This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to interface problems of related partial differential equations. We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problems. We introduce an energy norm stronger than the one used in [111]. Then we derive an estimate for the IFE interpolation error with this energy norm using patches of interface elements. We prove both the continuity and coercivity of the bilinear form in a partially penalized IFE (PPIFE) method. These properties allow us to derive an error bound for the PPIFE solution in the energy norm under the standard piecewise $H^2$ regularity assumption instead of the more stringent $H^3$ regularity used in [111]. As an important consequence, this new estimation further enables us to show the optimal convergence in the $L^2$ norm which could not be done by the analysis presented in [111]. Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. The first application is for the time-harmonic wave interface problem that involves the Helmholtz equation with a discontinuous coefficient. We design PPIFE and DGIFE schemes including the higher degree IFEs for Helmholtz interface problems. We present an error analysis for the symmetric linear/bilinear PPIFE methods. Under the standard piecewise $H^2$ regularity assumption for the exact solution, following Schatz's arguments, we derive optimal error bounds for the PPIFE solutions in both an energy norm and the usual $L^2$ norm provided that the mesh size is sufficiently small. {In the second group of applications, we focus on the error analysis for IFE methods developed for solving typical time-dependent interface problems associated with the second order elliptic operator with a discontinuous coefficient.} For hyperbolic interface problems, which are typical wave propagation interface problems, we reanalyze the fully-discrete PPIFE method in [143]. We derive the optimal error bounds for this PPIFE method for both an energy norm and the $L^2$ norm under the standard piecewise $H^2$ regularity assumption in the space variable of the exact solution. Simulations for standing and travelling waves are presented to corroborate the results of the error analysis. For parabolic interface problems, which are typical diffusion interface problems, we reanalyze the PPIFE methods in [113]. We prove that these PPIFE methods have the optimal convergence not only in an energy norm but also in the usual $L^2$ norm under the standard piecewise $H^2$ regularity. / Doctor of Philosophy / This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to a few types of interface problems. We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problem. We can show that the IFE methods for the elliptic interface problems converge optimally when the exact solution has lower regularity than that in the previous publications. Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. For interface problems of the Helmholtz equation which models time-Harmonic wave propagations, we design IFE schemes, including higher degree schemes, and derive error estimates for a lower degree scheme. For interface problems of the second order hyperbolic equation which models time dependent wave propagations, we derive better error estimates for the IFE methods and provides numerical simulations for both the standing and traveling waves. For interface problems of the parabolic equation which models the time dependent diffusion, we also derive better error estimates for the IFE methods.

The thesis presents a comprehensive study of a posteriori error estimation in the adaptive solution to some classical elliptic partial differential equations. Several new error estimators are proposed for diffusion problems with discontinuous coefficients and for convection-reaction-diffusion problems with dominated convection/reaction. The robustness of the new estimators is justified theoretically. Extensive numerical results demonstrate the robustness of the new estimators for challenging problems and indicate that, compared to the well-known residual-type estimators, the new estimators are much more accurate.

University of Macau / Faculty of Science and Technology / Department of Mathematics

Cette thèse porte sur différents modèles de propagation en épidémiologie évolutive. L'objectif est d'en faire une analyse mathématique rigoureuse puis d'en tirer des enseignements biologiques. Dans un premier temps nous envisageons le cas d'une population d'hôtes répartis de manière homogène dans un espace linéaire, dans laquelle se propage un pathogène pouvant muter entre deux phénotypes plus ou moins virulents. Ce phénomène de mutation est à l'origine d'une interaction entre les dynamiques évolutive et épidémiologique du pathogène. Nous étudions la vitesse de propagation de l'épidémie et l'existence de fronts progressifs, ainsi que l'influence sur la vitesse de différents facteurs biologiques, comme des effets stochastiques liés à la taille de la population d'hôtes (explorations numériques). Dans un deuxième temps nous envisageons une hétérogénéité spatiale périodique dans la population d'hôtes, et l'existence de fronts pulsatoires pour le système de réaction-diffusion (non-coopératif) associé. Enfin nous considérons un pathogène pouvant muter vers un grand nombre de phénotypes différents et étudions l'existence de fronts potentiellement singuliers, modélisant ainsi une concentration sur un trait optimal. / In this thesis we consider several models of propagation arising in evolutionary epidemiology. We aim at performing a rigorous mathematical analysis leading to new biological insights. At first we investigate the spread of an epidemic in a population of homogeneously distributed hosts on a straight line. An underlying mutation process can shift the virulence of the pathogen between two values, causing an interaction between epidemiology and evolution. We study the propagation speed of the epidemic and the influence of some biologically relevant quantities, like the effects of stochasticity caused by the hosts' finite population size (numerical explorations), on this speed. In a second part we take into account a periodic heterogeneity in the hosts' population and study the propagation speed and the existence of pulsating fronts for the associated (non-cooperative) reaction-diffusion system. Finally, we consider a model in which the pathogen is allowed to shift between a large number of different phenotypes, and construct possibly singular traveling waves for the associated nonlocal equation, thus modelling concentration on an optimal trait.

