted to the smallest scales of fluctuations in the velocity and pressure fields that have an impact on the solution. For this reason, techniques such as the large eddy simulation method (LES)are used where the entire scale range is not solved, but the effect of the smallest scales on the resolved scales are modeled. In this thesis, we are interested in the Smagorinksy model, one of the simplest and most widely used among the LES models. This model expresses the effect of the small scales through an additional diffusion term, whose turbulent viscosity coefficient is a function of the resolved scales. We consider this model for the time dependent Navier-Stokes problem in dimension two and for the stationary Navier-Stokes problem in dimensions two and three.These problems are analyzed by introducing the equivalent variational formulations. Then, the corresponding discrete problems based on the finite element method for space discretization and on Euler’s scheme for time discretization are introduced. An a posteriori estimate that measures the error between the solution of the original Navier-Stokes system and the discrete computed solution is established. This estimate only depends on the calculated solution, the geometry of the mesh and the data of the problem ;three types of error indicators are involved : the first is related to the space discretization, the second to the filtering of the LES method and the third to time discretization in the time dependent case and to linearization in the stationary case. Finally, numerical investigations are shown where the whole process is implemented with the FreeFem++ software.; La simulation numérique directe (DNS) à nombre de Reynolds élevé du comportement d’un fluide décrit par les équations de Navier-Stokes est particulièrement coûteuse, voire impossible, puisque les tailles de maille et de pas de temps doivent être adaptées aux plus petites échelles des fluctuations des champs de vitesse et de pression ayant un impact sur la solution. Pour cette raison, on utilise des techniques comme la méthode de simulation des grandes échelles (LES) où l’on n’a pas besoin de résoudre l’intégralité de toutes les échelles, mais où l’effet des plus petites échelles sur les échelles résolues sera modélisé.Dans cette thèse, on s’intéresse au modèle de Smagorinksy, un des modèles les plus simples de LES et parmi les plus utilisés dans les codes de calcul. Il exprime l’effet des petites échelles par un terme de diffusion supplémentaire dont le coefficient de viscosité turbulente est une fonction des échelles résolues.Nous considérons ce modèle pour les équations instationnaires en dimension deux et stationnaires en dimensions deux et trois.On analyse ces problèmes en introduisant les formulations variationnelles équivalentes. Ensuite on introduit les problèmes discrets correspondants en se basant sur la méthode des éléments finis pour la discrétisation en espace et sur le schéma d’Euler pour la discrétisation en temps. On établit une estimation d’erreur a posteriori entre la solution des équations de Navier-Stokes originelles et la solution discrète calculée. Cette estimation ne dépend que de la solution discrète calculée, de la géométrie du maillage et des données du problème ; elle fait apparaître trois types d’indicateurs d’erreur : de discrétisation en espace, de filtrage dû à la méthode LES et de discrétisation en temps dans la cas instationnaire ou de linéarisation dans le cas stationnaire. Enfin, on montre des résultats numériques de validation où l’ensemble est implémenté à l’aide du logiciel FreeFem++.

