The Boltzmann Transport Equation (BTE) has been the keystone of the kinetic theory, which is at the center of Statistical Mechanics bridging the gap between the atomic structures and the continuum-like behaviors. The existence of solutions has been a great mathematical challenge and still remains elusive. As a grazing limit of the Boltzmann operator, the Fokker-Planck-Landau (FPL) operator is of primary importance for collisional plasmas. We have worked on the following three different projects regarding the most important kinetic models, the BTE and the FPL Equations. (1). A Discontinuous Galerkin Solver for Nonlinear BTE. We propose a deterministic numerical solver based on Discontinuous Galerkin (DG) methods, which has been rarely studied. As the key part, the weak form of the collision operator is approximated within subspaces of piecewise polynomials. To save the tremendous computational cost with increasing order of polynomials and number of mesh nodes, as well as to resolve loss of conservations due to domain truncations, the following combined procedures are applied. First, the collision operator is projected onto a subspace of basis polynomials up to first order. Then, at every time step, a conservation routine is employed to enforce the preservation of desired moments (mass, momentum and/or energy), with only linear complexity. The asymptotic error analysis shows the validity and guarantees the accuracy of these two procedures. We applied the property of ``shifting symmetries" in the weight matrix, which consists in finding a minimal set of basis matrices that can exactly reconstruct the complete family of collision weight matrix. This procedure, together with showing the sparsity of the weight matrix, reduces the computation and storage of the collision matrix from O(N3) down to O(N^2). (2). Spectral Gap for Linearized Boltzmann Operator. Spectral gaps provide information on the relaxation to equilibrium. This is a pioneer field currently unexplored form the computational viewpoint. This work, for the first time, provides numerical evidence on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a ``collision matrix". The original spectral gap problem is then approximated by a constrained minimization problem, with objective function the Rayleigh quotient of the "collision matrix" and with constraints the conservation laws. A conservation correction then applies. We also study the convergence of the approximate Rayleigh quotient to the real spectral gap. (3). A Conservative Scheme for Approximating Collisional Plasmas. We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equations coupled with Poisson equations. The original problem is splitted into two subproblems: collisonless Vlasov problem and collisonal homogeneous Fokker-Planck-Landau problem. They are handled with different numerical schemes. The former is approximated using Runge-Kutta Discontinuous Galerkin (RKDG) scheme with a piecewise polynomial basis subspace covering all collision invariants; while the latter is solved by a conservative spectral method. To link the two different computing grids, a special conservation routine is also developed. All the projects are implemented with hybrid MPI and OpenMP. Numerical results and applications are provided. / text

My main research topic is about developing new domain decomposition algorithms for the solution of systems of partial differential equations. This was mainly applied to fluid dynamics problems (as compressible Euler or Stokes equations) and electromagnetics (time-harmonic and time-domain first order system of Maxwell's equations). Since the solution of large linear systems is strongly related to the application of a discretization method, I was also interested in developing and analyzing the application of high order methods (such as Discontinuos Galerkin methods) to Maxwell's equations (sometimes in conjuction with time-discretization schemes in the case of time-domain problems). As an active member of NACHOS pro ject (besides my main afiliation as an assistant professor at University of Nice), I had the opportunity to develop certain directions in my research, by interacting with permanent et non-permanent members (Post-doctoral researchers) or participating to supervision of PhD Students. This is strongly refflected in a part of my scientific contributions so far. This memoir is composed of three parts: the first is about the application of Schwarz methods to fluid dynamics problems; the second about the high order methods for the Maxwell's equations and the last about the domain decomposition algorithms for wave propagation problems.

Waves and circulation processes interact in daily wind and tide driven flows as well as in more extreme events such as hurricanes. Currents and water levels affect wave propagation and the location of wave-breaking zones, while wave forces induce setup and currents. Despite this interaction, waves and circulation processes are modeled separately using different approaches. Circulation processes are represented by the shallow water equations, which conserve mass and momentum. This approach for wind-generated waves is impractical for large geographic scales due to the fine resolution that would be required. Therefore, wind-waves are instead represented in a spectral sense, governed by the action balance equation, which propagates action density through both geographic and spectral space. Even though wind-waves and circulation are modeled separately, it is important to account for their interactions by coupling their respective models. In this dissertation we use discontinuous-Galerkin (DG) methods to couple spectral wave and circulation models to model wave-current interactions. We first develop, implement, verify and validate a DG spectral wave model, which allows for the implementation of unstructured meshes in geographic space and the utility of adaptive, higher-order approximations in both geographic and spectral space. We then couple the DG spectral wave model to an existing DG circulation model, which is run on the same geographic mesh and allows for higher order information to be passed between the two models. We verify and validate coupled wave/circulation model as well as analyzing the error of the coupled wave/circulation model. / text