Cette thèse est consacrée à l'étude de phénomènes de propagation dans des modèles d’EDP venant de la biologie. On étudie des équations cinétiques inspirées par le déplacement de colonies de bactéries ainsi que des équations de réaction-diffusion importantes en écologie afin de reproduire plusieurs phénomènes de dynamique et d'évolution des populations. La première partie étudie des phénomènes de propagation pour des équations cinétiques. Nous étudions l'existence et la stabilité d'ondes progressives pour des modèles ou la dispersion est donnée par un opérateur hyperbolique et non par une diffusion. Cela fait entrer en jeu un ensemble de vitesses admissibles, et selon cet ensemble, divers résultats sont obtenus. Dans le cas d'un ensemble de vitesses borné, nous construisons des fronts qui se propagent à une vitesse déterminée par une relation de dispersion. Dans le cas d'un ensemble de vitesses non borné, on prouve un phénomène de propagation accélérée dont on précise la loi d'échelle. On adapte ensuite à des équations cinétiques une méthode basée sur les équations de Hamilton-Jacobi pour décrire des phénomènes de propagation. On montre alors comment déterminer un Hamiltonien effectif à partir de l'équation cinétique initiale, et prouvons des théorèmes de convergence.La seconde partie concerne l'étude de modèles de populations structurées en espace et en phénotype. Ces modèles sont importants pour comprendre l'interaction entre invasion et évolution. On y construit d'abord des ondes progressives que l'on étudie qualitativement pour montrer l'impact de la variabilité phénotypique sur la vitesse et la distribution des phénotypes à l'avant du front. On met aussi en place le formalisme Hamilton-Jacobi pour l'étude de la propagation dans ces équations de réaction-diffusion non locales.Deux annexes complètent le travail, l'une étant un travail en cours sur la dispersion cinétique en domaine non-borné, l'autre étant plus numérique et illustre l’introduction. / This thesis is devoted to the study of propagation phenomena in PDE models arising from biology. We study kinetic equations coming from the modeling of the movement of colonies of bacteria, but also reaction-diffusion equations which are of great interest in ecology to reproduce several features of dynamics and evolution of populations. The first part studies propagation phenomena for kinetic equations. We study existence and stability of travelling wave solutions for models where the dispersal part is given by an hyperbolic operator rather than by a diffusion. A set of admissible velocities comes into the game and we obtain various types of results depending on this set. In the case of a bounded set of velocities, we construct travelling fronts that propagate according to a speed given by a dispersion relation. When the velocity set is unbounded, we prove an accelerating propagation phenomena, for which we give the spreading rate. Then, we adapt to kinetic equations the Hamilton-Jacobi approach to front propagation. We show how to derive an effective Hamiltonian from the original kinetic equation, and prove some convergence results.The second part is devoted to studying models for populations structured by space and phenotypical trait. These models are important to understand interactions between invasion and evolution. We first construct travelling waves that we study qualitatively to show the influence of the genetical variability on the speed and the distribution of phenotypes at the edge of the front. We also perform the Hamilton-Jacobi approach for these non-local reaction-diffusion equations.Two appendices complete this work, one deals with the study of kinetic dispersal in unbounded domains, the other one being numerical aspects of competition models.