The shallow water system plays an important role in the numerical simulation of oceanic models, coastal flows and dam-break floods. Several kinds of source terms can be taken into account in this model, such as the influence of bottom topography, Manning friction effects and Coriolis force. For large scale oceanic phenomena, the Coriolis force due to the Earth's rotation plays a central role since the atmospheric or oceanic circulations are frequently observed around the so-called geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis source term. The ability of numerical schemes to well capture thelake at rest, has been widely studied. However, the geostrophic equilibrium issue, including the divergence free constraint on the velocity, is much more complex and only few works have been devoted to its preservation. In this manuscript, we design finite volume schemes that preserve the discrete geostrophic equilibrium in order to improve significantly the accuracy of numerical simulations of perturbations around this equilibrium. We first develop collocated and staggered schemes on rectangular and triangular meshes for a linearized model of the original shallow water system. The crucial common point of the various methods is to adapt and combine several strategies known as the Apparent Topography, the Low Mach and the Divergence Penalisation methods, in order to handle correctly the numerical diffusions involved in the schemes on different cell geometries, so that they do not destroy geostrophic equilibria. Finally, we extend these strategies to the non-linear case and show convincing numerical results.; Le système de Saint-Venant joue un rôle important dans la simulation de modèles océaniques, d'écoulements côtiers et de ruptures de barrages. Plusieurs sortes de termes sources peuvent être pris en compte dans ce modèle, comme la topographie, les effets de friction de Manning et la force de Coriolis. Celle-ci joue un rôle central dans les phénomènes à grande échelle spatiale car les circulations atmosphériques ou océaniques sont souvent observées autour de l'équilibre géostrophique qui correspond à l'équilibre du gradient de pression et de cette force. La capacité des schémas numériques à bien reproduire le lac au repos a été largement étudiée; en revanche, la question de l'équilibre géostrophique (incluant la contrainte de vitesse à divergence nulle) est beaucoup plus complexe et peu de travaux lui ont été consacrés.Dans cette thèse, nous concevons des schémas volumes finis qui préservent les équilibres géostrophiques discrets dans le but d'améliorer significativement la précision des simulations numériques de perturbations autour de ces équilibres. Nous développons tout d'abord des schémas colocalisés et décalés sur des maillages rectangulaires ou triangulaires pour une linéarisation du modèle d'origine. Le point commun décisif de ces méthodes est d'adapter et de combiner les stratégies dites "topographie apparente", "bas Mach" et "pénalisation de divergence" pour contrôler l'effet de la diffusion numérique contenue dans les schémas, de telle sorte qu'elle ne détruise pas les équilibres géostrophiques. Enfin, nous étendons ces stratégies au cas non-linéaire et montrons des résultats prometteurs.

Le système de Saint-Venant joue un rôle important dans la simulation de modèles océaniques, d’écoulements côtiers et de ruptures de barrages. Plusieurs sortes de termes sources peuvent être pris en compte dans ce modèle, comme la topographie, les effets de friction de Manning et la force de Coriolis. Celle-ci joue un rôle central dans les phénomènes à grande échelle spatiale car les circulations atmosphériques ou océaniques sont souvent observées autour de l’équilibre géostrophique qui correspond à l’équilibre du gradient de pression et de cette force. La capacité des schémas numériques à bien reproduire le lac au repos a été largement étudiée; en revanche, la question de l’équilibre géostrophique (incluant la contrainte de vitesse à divergence nulle) est beaucoup plus complexe et peu de travaux lui ont été consacrés. Dans cette thèse, nous concevons des schémas volumes finis qui préservent les équilibres géostrophiques discrets dans le but d’améliorer significativement la précision des simulations numériques de perturbations autour de ces équilibres. Nous développons tout d’abord des schémas colocalisés et décalés sur des maillages rectangulaires ou triangulaires pour une linéarisation du modèle d’origine. Le point commun décisif de ces méthodes est d’adapter et de combiner les stratégies dites "topographie apparente", "bas Mach" et "pénalisation de divergence" pour contrôler l’effet de la diffusion numérique contenue dans les schémas, de telle sorte qu’elle ne détruise pas les équilibres géostrophiques. Enfin, nous étendons ces stratégies au cas non-linéaire et montrons des résultats prometteurs.; The shallow water system plays an important role in the numerical simulation of oceanic models, coastal flows and dam-break floods. Several kinds of source terms can be taken into account in this model, such as the influence of bottom topography, Manning friction effects and Coriolis force. For large scale oceanic phenomena, the Coriolis force due to the Earth’s rotation plays a central role since the atmospheric or oceanic circulations are frequently observed around the so-called geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis source term. The ability of numerical schemes to well capture the lake at rest, has been widely studied. However, the geostrophic equilibrium issue, including the divergence free constraint on the velocity, is much more complex and only few works have been devoted to its preservation. In this manuscript, we design finite volume schemes that preserve the discrete geostrophic equilibriuminordertoimprovesignificantlytheaccuracyofnumericalsimulationsofperturbations around this equilibrium. We first develop collocated and staggered schemes on rectangular and triangular meshes for a linearized model of the original shallow water system. The crucial common point of the various methods is to adapt and combine several strategies known as the Apparent Topography, the Low Mach and the Divergence Penalisation methods, in order to handle correctly the numerical diffusions involved in the schemes on different cell geometries, so that they do not destroy geostrophic equilibria. Finally, we extend these strategies to the non-linear case and show convincing numerical results.