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

Indiana University-Purdue University Indianapolis (IUPUI) / The interplay between biomechanics and blood perfusion in the optic nerve head (ONH) has a critical role in ocular pathologies, especially glaucomatous optic neuropathy. Elucidating the complex interactions of ONH perfusion and tissue structure in health and disease using current imaging methodologies is difficult, and mathematical modeling provides an approach to address these limitations. The biophysical phenomena governing the ONH physiology occur at different scales in time and space and porous media theory provides an ideal framework to model them. We critically review fundamentals of porous media theory, paying particular attention to the assumptions leading to a continuum biphasic model for the phenomenological description of fluid flow through biological tissues exhibiting viscoelastic behavior. The resulting system of equations is solved via a numerical method based on a novel hybridizable discontinuous Galerkin finite element discretization that allows accurate approximations of stresses and discharge velocities, in addition to solid displacement and fluid pressure. The model is used to theoretically investigate the influence of tissue viscoelasticity on the blood perfusion of the lamina cribrosa in the ONH. Our results suggest that changes in viscoelastic properties of the lamina may compromise tissue perfusion in response to sudden variations of intraocular pressure, possibly leading to optic disc hemorrhages.

Incompressible flows -- flows in which variations in the density of a fluid are negligible -- arise in a wide variety of applications, from hydraulics to aerodynamics. The incompressible Navier-Stokes equations which govern such flows are also of fundamental physical and mathematical interest. They are believed to hold the key to understanding turbulent phenomena; precise conditions for the existence and uniqueness of solutions remain unknown -- and establishing such conditions is the subject of one of the Clay Mathematics Institute's Millennium Prize Problems. Typical solutions of incompressible flow problems involve both fine- and large-scale phenomena, so that a uniform finite element mesh of sufficient granularity will at best be wasteful of computational resources, and at worst be infeasible because of resource limitations. Thus adaptive mesh refinements are required. In industry, the adaptivity schemes used are ad hoc, requiring a domain expert to predict features of the solution. A badly chosen mesh may cause the code to take considerably longer to converge, or fail to converge altogether. Typically, the Navier-Stokes solve will be just one component in an optimization loop, which means that any failure requiring human intervention is costly. Therefore, I pursue technological foundations for a solver of the incompressible Navier-Stokes equations that provides robust adaptivity starting with a coarse mesh. By robust, I mean both that the solver always converges to a solution in predictable time, and that the adaptive scheme is independent of the problem -- no special expertise is required for adaptivity. The cornerstone of my approach is the discontinuous Petrov-Galerkin (DPG) finite element methodology developed by Leszek Demkowicz and Jay Gopalakrishnan. For a large class of problems, DPG can be shown to converge at optimal rates. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. Several approximations to Navier-Stokes are of interest, and I study each of these in turn, culminating in the study of the steady 2D incompressible Navier-Stokes equations. The Stokes equations can be obtained by neglecting the convective term; these are accurate for "creeping" viscous flows. The Oseen equations replace the convective term, which is nonlinear, with a linear approximation. The steady-state incompressible Navier-Stokes equations approximate the transient equations by neglecting time variations. Crucial to this work is Camellia, a toolbox I developed for solving DPG problems which uses the Trilinos numerical libraries. Camellia supports 2D meshes of triangles and quads of variable polynomial order, allows simple specification of variational forms, supports h- and p-refinements, and distributes the computation of the stiffness matrix, among other features. The central contribution of this dissertation is design and development of mathematical techniques and software, based on the DPG method, for solving the 2D incompressible Navier-Stokes equations in the laminar regime (Reynolds numbers up to about 1000). Along the way, I investigate approximations to these equations -- the Stokes equations and the Oseen equations -- followed by the steady-state Navier-Stokes equations. / text

