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We obtain fully computable constant free a posteriori error bounds on simplicial meshes for: a nonconforming finite element approximations for a Stokes problem and a low-order conform- ing and low-order stabilized conforming finite element approximations for Poisson, Stokes and Advection-Reaction-Diffusion problems. All the estimators are completely free of unknown con- stants and provide guaranteed numerical bounds on natural norms, in terms of a lower bound for the inf-sup constant of the underlying continuous problem in the Stokes case. These estimators are also shown to provide a lower bound for the natural norms of the error up to a constant and higher order data oscillation terms. In the Stokes problem, the adaptive selection of the stabilization parameter appears as an application. Numerical results are presented illustrating the theory and the performance of the error estimators.

This thesis consists of two parts. We start with some background theory that will be used throughout the thesis. Then, in the first part, we investigate the invariant measures of stochastic evolution equations of pure jump type and obtain a characterization of invariant measures. As an application, it is shown that the equation has a unique invariant probability measure under some reasonable conditions.

This thesis deals with infinite energy solutions of the Navier-Stokes and Boussinesq equations in a strip. Here, the properly chosen Uniformly local Sobolev Spaces of functions are used as the phase spaces for the problem considered. The global well-posedness and dissipativity of the Navier-Stokes equations was first established in a paper by Zelik [37] on Spatially Nondecaying Solutions of the 2D Navier-Stokes equations in a strip. However, the proof given there contains error which emmanated from wrong estimation of the solutions of the auxiliary non-autonomous linear Stokes problem with non-homogeneous divergence. In this thesis, we correct the aforementioned error and show that the main results of [37], i.e the well-posedness of the Navier-Stokes problem in uniformly local spaces, remains true. Albeit, only a weaker version of the postulated results in [37] was amenable; therefore, we reworked most part of the non-linear theory as well as to show that they are sufficient for the well-posedness of the non-linear system. We also extended these results to the thermal convection problem in a strip and associated Boussinesq equations. We considered the temperature equation and proved, using maximum principle, the well-posedness of the full Boussinesq system in a Strip. 1

PhD (Applied Mathematics) / Department of Mathematics and Applied Mathematics / The work presented in this thesis is the application of the recently introduced numerical techniques, namely the spectral quasi-linearization method (SQLM) and the bivariate spectral quasi-linearization method (BSQLM), in solving problems arising in fluid flow. Firstly, we use the SQLM to solve the highly non-linear one dimensional Bratu problem. The results obtained are compared with exact solution and previously published results using the B-spline method, Picard’s Green’s Embedded Method and the iterative finite difference method. The results obtained show that the SQLM is highly accurate and computationally efficient. Secondly, we use the bivariate spectral quasi-linearization method to solve the two dimensional Bratu problem. Since the exact solution of the two-dimensional Bratu problem is unknown, the results obtained are compared with those previously published results using the finite difference method and the weighted residual method. Thirdly, we use the BSQLM to study numerically the boundary layer flow of a third grade non-Newtonian fluid past a vertical porous plate. We use the Jeffrey fluid as a typical fluid which shows non-Newtonian characteristics. Similarity transformations are used to transform a system of coupled nonlinear partial differential equations into a system of linear partial differential equations which are then solved using BSQLM. The influence of some thermo-physical parameters namely, the ratio relaxation to retardation times parameter, Prandtl number, Schmidt number and the Deborah number is investigated. Also investigated is the influence of the ratio of relaxation to retardation times, Schmidt number and the Prandtl number on the skin friction, heat transfer rate and the mass transfer rate. The results obtained show that increasing the Schmidt number decelerates the fluid flow, reduces the skin friction, heat and mass transfer rates and strongly depresses the fluid concentration whilst the temperature is increased. The fluid velocity, the skin friction, heat and mass transfer rates are increased with increasing values of the relaxation to retardation parameter whilst the fluid temperature and concentration are reduced. Using the the solution based errors, it was shown that the BSQLM converges to the solution only after 5 iterations. The residual error infinity norms showed that BSQLM is very accurate by giving an error of order of 10−4 within 5 iterations. Lastly we propose a model of the non-Newtonian fluid flow past a vertical porous plate in the presence of thermal radiation and chemical reaction. Similarity transformations are used to transform a system of coupled nonlinear partial differential equations into a system of linear partial differential equations. The BSQLM is used to solve the system of equations. We investigate the influence of the ratio of relaxation to retardation parameter, Schmidt number, Prandtl number, thermal radiation parameter, chemical reaction iv parameter, Nusselt number, Sherwood number, local skin fiction coefficient on the fluid concentration, fluid temperature as well as the fluid velocity. From the study, it is noted that the fluid flow velocity, the local skin friction coefficient, heat and mass transfer rate are increased with increasing ratio of relaxation to retardation times parameter whilst the fluid concentration is depressed. Increasing the Prandtl number causes a reduction in the velocity and temperature of the fluid whilst the concentration is increased. Also, the local skin friction coefficient and the mass transfer rates are depressed with an increase in the Prandtl number. An increase in the chemical reaction parameter decreases the fluid velocity, temperature and the concentration. Increasing the thermal radiation parameter has an effect of decelerating the fluid flow whilst the temperature and the concentration are slightly enhanced. The infinity norms were used to show that the method converges fast. The method converges to the solution within 5 iterations. The accuracy of the solution is checked using residual errors of the functions f, and . The errors show that the BSQLM is accurate, giving errors of less than 10−4, 10−7 and 10−8 for f, and , respectively, within 5 iterations. / NRF