Submitted by Biblioteca do Instituto de Física (bif@ndc.uff.br) on 2017-03-07T18:35:27Z No. of bitstreams: 1 TESE.pdf: 797081 bytes, checksum: 36b77c687969ac7b12aeef2589d1d766 (MD5) / Made available in DSpace on 2017-03-07T18:35:27Z (GMT). No. of bitstreams: 1 TESE.pdf: 797081 bytes, checksum: 36b77c687969ac7b12aeef2589d1d766 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / O confinamento de quarks e glúons continua sendo um dos maiores problemas da Física atual, mesmo depois de passados 50 anos da criação da cromodinâmica quântica. Existem diversas abordagens que procuram uma explicação para este comportamento. Um destes cenários consiste na supercondutividade dual, proposta por G. t’Hooft em 1978. Aqui, ele discute como a condensação de objetos cromomagnéticos poderia originar um potencial linear entre cargas cromoelétricas. Este mecanismo é um dos mais aceitos atualmente e nos dirige à algumas perguntas cruciais: como estes objetos poderiam se tornar relevantes em teorias de Yang-Mills puras? quais os tipos de objetos que devemos levar em consideração para gerar as propriedades do potencial confinante? Embora a primeira pergunta seja difícil de responder, a segunda pode ser atacada por técnicas diferentes, suportadas pelas descrições na rede e por descrições efetivas de ensembles 1. Nesta tese, me dedico a estudar uma classe de objetos que s˜ao bons candidatos a resolverem a segunda questão: monopólos e vórtices de centro. Quando estamos lidando com as teorias de Yang-Mills puras SU(N), o problema consiste que, em nível clássico, estes defeitos são singulares. Porém, recebendo suporte da rede (nosso laboratório em teoria quântica de campos), podemos imaginar que, devido a flutuações quânticas do vácuo, estes objetos poderiam adquirir algumas propriedades dimensionais, como tensão,rigidez e interações que ajudariam a caracterizar o ensemble magnético nos levando a descrições de campos efetivas, que podem ser utilizadas para extrair a corda elétrica confinante. Utilizando técnicas oriundas da física de polímeros obtivemos equações de difusão que representam objetos unidimensionais, como vórtices de centro em 3D ou monopólos em 4D. O surgimento de uma derivada covariante abeliana, no caso do ensemble de vórtices de centro e instantons correlacionados em 3D, e de uma derivada covariante não abeliana, no caso do ensemble de monopólos coloridos em 4D, foi fundamental paragerar os modelos efetivos correspondentes. Acreditamos que estas equações de difusão poderão ser úteis, no futuro, para relacionar as propriedades do potencial entre quarks e aquelas de seus possíveis ensembles correspondentes. / Nowadays, quark and gluon confinement continues to be one of the most important problems in Physics. It remains unsolved, although 50 years have passed since the foundations of quantum chromodynamics. There are various approaches aimed at explaining this behaviour. One of them is the dual superconductor scenario proposed by G. t’Hooft in 1978. The general idea is that the condensation of chromomagnetics objects could originate a linear potential between chromoelectric charges. This is a promising mechanism that posses some crucial questions: how could these objects be relevant in pure YangMills? what type of object would be needed in order to generate the properties of the confining potential? While the first question is very difficult, the second one can be approached by different techniques, guided by the lattice and effective ensemble descriptions. In this thesis, I’ve been working on some good candidates to solve the second question: monopoles and center vortices. When dealing with pure SU(N) Yang-Mills theory, the problem is that at the classical level these magnetic defects are singular. Nevertheless, supported by the lattice (our laboratory in quantum field theory), we can imagine that, due to quantum vacuum fluctuations, they could acquire dimensionful properties. The tension, stiffness, as well as possible interactions that characterize the magnetic ensemble lead to effective field descriptions, that could be used to extract the corresponding confining electric string. Based on techniques borrowed from the physics of polymers, we obtained diffusion equations that describe magnetic one-dimensional objects, such as center vortices in 3D and monopoles in 4D. The appearance of an Abelian covariant derivative, for an ensemble of chains in 3D, and a non Abelian one, in the case of coloured loops in 4D, was essential to generate the corresponding effective descriptions. We believe that these diffusion equations could be helpful in the future, to relate the properties of the interquark potential and those of the possible underlying ensembles.