In this thesis, we explore a novel approach to solving the Helmholtz equation,the WaveHolz iteration method. This method aims to overcome some ofthe difficulties with solving the Helmholtz equation by providing a highlyparallelizable iterative method based on solving the time-dependent Waveequation. If this method proves reliable and computationally feasible it wouldhave great value for future application. Therefore, it is of interest to evaluatethe performance and properties of this method. To fully evaluate this method,the method was implemented and conclusions were based on results fromsimulations of the method. The method was able to solve problems in threedimensions and it seems that the method is very well suited for parallelized computations. To replicate real-world scenarios simulations of problems ininfinite and curvilinear domains were conducted. Based on the result presentedhere it is possible to further refine the method, especially regarding the setupof the domain and the implementation of boundary conditions for infinitedomains. / I detta examensarbete presenteras en ny metod för att lösa Helmholtz ekvation, WaveHoltz iterativa metod. Målet med denna metod är att undkomma vissa problem som uppstår med andra metoder för att lösa Helmholtz ekvation genom att tillhandahålla iterativ metod som baseras på lösningar av den tidsberoende vågekvationen samt kan parallelliseras effektivt. Om denna metod visar sig vara stabil och effektiv beräkningsmässigt skulle detta medföra stor potential för framtida tillämpningar. Av denna anledning undersöks metoden och dess egenskaper. För att fullt ut kunna evaluera denna method implementerades den vartefter simuleringar genomfördes och slutsatser drogs. Med metoden var att det var möjligt att lösa problem i tre dimensioner och metoden visade sig vara lämplig för parallella beräkningar. För att återskapa verklighetstrogna scenarion beräknades problem i oändliga och kroklinjiga domäner. Baserat på resultaten som presenteras i denna rapport är det möjligt att förfina metoden, speciellt vid konstruktionen av komplicerade beräkningsnät och randvillkoren för de oändliga problemen.

L’industrie pétrolière s’intéresse désormais à des régions de la terre de plus en plus difficiles d’accès et il est essentiel de proposer des techniques permettant de garantir l’efficaité d’un forage. Parmi ces techniques, la Reverse Time Migration (RTM) est connue pour sa précision. Elle utilise les ondes réfléchies pour reconstruire une carte du sous-sol représentant les interfaces géophysiques. Elle peut être décrite en trois étapes : (i) propager le champs émis par les sources durant la campagne d’acquisition; (ii) pour chaque source, propager le champ enregistré par les récepteurs; (iii) obtenir une image du sous-sol en appliquant une condition d’imagerie à chaque pas de temps et pour chaque source. Cette technique requiert de très grosses capacités de calcul et il est encore difficile d’imager des milieux réalistes 3D, même avec l’aide du calcul haute performance. Nous avons choisi la méthode de Galerkine discontinue pour modéliser la partie propagation car elle permet d’obtenir des solutions précises et est adaptable au calcul parallèle. La quantité d’information à sauvegarder pour faire une corrélation étant importante, on se doit de trouver un algorithme de calcul d’images du sous-sol réduisant ce coût. Nous avons utilisé l’algorithme de Griewank, appelé “Optimal Checkpointing”. Ce problème de coût étant réglé, on se doit de considérer l’efficacité des ondes élastiques incluant des champs multiples pour améliorer la précision de l’image. La condition traditionnelle de J. Claerbout ne prend pas en compte les conversions d’ondes, et n’est alors surtout utile que dans le cas acoustique. De plus, les ondes P et S interagissant entre elles, il est intéressant de trouver une condition d’imagerie utilisant ce fait. Cela a été abordé dans le cadre de la méthode de l’état adjoint dans les travaux de A. Tarantola et J. Tromp et ce travail en propose utilisation dans le cadre de la RTM. Nous proposons une nouvelle condition d’imagerie prenant en compte les paramètres élastiques du milieu considéré et permettant de supprimer les artefacts numériques. Nous illustrons les images sur des cas industriels / Since a large number of sedimentary basins have been explored, oil exploration is now interested in investigating regions of the Earth which are hostile. Among existing methods for seismic imaging, Reverse Time Migration (RTM) is a technique known by industry to be efficient. The RTM uses reflected waves and is able to construct a map of the subsurface which is depicted by the interfaces limiting the geophysical layers. The algorithm of RTM can be described as a three-step procedure: (i) compute the wavefields emitted by the sources used during the seismic acquisition campaign; (ii) for each source, compute the so-called “backpropagated wavefield”, which is the wavefield obtained by using as sources the signals recorded at the receivers during the acquisition campaign and by reversing the time; (iii) get an image of the subsurface by applying an imaging condition combining the propagated and the backpropagated wavefields at each time step of the numerical scheme and for each source. This technique is computationnaly intensive and it is still difficult to image realistic 3D elastic media, even with the help of HPC. We have thus chosen to consider high-order Discontinuous Galerkin Methods which are known to be well-adapted to provide accurate solutions based upon parallel computing. As we need to correlate a lot of wavefields, we need to find an algorithm reducing the CPU time and the storage : this is the Griewank’s algorithm, so-called “Optimal Checkpointing”. The traditional imaging condition, proposed by J. Claerbout, does not take wave conversions into account and since P-wave and S-wave interact with each other, it might be relevant to use an imaging condition including these interactions. In fact, this has been done successfully by A. Tarantola and J. Tromp for seismology applications based upon the inversion of the global Earth. In this work, we propose a new imaging condition using the elastic parameters which attenuates numerical artifacts. We illustrate the properties of the new imaging condition on industrial benchmarks like the Marmousi model. In particular, we compare the new imaging condition with other imaging conditions by using as criteria the quality of the image.

