Dans la première partie de cette thèse, nous avons considéré les équations de Maxwell en temps et construit une formulation discontinue de Galerkin (DG). On a montré que cette formulation est bien posée et ensuite on a établi des estimateurs a priori pour cette formulation. On a obtenu des résultats numériques pour valider les estimateurs a priori obtenus théoriquement. Dans la deuxième partie de cette thèse, des estimateurs d'erreur a posteriori de cette formulation sont établis, pour le cas semi-discret et pour le système complètement discrétisé. Dans la troisième partie de cette thèse, on considére les équations de Maxwell en régime harmonique. On a développé une formulation discontinue de Galerkin mixte. On a établi des estimations d'erreur a posteriori pour cette formulation. / In the first part of this thesis, we have considered the time-dependent Maxwell's equations in second-order form and constructed discontinuous Galerkin (DG) formulation. We have established a priori error estimates for this formulation and carried out the numerical analysis to confirm our theoretical results. In the second part of this thesis, we have established a posteriori error estimates of this formulation for both semi discrete and fully discrete case. In the third part of the thesis we have considered the time-harmonic Maxwell's equations and we have developed mixed discontinuous Galerkin formulation. We showed the well posedness of this formulation and have established a posteriori error estimates.

Cette thèse traite des méthodes d’éléments finis Galerkin discontinus d’ordre élevé pour la résolution d’équations aux dérivées partielles, avec un intérêt particulier pour l’équation de transport des neutrons. Nous nous intéressons tout d’abord à une méthode de pré-traitement de matrices creuses par blocs, qu’on retrouve dans les méthodes Galerkin discontinues, avant factorisation par un solveur multifrontal. Des expériences numériques conduites sur de grandes matrices bi- et tri-dimensionnelles montrent que cette méthode de pré-traitement permet une réduction significative du ’fill-in’, par rapport aux méthodes n’exploitant pas la structure par blocs. Ensuite, nous proposons une méthode d’éléments finis Galerkin discontinus, employant des éléments d’ordre élevé en espace comme en angle, pour résoudre l’équation de transport des neutrons. Nous considérons des solveurs parallèles basés sur les sous-espaces de Krylov à la fois pour des problèmes ’source’ et des problèmes aux valeur propre multiplicatif. Dans cet algorithme, l’erreur est décomposée par projection(s) afin d’équilibrer les contraintes numériques entre les parties spatiales et angulaires du domaine de calcul. Enfin, un algorithme HP-adaptatif est présenté ; les résultats obtenus démontrent une nette supériorité par rapport aux algorithmes h-adaptatifs, à la fois en terme de réduction de coût de calcul et d’amélioration de la précision. Les valeurs propres et effectivités sont présentées pour un panel de cas test industriels. Une estimation précise de l’erreur (avec effectivité de 1) est atteinte pour un ensemble de problèmes aux domaines inhomogènes et de formes irrégulières ainsi que des groupes d’énergie multiples. Nous montrons numériquement que l’algorithme HP-adaptatif atteint une convergence exponentielle par rapport au nombre de degrés de liberté de l’espace éléments finis. / We consider high order discontinuous-Galerkin finite element methods for partial differential equations, with a focus on the neutron transport equation. We begin by examining a method for preprocessing block-sparse matrices, of the type that arise from discontinuous-Galerkin methods, prior to factorisation by a multifrontal solver. Numerical experiments on large two and three dimensional matrices show that this pre-processing method achieves a significant reduction in fill-in, when compared to methods that fail to exploit block structures. A discontinuous-Galerkin finite element method for the neutron transport equation is derived that employs high order finite elements in both space and angle. Parallel Krylov subspace based solvers are considered for both source problems and $k_{eff}$-eigenvalue problems. An a-posteriori error estimator is derived and implemented as part of an h-adaptive mesh refinement algorithm for neutron transport $k_{eff}$-eigenvalue problems. This algorithm employs a projection-based error splitting in order to balance the computational requirements between the spatial and angular parts of the computational domain. An hp-adaptive algorithm is presented and results are collected that demonstrate greatly improved efficiency compared to the h-adaptive algorithm, both in terms of reduced computational expense and enhanced accuracy. Computed eigenvalues and effectivities are presented for a variety of challenging industrial benchmarks. Accurate error estimation (with effectivities of 1) is demonstrated for a collection of problems with inhomogeneous, irregularly shaped spatial domains as well as multiple energy groups. Numerical results are presented showing that the hp-refinement algorithm can achieve exponential convergence with respect to the number of degrees of freedom in the finite element space