Cette thèse est consacrée à l'étude de problèmes de stabilisation exponentielle par retour d'état ou "feedback" des équations de Navier-Stokes dans un domaine borné Ω ⊂ Rd, d = 2 ou 3. Le cas d'un contrôle localisé sur la frontière du domaine est considéré. Le contrôle s'exprime en fonction du champ de vitesse à l'aide d'une loi de feedback non-linéaire. Celle-ci est fournie grâce aux techniques d'estimation a priori via la procédure de Faedo-Galerkin laquelle consiste à construire une suite de solutions approchées en utilisant une base de Galerkin adéquate. Cette loi de feedback assure la décroissance exponentielle de l'énergie du problème discret correspondant et grâce au résultat de compacité, nous passons à la limite dans le système satisfait par les solutions approchées. Le chapitre 1 étudie le problème de stabilisation des équations de Navier- Stokes autour d'un état stationnaire donné, tandis que le chapitre 2 examine le problème de stabilisation autour d'un état non-stationnaire prescrit. Le chapitre 3 est consacré à l'étude de la stabilisation du problème de Navier-Stokes avec des conditions aux bords mixtes (Dirichlet- Neumann) autour d'un état d'équilibre donné. Enfin, nous présentons dans le chapitre 4, des résultats numériques dans le cas d'un écoulement autour d'un obstacle circulaire / In this thesis we study the exponential stabilization of the two and three-dimensional Navier- Stokes equations in a bounded domain Ω, by means of a boundary control. The Control is expressed in terms of the velocity field by using a non-linear feedback law. In order to determine a feedback law, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the domain boundary. While most traditional approaches apply a feedback controller via an algebraic Riccati equation, the Stokes-Oseen operator or extension operators, a Galerkin method is proposed instead in this study. The Galerkin method permits to construct a stabilizing boundary control and by using energy a priori estimation technics, the exponential decay is obtained. A compactness result then allows us to pass to the limit in the nonlinear system satisfied by the approximated solutions. Chapter 1 deals with the stabilization problem of the Navier-Stokes equations around a given steady state, while Chapter 2 examines the stabilization problem around a prescribed non-stationary state. Chapter 3 is devoted to the stabilization of the Navier-Stokes problem with mixed-boundary conditions (Dirichlet-Neumann), around to a given steady-state. Finally, we present in Chapter 4, numerical results in the case of a flow around a circular obstacle