The shallow water system plays an important role in the numerical simulation of oceanic models, coastal flows and dam-break floods. Several kinds of source terms can be taken into account in this model, such as the influence of bottom topography, Manning friction effects and Coriolis force. For large scale oceanic phenomena, the Coriolis force due to the Earth's rotation plays a central role since the atmospheric or oceanic circulations are frequently observed around the so-called geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis source term. The ability of numerical schemes to well capture thelake at rest, has been widely studied. However, the geostrophic equilibrium issue, including the divergence free constraint on the velocity, is much more complex and only few works have been devoted to its preservation. In this manuscript, we design finite volume schemes that preserve the discrete geostrophic equilibrium in order to improve significantly the accuracy of numerical simulations of perturbations around this equilibrium. We first develop collocated and staggered schemes on rectangular and triangular meshes for a linearized model of the original shallow water system. The crucial common point of the various methods is to adapt and combine several strategies known as the Apparent Topography, the Low Mach and the Divergence Penalisation methods, in order to handle correctly the numerical diffusions involved in the schemes on different cell geometries, so that they do not destroy geostrophic equilibria. Finally, we extend these strategies to the non-linear case and show convincing numerical results., Le système de Saint-Venant joue un rôle important dans la simulation de modèles océaniques, d'écoulements côtiers et de ruptures de barrages. Plusieurs sortes de termes sources peuvent être pris en compte dans ce modèle, comme la topographie, les effets de friction de Manning et la force de Coriolis. Celle-ci joue un rôle central dans les phénomènes à grande échelle spatiale car les circulations atmosphériques ou océaniques sont souvent observées autour de l'équilibre géostrophique qui correspond à l'équilibre du gradient de pression et de cette force. La capacité des schémas numériques à bien reproduire le lac au repos a été largement étudiée; en revanche, la question de l'équilibre géostrophique (incluant la contrainte de vitesse à divergence nulle) est beaucoup plus complexe et peu de travaux lui ont été consacrés.Dans cette thèse, nous concevons des schémas volumes finis qui préservent les équilibres géostrophiques discrets dans le but d'améliorer significativement la précision des simulations numériques de perturbations autour de ces équilibres. Nous développons tout d'abord des schémas colocalisés et décalés sur des maillages rectangulaires ou triangulaires pour une linéarisation du modèle d'origine. Le point commun décisif de ces méthodes est d'adapter et de combiner les stratégies dites "topographie apparente", "bas Mach" et "pénalisation de divergence" pour contrôler l'effet de la diffusion numérique contenue dans les schémas, de telle sorte qu'elle ne détruise pas les équilibres géostrophiques. Enfin, nous étendons ces stratégies au cas non-linéaire et montrons des résultats prometteurs.

Le système de Saint-Venant joue un rôle important dans la simulation de modèles océaniques, d’écoulements côtiers et de ruptures de barrages. Plusieurs sortes de termes sources peuvent être pris en compte dans ce modèle, comme la topographie, les effets de friction de Manning et la force de Coriolis. Celle-ci joue un rôle central dans les phénomènes à grande échelle spatiale car les circulations atmosphériques ou océaniques sont souvent observées autour de l’équilibre géostrophique qui correspond à l’équilibre du gradient de pression et de cette force. La capacité des schémas numériques à bien reproduire le lac au repos a été largement étudiée; en revanche, la question de l’équilibre géostrophique (incluant la contrainte de vitesse à divergence nulle) est beaucoup plus complexe et peu de travaux lui ont été consacrés. Dans cette thèse, nous concevons des schémas volumes finis qui préservent les équilibres géostrophiques discrets dans le but d’améliorer significativement la précision des simulations numériques de perturbations autour de ces équilibres. Nous développons tout d’abord des schémas colocalisés et décalés sur des maillages rectangulaires ou triangulaires pour une linéarisation du modèle d’origine. Le point commun décisif de ces méthodes est d’adapter et de combiner les stratégies dites "topographie apparente", "bas Mach" et "pénalisation de divergence" pour contrôler l’effet de la diffusion numérique contenue dans les schémas, de telle sorte qu’elle ne détruise pas les équilibres géostrophiques. Enfin, nous étendons ces stratégies au cas non-linéaire et montrons des résultats prometteurs.; The shallow water system plays an important role in the numerical simulation of oceanic models, coastal flows and dam-break floods. Several kinds of source terms can be taken into account in this model, such as the influence of bottom topography, Manning friction effects and Coriolis force. For large scale oceanic phenomena, the Coriolis force due to the Earth’s rotation plays a central role since the atmospheric or oceanic circulations are frequently observed around the so-called geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis source term. The ability of numerical schemes to well capture the lake at rest, has been widely studied. However, the geostrophic equilibrium issue, including the divergence free constraint on the velocity, is much more complex and only few works have been devoted to its preservation. In this manuscript, we design finite volume schemes that preserve the discrete geostrophic equilibriuminordertoimprovesignificantlytheaccuracyofnumericalsimulationsofperturbations around this equilibrium. We first develop collocated and staggered schemes on rectangular and triangular meshes for a linearized model of the original shallow water system. The crucial common point of the various methods is to adapt and combine several strategies known as the Apparent Topography, the Low Mach and the Divergence Penalisation methods, in order to handle correctly the numerical diffusions involved in the schemes on different cell geometries, so that they do not destroy geostrophic equilibria. Finally, we extend these strategies to the non-linear case and show convincing numerical results.