L'étude de la susceptibilité électromagnétique de circuits électroniques nécessite l'utilisation d'un outil de simulation rapide, précis et suffisamment flexible pour intégrer les dernières innovations technologiques. La méthode itérative basée sur le concept d'onde (notée WCIP pour Wave Concept Iterative Procedure) initialement proposée par H. Baudrand est particulièrement adaptée pour la modélisation numérique de circuits multicouches à plusieurs niveaux de métallisation. Pour ce type de circuits, elle se révèle être l'une des méthodes qui utilise le plus petit nombre d'inconnues pour atteindre une précision donnée. Néanmoins, la WCIP n'est pas adaptée à la prise en compte des diélectriques inhomogènes et des trous d'interconnexion. L'objectif de la thèse est de s'affranchir de ces limitations par un couplage avec des méthodes numériques volumiques. En premier lieu, l'hybridation a été mise en œuvre avec une méthode basée sur la théorie des lignes de transmission pour des raisons de correspondance de maillages. Par la suite, le couplage avec une technique d'éléments finis de type Galerkin Discontinu (notée GD) Hybridée permet d'atteindre des objectifs de précision et de rapidité car GD apporte une flexibilité dans la discrétisation. En effet, c'est une méthode d'éléments finis non conforme qui permet notamment de faire varier d'un élément à l'autre l'ordre polynomial d'approximation. On a ainsi développé une nouvelle méthode numérique hybride couplant la WCIP avec des méthodes volumiques qui offrent plus de souplesse pour la prise en compte des milieux complexes. Enfin, une stratégie de résolution par décomposition de domaines est également abordée à la fin du manuscrit. / Electromagnetic susceptibility study of electronic circuits requires the use of a simulation tool which is fast, accurate and flexible enough to incorporate the latest technological innovations. The Wave Concept Iterative Procedure (WCIP) initially proposed by H. Baudrand is particularly adapted for numerical modeling of multilayered circuits with multilevel metallization. For this kind of circuits, it turns out to be one of the methods that uses the smallest number of unknowns to reach a given accuracy. However, the WCIP is not appropriate for inhomogeneous dielectric substrates and metallized via holes. The aim of this PhD thesis is to overcome these limitations coupling the WCIP with volume numerical methods. First, hybridization is carried out with the Frequency Domain Transmission Line Matrix (denoted FDTLM) assuming matching meshes at the interface between computational domains of both methods. Subsequently, the coupling with a finite element technique like a Hybridized Discontinuous Galerkin (denoted DG) method is considered to achieve the objectives of accuracy and speed because DG brings flexibility in the discretization. Indeed, it is a nonconforming finite element method which allows in particular changing the polynomial approximation order from one element to another. Therefore, a new hybrid method is developed coupling the WCIP with volume numerical methods which offer more flexibility for dealing with complex environments. Finally, a domain decomposition solution strategy is also discussed at the end of the manuscript.

Cette thèse se penche sur la propagation d’ondes au sein du système coupléTerre-océan-atmosphère. La compréhension de ces phénomènes a une importance majeure pour l’étude de perturbations sismiques et d’explosions atmosphériques notamment dans le cadre de missions spatiales planétaires. Les formes d’ondes issues du couplage fluide-solide permettent d’obtenir de précieuses informations sur la source du signal ou les propriétés des milieux de propagation. On développe donc deux outils numériques d’ordre élevé pour la propagation d’ondes acoustiques et de gravité. L'u en différences finies et se concentre sur le milieu atmosphérique et la propagation d’ondes linéaires dans un milieu stratifié visqueux et avec du vent. Cette méthode linéaire est validée par des solutions quasi-analytiques reposant sur les équations de dispersion dans une atmosphère stratifiée. Elle est aussi appliquée à deux cas d’études : la propagation d’ondes liée à l’impact d’une météorite à la surface de Mars (mission NASA INSIGHT), et la propagation d’ondes atmosphériques liées au tsunami de Sumatra en 2004. La seconde méthode résout la propagation non-linéaire d’ondes gravito-acoustiques dans une atmosphère couplée, avec topographie, à la propagation d’ondes élastiques dans un solide visco-élastique. Cette méthode repose sur sur le couplage d’une formulation en éléments finis discontinus, pour résoudre les équations de Navier-Stokes la partie fluide, par éléments finis continus pour résoudre les équations de l’élastodynamique dans la partie solide. Elle a été validée grâce à des solutions analytiques ainsi que par des comparaisons avec les résultats de la méthode par différences finies. / This thesis deals with the wave propagation problem within the Earth-ocean-atmosphere coupled system. A good understanding of the these phenomena has a major importance for seismic and atmospheric explosion studies, especially for planetary missions. Atmospheric wave-forms generated by explosions or surface oscillations can bring valuable information about the source mechanism or the properties of the various propagation media. We develop two new numerical full-wave high-order modeling tools to model the propagation of acoustic and gravity waves in realistic atmospheres. The first one relies on a high-order staggered finite difference method and focus only on the atmosphere. It enables the simultaneous propagation of linear acoustic and