Mes travaux concernent deux systèmes d’équations utilisés dans la modélisation mathématique de semi-conducteurs et de plasmas : le système d’Euler-Poisson et le système d’Euler-Maxwell. Le premier système est constitué des équations d’Euler pour la conservation de la masse et de la quantité de mouvement couplées à l’équation de Poisson pour le potentiel électrostatique. Le second système décrit le phénomène d’électro-magnétisme. C’est un système couplé, qui est constitué des équations d’Euler pour la conservation de la masse et de la quantité de mouvement et les équations de Maxwell, aussi appelées équations de Maxwell-Lorentz. Les équations de Maxwell sont dues aux lois fondamentales de la physique. Elles constituent les postulats de base de l’électromagnétisme, avec l’expression de la force électromagnétique de Lorentz. En utilisant une technique de développement asymptotique, nous étudions les limites en zéro du système d’Euler-Poisson dans les modèles unipolaire et bipolaire. Il est bien connu que la limite formelle du système d’Euler-Poisson est gouvernée par les équations de dérive-diffusion lorsque le temps de relaxation tend vers zéro. Par des estimations d’énergie aux systèmes hyperboliques symétriques, nous justifions rigoureusement cette limite lorsque les conditions initiales sont bien préparées. Le phénomène des conditions initiales mal préparées est interprété par l’apparition de couches initiales. Dans ce cas, nous faisons une analyse mathématique de ces couches initiales en ajoutant des termes de correction dans le développement asymptotique. En utilisant les techniques itératives des systèmes hyperboliques symétrisables et la technique de développement asymptotique, nous étudions la limite de relaxation en zéro du système d’Euler-Maxwell, avec des conditions initiales bien préparées ainsi que l’étude des couches initiales, dans le modèle évolutif bipolaire et unipolaire. / My work is concerned with two different systems of equations used in the mathematical modeling of semiconductors and plasmas : the Euler-Poisson system and the Euler-Maxwell system. The first is given by the Euler equations for the conservation of the mass and momentum, with a Poisson equation for the electrostatic potential. The second system describes the phenomenon of electromagnetism. It is given by the Euler equations for the conservation of the mass and momentum, with a Maxwell equations for the electric field and magnetic field which are coupled to the electron density through the Maxwell equations and act on electrons via the Lorentz force. Using an asymptotic expansion method, we study the zero relaxation limit of unipolar Euler-Poisson system and of two-fluid multidimensional Euler-Poisson equations, we prove the existence and uniqueness of profiles to the asymptotic expansion and some error estimate. By employing the classical energy estimate for symmetrizable hyperbolic equations, we justify rigorously the convergence of Euler-Poisson system with well-prepared initial data. For ill-prepared initial data, the phenomenon of initial layers occurs. In this case, we also add the correction terms in the asymptotic expansion. Using an iterative method of symmetrizable hyperbolic systems and asymptotic expansion method, we study the zero-relaxation limit of unipolar and bipolar Euler-Maxwell system. For well-prepared initial data, we construct an approximate solution by an asymptotic expansion up to any order. For ill-prepared initial data, we also construct initial layer corrections in the asymptotic expansion.