International audience; We present an a posteriori error estimate for the linear elasticity problem. The estimate is based on an equilibrated reconstruction of the Cauchy stress tensor, which is obtained from mixed finite element solutions of local Neumann problems. We propose two different reconstructions, one using Arnold–Winther mixed finite element spaces providing a symmetric stress tensor and one using Arnold–Falk– Winther mixed finite element spaces with a weak symmetry constraint. The performance of the estimate is illustrated on a numerical test with analytical solution.

International audience; A fractional step method that provides approximate solutions of a three-phase flow model is presented herein. The three-fluid model enables to handle smooth or discontinuous unsteady solutions. The numerical method is grounded on the use of the entropy inequality that governs smooth solutions of the set of PDEs. The evolution step relies on an explicit scheme, while implicit schemes are embedded in the relaxation step. The main properties of the scheme are given. Numerical approximations of two basic Riemann problems are eventually presented.

International audience; In this work, we introduce a novel algorithm for the quasi-static nonlin-ear poroelasticity problem describing Darcian flow in a deformable saturated porous medium. The nonlinear elasticity operator is discretized using a Hybrid High-Order method while the heterogeneous diffusion part relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, allows arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle rough variations of the permeability coefficient and vanishing specific storage coefficient. Numerical tests demonstrating the performance of the method are provided.

International audience; The so-called mixed flows in pipes include two-phase stratified regimes as well as single-phase pressurized regimes with transitions. It is proposed to handle those configurations numerically with the compressible two-layer model developed in [7]. Thus, a fractional step method is proposed to deal explicitly with the slow propagation phenomena and implicitly with the fast ones. It results in a large time-step scheme accurate in both regimes. Numerical experiments are performed including convergence results and academical test cases.

We consider here the two-dimensional version of the Hybrid High-Order (HHO) method for the steady incompressible Navier–Stokes equations originally introduced in [Di Pietro, Krell, A Hybrid High-Order method for the steady incompressible Navier–Stokes problem, preprint arXiv:1607.08159 math.NA]. This method displays several advantageous features: it is inf-sup stable on general meshes including polyhedral elements and nonmatching interfaces, it supports arbitrary approximation order, and has a reduced computational cost thanks to the possibility of statically condensing a subset of both velocity and pressure degrees of freedom (DOFs) at each nonlinear iteration.

Saint-Venant equations can be generalized to account for a viscoelastic rheology in shallow flows. A Finite-Volume discretization for the 1D Saint-Venant system generalized to Upper-Convected Maxwell (UCM) fluids was proposed in Bouchut and Boyaval (M3AS 23(08): 1479–1526, 2013, [6]), which preserved a physically-natural stability property (i.e. free-energy dissipation) of the full system. It invoked a relaxation scheme of Suliciu type for the numerical computation of approximate solutions to Riemann problems. Here, the approach is extended to the 1D Saint-Venant system generalized to the finitely-extensible nonlinear elastic fluids of Peterlin (FENE-P). We are currently not able to ensure all stability conditions a priori, but the scheme is fully computable. And, using numerical simulations, it may help understand the famous High-Weissenberg number problem (HWNP) well-known in computational rheology.