gravity waves in stratified viscous and windy atmosphere. This method is validated against quasi-analytical solutions based on the dispersion equations for a stratified atmosphere. It has also been employed to investigate two cases : the atmospheric propagation generated by a meteor impact on Mars for the INSIGHT NASA mission and for the study of tsunami-induced acoutic and gravity waves following the 2004 Sumatra tsunami. The second numerical method resolves the non-linear acoustic and gravity wave propagation in a realistic atmosphere coupled, with topography, to the elastic wave propagation in a visco-elastic solid. This numerical tool relies on a discontinuous Galerkin method to solve the full Navier-Stokes equations in the fluid domain and a continuous Galerkin method to solve the elastodynamics equations in the solid domain. It is validated against analytical solutions and numerical results provided by the finite-difference method.

Estuda-se o problema de equilíbrio sem força de corpo de uma esfera anisotrópica sob compressão radial uniformemente distribuída sobre o seu contorno no contexto da teoria da elasticidade linear clássica. A solução deste problema prediz o fenômeno inaceitável da auto-intersecção em uma região próxima ao centro da esfera para uma dada faixa de parâmetros materiais. Sob o contexto de uma teoria de minimização do funcional de energia potencial total da elasticidade linear clássica com a restrição de que o determinante do gradiente da função mudança de configuração seja injetivo, este fenômeno é eliminado. Aplicam-se duas formulações do Método dos Elementos Finitos de Galerkin Descontínuo (MEFGD) para obter soluções aproximadas para o problema de equilíbrio da esfera sem restrição. A primeira formulação do MEFGD aproxima diretamente os campos de deslocamento e deformação infinitesimal. A consideração do campo adicional de deformação na formulação do MEFGD aumenta o número de graus de liberdade associados aos nós da malha de elementos finitos e, consequentemente, o custo computacional. Com o objetivo de reduzir o número de graus de liberdade, introduz-se neste trabalho uma formulação alternativa do MEFGD. Nesta formulação, o campo de deformação infinitesimal não é obtido diretamente da inversão do sistema de equações resultante, mas sim por pós-processamento, a partir do campo de deslocamento aproximado. As soluções aproximadas obtidas com ambas as formulações do MEFGD são comparadas com a solução exata do problema sem restrição e com soluções aproximadas obtidas com o Método dos Elementos Finitos de Galerkin Clássico (MEFGC). Ambas as formulações do MEFGD fornecem melhores aproximações para a solução exata do que as aproximações obtidas com o MEFGC. Os erros entre a solução exata e as soluções aproximadas obtidas com a formulação alternativa do MEFGD são um pouco maiores do que os erros correspondentes obtidos com a formulação original do MEFGD. Este aumento nos erros é compensado pelo menor esforço computacional exigido pela formulação alternativa. Este trabalho serve de base para o estudo de problemas com restrição de injetividade utilizando o método de Galerkin descontínuo. / The equilibrium problem without body force of an anisotropic sphere under radial compression that is uniformly distributed on the sphere\'s boundary is investigated in the context of the classical linear elasticity theory. The solution of this problem predicts the unacceptable phenomenon of self-intersection in a vicinity of the center of the sphere for a given range of material parameters. This phenomenon can be eliminated in the context of a theory that minimizes the total potential energy of classical linear elasticity subjected to the restriction that the deformation field be injective. Two formulations of the Finite Element Method using Discontinuous Galerkin (MEFGD) are used to obtain approximate solutions for the unconstrained problem. The first formulation of the MEFGD approximates both the displacement and the strain fields. The consideration of the strain as an additional field in the formulation of the MEFGD increases the number of degrees of freedom associated to the finite elements and, therefore, the computational cost. With the objective of reducing the number of degrees of freedom, an alternative formulation of the MEFGD is introduced in this work. In this formulation, the strain field is not obtained directly from the inversion of the resulting linear system of equations, but from a post-processing calculation using the approximate displacement field. The approximate solutions obtained with both formulations of the MEFGD are compared with the exact solution of the problem without restriction and with approximate solutions obtained with the Finite Element Method using Classical Galerkin (MEFGC). Both formulations of the MEFGD yield better approximations for the exact solution than the approximations obtained with the MEFGC. The errors between the exact solution and the approximate solutions obtained with the alternative formulation of the MEFGD are slightly higher than the corresponding errors obtained with the original formulation of the MEFGD. These errors are compensated by the fact that the alternative formulation requires less computational effort than the computational effort required by the original formulation. This work serves as a basis for the study of problems with the injectivity restriction using the discontinuous Galerkin method.