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

Le contexte scientifique de cette thèse est l'imagerie sismique dont le but est de reconstituer la structure du sous-sol de la Terre. Comme le forage a un coût assez élevé, l'industrie pétrolière s'intéresse à des méthodes capables de reconstituer les images de la structure terrestre interne avant de le faire. La technique d'imagerie sismique la plus utilisée est la technique de sismique-réflexion qui est basée sur le modèle de l'équation d'ondes. L'imagerie sismique est un problème inverse qui requiert de résoudre un grand nombre de problèmes directs. Dans ce contexte, nous nous intéressons dans cette thèse à la résolution du problème direct en régime harmonique, soit à la résolution des équations d'Helmholtz. L'objectif principal est de proposer et de développer un nouveau type de solveur élément fini (EF) caractérisé par un opérateur discret de taille réduite (comparée à la taille des solveurs déjà existants) sans pour autant altérer la précision de la solution numérique. Nous considérons les méthodes de Galerkine discontinues (DG). Comme les méthodes DG classiques sont plus coûteuses que les méthodes EF continues si l'on considère un même problème à cause d'un grand nombre de degrés de liberté couplés, résultat des approximations discontinues, nous développons une nouvelle classe de méthode DG réduisant ce problème : la méthode DG hybride (HDG). Pour valider l'efficacité de la méthode HDG proposée, nous comparons les résultats obtenus avec ceux obtenus avec une méthode DG basée sur des flux décentrés en 2D. Comme l'industrie pétrolière s'intéresse au traitement de données réelles, nous développons ensuite la méthode HDG pour les équations élastiques d'Helmholtz 3D. / The scientific context of this thesis is seismic imaging which aims at recovering the structure of the earth. As the drilling is expensive, the petroleum industry is interested by methods able to reconstruct images of the internal structures of the earth before the drilling. The most used seismic imaging method in petroleum industry is the seismic-reflection technique which uses a wave equation model. Seismic imaging is an inverse problem which requires to solve a large number of forward problems. In this context, we are interested in this thesis in the modeling part, i.e. the resolution of the forward problem, assuming a time-harmonic regime, leading to the so-called Helmholtz equations. The main objective is to propose and develop a new finite element (FE) type solver characterized by a reduced-size discrete operator (as compared to existing such solvers) without hampering the accuracy of the numerical solution. We consider the family of discontinuous Galerkin (DG) methods. However, as classical DG methods are much more expensive than continuous FE methods when considering steady-like problems, because of an increased number of coupled degrees of freedom as a result of the discontinuity of the approximation, we develop a new form of DG method that specifically address this issue: the hybridizable DG (HDG) method. To validate the efficiency of the proposed HDG method, we compare the results that we obtain with those of a classical upwind flux-based DG method in a 2D framework. Then, as petroleum industry is interested in the treatment of real data, we develop the HDG method for the 3D elastic Helmholtz equations.