Οι μερικές διαφορικές εξισώσεις (ΜΔΕ), σε πολλές περιπτώσεις πρακτικού ενδιαφέροντος, δέχονται μια επίλυση μέσω της "μεθόδου ομοιότητας". Σε αντίθεση με τη γνωστή μέθοδο του χωρισμού των μεταβλητών, η μέθοδος ομοιότητας βασίζεται σε έναν εύστοχο συνδυασμό των μεταβλητών, μετατρέποντας την αρχική ΜΔΕ σε μια συνήθη διαφορική εξίσωση (ΣΔΕ) μιας και μοναδικής συνδυασμένης μεταβλητής. Η μετατροπή αυτή όχι μόνο αποτελεί το πλέον καθοριστικό βήμα προς τη ζητούμενη λύση αλλά ταυτόχρονα μας αποκαλύπτει και πολλά χαρακτηριστικά από την ουσιαστική φύση του ίδιου του προβλήματος. Στην παρούσα διπλωματική εργασία μελετάμε μερικά αντιπροσωπευτικά παραδείγματα της μεθόδου από τη Μηχανική των Ρευστών, όπως το "πρώτο πρόβλημα του Stokes" (περί της ροής πάνω από μια οριζόντια πλάκα που μέσω μιας ξαφνικής ώθησης τίθεται σε μια ευθύγραμμη ομαλή κίνηση) και το "πρόβλημα του Blasius" περί του συνοριακού στρώματος. Τέλος, στο πλαίσιο του βασικού ερωτήματος πότε ένα πρόβλημα ΜΔΕ δέχεται μια λύση ομοιότητας και πότε όχι, θα συζητήσουμε και το "δεύτερο πρόβλημα του Stokes", περί της ροής πάνω από μια οριζόντια πλάκα που ταλαντώνεται περιοδικά από τα αριστερά προς τα δεξιά και αντίστροφα. Αυτό το πρόβλημα αποτελεί ένα αντιπαράδειγμα για την καθολικότητα της μεθόδου, δείχνοντας ότι αυτή δεν εφαρμόζεται όταν υπάρχουν προκαθορισμένες χωρικές ή χρονικές κλίμακες στο σύστημα. / --

Cette thèse se situe dans le cadre de l'analyse théorique et numérique de quelques généralisations de l'équation de Cahn-Hilliard. On étudie l'existence, l'unicité et la régularité de la solution de ces modèles ainsi que son comportement asymptotique en terme d'existence d'un attracteur global de dimension fractale finie. La première partie de la thèse concerne des modèles appliqués à la retouche d'images. D'abord, on étudie la dynamique de l'équation de Bertozzi-Esedoglu-Gillette-Cahn-Hilliard avec des conditions de type Neumann sur le bord et une nonlinéarité régulière de type polynomial et on propose un schéma numérique avec une méthode de seuil efficace pour le problème de la retouche et très rapide en terme de temps de convergence. Ensuite, on étudie ce modèle avec des conditions de type Neumann sur le bord et une nonlinéarité singulière de type logarithmique et on donne des simulations numériques avec seuil qui confirment que les résultats obtenus avec une nonlinéarité de type logarithmique sont meilleurs que ceux obtenus avec une nonlinéarité de type polynomial. Finalement, on propose un modèle basé sur le système de Cahn-Hilliard pour la retouche d'images colorées. La deuxième partie de la thèse est consacrée à des applications en biologie et en chimie. On étudie la convergence de la solution d'une généralisation de l'équation de Cahn-Hilliard avec un terme de prolifération, associée à des conditions aux limites de type Neumann et une nonlinéarité régulière. Dans ce cas, on démontre que soit la solution explose en temps fini soit elle existe globalement en temps. Par ailleurs, on donne des simulations numériques qui confirment les résultats théoriques obtenus. On termine par l'étude de l'équation de Cahn-Hilliard avec un terme source et une nonlinéarité régulière. Dans cette étude, on considère le modèle à la fois avec des conditions aux limites de type Neumann et de type Dirichlet. / This thesis is situated in the context of the theoretical and numerical analysis of some generalizations of the Cahn-Hilliard equation. We study the well-possedness of these models, as well as the asymptotic behavior in terms of the existence of finite-dimenstional (in the sense of the fractal dimension) attractors. The first part of this thesis is devoted to some models which, in particular, have applications in image inpainting. We start by the study of the dynamics of the Bertozzi-Esedoglu-Gillette-Cahn-Hilliard equation with Neumann boundary conditions and a regular nonlinearity. We give numerical simulations with a fast numerical scheme with threshold which is sufficient to obtain good inpainting results. Furthermore, we study this model with Neumann boundary conditions and a logarithmic nonlinearity and we also give numerical simulations which confirm that the results obtained with a logarithmic nonlinearity are better than the ones obtained with a polynomial nonlinearity. Finally, we propose a model based on the Cahn-Hilliard system which has applications in color image inpainting. The second part of this thesis is devoted to some models which, in particular, have applications in biology and chemistry. We study the convergence of the solution of a Cahn-Hilliard equation with a proliferation term and associated with Neumann boundary conditions and a regular nonlinearity. In that case, we prove that the solutions blow up in finite time or exist globally in time. Furthermore, we give numericial simulations which confirm the theoritical results. We end with the study of the Cahn-Hilliard equation with a mass source and a regular nonlinearity. In this study, we consider both Neumann and Dirichlet boundary conditions.