Pour la conception des cœurs de réacteurs de 4ème génération, une précision accrue est requise pour les calculs des différents paramètres neutroniques. Les ressources mémoire et le temps de calcul étant limités, une solution consiste à utiliser des méthodes de raffinement de maillage afin de résoudre l'équation de transport des neutrons. Le flux neutronique, solution de cette équation, dépend de l'énergie, l'angle et l'espace. Les différentes variables sont discrétisées de manière successive. L'énergie avec une approche multigroupe, considérant les différentes grandeurs constantes sur chaque groupe, l'angle par une méthode de collocation, dite approximation Sn. Après discrétisation énergétique et angulaire, un système d'équations hyperboliques couplées ne dépendant plus que de la variable d'espace doit être résolu. Des éléments finis discontinus sont alors utilisés afin de permettre la mise en place de méthodes de raffinement dite hp. La précision de la solution peut alors être améliorée via un raffinement en espace (h-raffinement), consistant à subdiviser une cellule en sous-cellules, ou en ordre (p-raffinement) en augmentant l'ordre de la base de polynômes utilisée.Dans cette thèse, les propriétés de ces méthodes sont analysées et montrent l'importance de la régularité de la solution dans le choix du type de raffinement. Ainsi deux estimateurs d'erreurs permettant de mener le raffinement ont été utilisés. Le premier, suppose des hypothèses de régularité très fortes (solution analytique) alors que le second utilise seulement le fait que la solution est à variations bornées. La comparaison de ces deux estimateurs est faite sur des benchmarks dont on connaît la solution exacte grâce à des méthodes de solutions manufacturées. On peut ainsi analyser le comportement des estimateurs au regard de la régularité de la solution. Grâce à cette étude, une stratégie de raffinement hp utilisant ces deux estimateurs est proposée et comparée à d'autres méthodes rencontrées dans la littérature. L'ensemble des comparaisons est réalisé tant sur des cas simplifiés où l'on connaît la solution exacte que sur des cas réalistes issus de la physique des réacteurs.Ces méthodes adaptatives permettent de réduire considérablement l'empreinte mémoire et le temps de calcul. Afin d'essayer d'améliorer encore ces deux aspects, on propose d'utiliser des maillages différents par groupe d'énergie. En effet, l'allure spatiale du flux étant très dépendante du domaine énergétique, il n'y a a priori aucune raison d'utiliser la même décomposition spatiale. Une telle approche nous oblige à modifier les estimateurs initiaux afin de prendre en compte le couplage entre les différentes énergies. L'étude de ce couplage est réalisé de manière théorique et des solutions numériques sont proposées puis testées. / The different neutronic parameters have to be calculated with a higher accuracy in order to design the 4th generation reactor cores. As memory storage and computation time are limited, adaptive methods are a solution to solve the neutron transport equation. The neutronic flux, solution of this equation, depends on the energy, angle and space. The different variables are successively discretized. The energy with a multigroup approach, considering the different quantities to be constant on each group, the angle by a collocation method called Sn approximation. Once the energy and angle variable are discretized, a system of spatially-dependent hyperbolic equations has to be solved. Discontinuous finite elements are used to make possible the development of $hp-$refinement methods. Thus, the accuracy of the solution can be improved by spatial refinement (h-refinement), consisting into subdividing a cell into subcells, or by order refinement (p-refinement), by increasing the order of the polynomial basis.In this thesis, the properties of this methods are analyzed showing the importance of the regularity of the solution to choose the type of refinement. Thus, two error estimators are used to lead the refinement process. Whereas the first one requires high regularity hypothesis (analytical solution), the second one supposes only the minimal hypothesis required for the solution to exist. The comparison of both estimators is done on benchmarks where the analytic solution is known by the method of manufactured solutions. Thus, the behaviour of the solution as a regard of the regularity can be studied. It leads to a hp-refinement method using the two estimators. Then, a comparison is done with other existing methods on simplified but also realistic benchmarks coming from nuclear cores.These adaptive methods considerably reduces the computational cost and memory footprint. To further improve these two points, an approach with energy-dependent meshes is proposed. Actually, as the flux behaviour is very different depending on the energy, there is no reason to use the same spatial discretization. Such an approach implies to modify the initial estimators in order to take into account the coupling between groups. This study is done from a theoretical as well as from a numerical point of view.