Cette thèse est dédiée au développement et à l’analyse de schémas numériques de type volumes finis pour des équations de convection-diffusion, qui apparaissent notamment dans des modèles issus de la physique ou de la biologie. Nous nous intéressons plus particulièrement à la préservation de comportements asymptotiques au niveau discret. Ce travail s’articule en trois parties, composées chacune de deux chapitres. Dans la première partie, nous considérons la discrétisation du système de dérive diffusion linéaire pour les semi-conducteurs par le schéma de Scharfetter-Gummel implicite en temps. Nous nous intéressons à la préservation par ce schéma de deux types d’asymptotiques : l’asymptotique en temps long et la limite quasi-neutre. Nous démontrons des estimations d’énergie–dissipation d’énergie discrètes qui permettent de prouver d’une part la convergence en temps long de la solution approchée vers une approximation de l’équilibre thermique, d’autre part la stabilité à la limite quasi-neutre du schéma. Dans la deuxième partie, nous nous intéressons à des schémas volumes finis préservant l’asymptotique en temps long dans un cadre plus général. Plus précisément, nous considérons des équations de type convection-diffusion non linéaires qui apparaissent dans plusieurs contextes physiques : équations des milieux poreux, système de dérive-diffusion pour les semi-conducteurs... Nous proposons deux discrétisations en espace permettant de préserver le comportement en temps long des solutions approchées. Dans un premier temps, nous étendons la définition du flux de Scharfetter-Gummel pour une diffusion non linéaire. Ce schéma fournit des résultats numériques satisfaisants si la diffusion ne dégénère pas. Dans un second temps, nous proposons une discrétisation dans laquelle nous prenons en compte ensemble les termes de convection et de diffusion, en réécrivant le flux sous la forme d’un flux d’advection. Le flux numérique est défini de telle sorte que les états d’équilibre soient préservés, et nous utilisons une méthode de limiteurs de pente pour obtenir un schéma précis à l’ordre deux en espace, même dans le cas dégénéré. Enfin, la troisième et dernière partie est consacrée à l’étude d’un schéma numérique pour un modèle de chimiotactisme avec diffusion croisée pour lequel les solutions n’explosent pas en temps fini, quelles que soient les données initiales. L’étude de la convergence du schéma repose sur une estimation d’entropie discrète nécessitant l’utilisation de versions discrètes d’inégalités fonctionnelles telles que les inégalités de Poincaré-Sobolev et de Gagliardo-Nirenberg-Sobolev. La démonstration de ces inégalités fait l’objet d’un chapitre indépendant dans lequel nous proposons leur étude dans un contexte assez général, incluant notamment le cas de conditions aux limites mixtes et une généralisation au cadre des schémas DDFV. / This dissertation is dedicated to the development and analysis of finite volume numericals chemes for convection-diffusion equations, which notably occur in models arising from physics and biology. We are more particularly interested in preserving asymptotic behavior at the discrete level. This dissertation is composed of three parts, each one including two chapters. In the first part, we consider the discretization of the linear drift-diffusion system for semiconductors with the implicit Scharfetter-Gummel scheme. We focus on preserving two kinds of asymptotics with this scheme : the long-time asymptotic and the quasineutral limit. We show discrete energy–energy dissipation estimates which constitute the main point to prove first the large time convergence of the approximate solution to an approximation of the thermal equilibrium, and then the stability at the quasineutral limit. In the second part, we are interested in designing finite volume schemes which preserve the long time behavior in a more general framework. More precisely, we consider nonlinear convection-diffusion equations arising in various physical models : porous media equation, drift-diffusion system for semiconductors... We propose two spatial discretizations which preserve the long time behavior of the approximate solutions. We first generalize the Scharfetter-Gummel flux for a nonlinear diffusion. This scheme provides satisfying numerical results if the diffusion term does not degenerate. Then we propose a discretization which takes into account together the convection and diffusion terms by rewriting the flux as an advective flux. The numerical flux is then defined in such a way that equilibrium states are preserved, and we use a slope limiters method so as to obtain second order space accuracy, even in the degenerate case. Finally, the third part is devoted to the study of a numerical scheme for a chemotaxis model with cross diffusion, for which the solutions do not blow up in finite time, even for large initial data. The proof of convergence is based on a discrete entropy estimate which requires the use of discrete functional inequalities such as Poincaré-Sobolev and Gagliardo-Nirenberg-Sobolev inequalities. The demonstration of these inequalities is the subject of an independent chapter in which we propose a study in quite a general framework, including mixed boundary conditions and generalization to DDFV schemes.

Philosophiae Doctor - PhD / In this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with those obtained by other researchers.

The retina is the ocular tissue responsible for the detection of light. Its extensive demand for oxygen, coupled with a concomitant elevated supply, renders this tissue prone to both hypoxia and hyperoxia. In this thesis, we construct mathematical models of the retina, formulated as systems of reaction-diffusion equations, investigating its oxygen-related dynamics in healthy and diseased states. In the healthy state, we model the oxygen distribution across the human retina, examining the efficacy of the protein neuroglobin in the prevention of hypoxia. It has been suggested that neuroglobin could prevent hypoxia, either by transporting oxygen from regions where it is rich to those where it is poor, or by storing oxygen during periods of diminished supply or increased uptake. Numerical solutions demonstrate that neuroglobin may be effective in preventing or alleviating hypoxia via oxygen transport, but that its capacity for oxygen storage is essentially negligible, whilst asymptotic analysis reveals that, contrary to the prevailing assumption, neuroglobin's oxygen affinity is near optimal for oxygen transport. A further asymptotic analysis justifies the common approximation of a piecewise constant oxygen uptake across the retina, placing existing models upon a stronger theoretical foundation. In the diseased state, we explore the effect of hyperoxia upon the progression of the inherited retinal diseases, known collectively as retinitis pigmentosa. Both numerical solutions and asymptotic analyses show that this mechanism may replicate many of the patterns of retinal degeneration seen in vivo, but that others are inaccessible to it, demonstrating both the strengths and weaknesses of the oxygen toxicity hypothesis. It is shown that the wave speed of hyperoxic degeneration is negatively correlated with the local photoreceptor density, high density regions acting as a barrier to the spread of photoreceptor loss. The effects of capillary degeneration and treatment with antioxidants or trophic factors are also investigated, demonstrating that each has the potential to delay, halt or partially reverse photoreceptor loss. In addition to answering questions that are not accessible to experimental investigation, these models generate a number of experimentally testable predictions, forming the first loop in what has the potential to be a fruitful experimental/modelling cycle.