This thesis is concerned with the analysis of the finite element method and the discontinuous Galerkin method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity, if the weak solution is nonzero on a large part of the boundary. If the weak solution is zero on the whole boundary, the nonlinearity only slows down the convergence of the function values but not the convergence of the gradient. The same analysis is carried out for approximate solutions obtained with numerical integration. The theoretical results are verified by numerical experiments. 1

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A triangular B-spline (DMS-spline)-based finite element method (TBS-FEM) is proposed along with possible enrichment through discontinuous Galerkin, continuous-discontinuous Galerkin finite element (CDGFE) and stabilization techniques. The developed schemes are also numerically explored, to a limited extent, for weak discretizations of a few second order partial differential equations (PDEs) of interest in solid mechanics. The presently employed functional approximation has both affine invariance and convex hull properties. In contrast to the Lagrangian basis functions used with the conventional finite element method, basis functions derived through n-th order triangular B-splines possess (n ≥ 1) global continuity. This is usually not possible with standard finite element formulations. Thus, though constructed within a mesh-based framework, the basis functions are globally smooth (even across the element boundaries). Since these globally smooth basis functions are used in modeling response, one can expect a reduction in the number of elements in the discretization which in turn reduces number of degrees of freedom and consequently the computational cost. In the present work that aims at laying out the basic foundation of the method, we consider only linear triangular B-splines. The resulting formulation thus provides only a continuous approximation functions for the targeted variables. This leads to a straightforward implementation without a digression into the issue of knot selection, whose resolution is required for implementing the method with higher order triangular B-splines. Since we consider only n = 1, the formulation also makes use of the discontinuous Galerkin method that weakly enforces the continuity of first derivatives through stabilizing terms on the interior boundaries. Stabilization enhances the numerical stability without sacrificing accuracy by suitably changing the weak formulation. Weighted residual terms are added to the variational equation, which involve a mesh-dependent stabilization parameter. The advantage of the resulting scheme over a more traditional mixed approach and least square finite element is that the introduction of additional unknowns and related difficulties can be avoided. For assessing the numerical performance of the method, we consider Navier’s equations of elasticity, especially the case of nearly-incompressible elasticity (i.e. as the limit of volumetric locking approaches). Limited comparisons with results via finite element techniques based on constant-strain triangles help bring out the advantages of the proposed scheme to an extent.

Voluntary coughs are used as a diagnostic tool to detect lung diseases. Understanding the mechanics of a cough is therefore crucial to accurately interpreting the test results. A cough is characterised by a dynamic compression of the airways, resulting in large flow velocities and producing transient peak expiratory flows. Existing models for pulmonary flow have one or more of the following limitations: 1) they assume quasi-steady flows, 2) they assume low speed flows, 3) they assume a symmetrical branching airway system. The main objective of this thesis is to develop a model for a cough in the branching pulmonary airway system. First, the time-dependent one-dimensional equations for flow in a compliant tube is used to simulate a cough in a single airway. Using anatomical and physiological data, the tube law coupling the fluid and airway mechanics is constructed to accurately mimic the airway behaviour in its inflated and collapsed states. Next, a novel model for air flow in an airway bifurcation is constructed. The model is the first to capture successfully subcritical and supercritical flows across the bifurcation and allows for free time evolution from one case to another. The model is investigated by simulating a cough in both symmetric and asymmetric airway bifurcations. Finally, a cough model for the complete branching airway system is developed. The model takes into account the key factors involved in a cough; namely, the compliance of the lungs and the airways, the coughing effort and the sudden opening of the glottis. The reliability of the model is assessed by comparing the model predictions with previous experimental results. The model captures the main characteristics of forced expiatory flows; namely, the flow limitation phenomenon (the flow out of the lungs becomes independent of the applied expiratory effort) and the negative effort dependence phenomenon (the flow out of the lungs decreases with increasing expiratory effort). The model also gives a good qualitative agreement with the measured values of airway resistance. The location of the collapsed airway segment during forced expiration is, however, inconsistent with previous experimental results. The effect of changing the model parameters on the model predictions is therefore discussed.