Ziel dieser Arbeit ist es zwei verschiedene Klassen von Systemen nichtlinearer parabolischer Gleichungen zu homogenisieren, und zwar Reaktions-Diffusions-Systeme mit verschiedenen Diffusionslängenskalen und Gleichungen vom Typ Cahn-Hilliard. Wir betrachten parabolische Gleichungen mit periodischen Koeffizienten, wobei die Periode dem Verhältnis der charakteristischen mikroskopischen zu der makroskopische Längenskala entspricht. Unser Ziel ist es, effektive Gleichungen rigoros herzuleiten, um die betrachteten Systeme besser zu verstehen und den Simulationsaufwand zu minimieren. Wir suchen also einen Konvergenzbegriff, mit dem die Lösung des Ausgangsmodells im Limes der Periode gegen Null gegen die Lösung des effektiven Modells konvergiert. Um die periodische Mikrostruktur und die verschiedenen Diffusivitäten zu erfassen, verwenden wir die Zwei-Skalen Konvergenz mittels periodischer Auffaltung. Der erste Teil der Arbeit handelt von Reaktions-Diffusions-Systemen, in denen einige Spezies mit der charakteristischen Diffusionslänge der makroskopischen Skala und andere mit der mikroskopischen diffundieren. Die verschiedenen Diffusivitäten führen zu einem Verlust der Kompaktheit, sodass wir nicht direkt den Grenzwert der nichtlinearen Terme bestimmen können. Wir beweisen mittels starker Zwei-Skalen Konvergenz, dass das effektive Modell ein zwei-skaliges Modell ist, welches von der makroskopischen und der mikroskopischen Skale abhängt. Unsere Methode erlaubt es uns, explizite Raten für die Konvergenz der Lösungen zu bestimmen. Im zweiten Teil betrachten wir Gleichungen vom Typ Cahn-Hilliard, welche ortsabhängige Mobilitätskoeffizienten und allgemeine Potentiale beinhalten. Wir beweisen evolutionäre Gamma-Konvergenz der zugehörigen Gradientensysteme basierend auf der Gamma-Konvergenz der Energien und der Dissipationspotentiale. / The aim of this thesis is to derive homogenization results for two different types of systems of nonlinear parabolic equations, namely reaction-diffusion systems involving different diffusion length scales and Cahn-Hilliard-type equations. The coefficient functions of the considered parabolic equations are periodically oscillating with a period which is proportional to the ratio between the charactersitic microscopic and macroscopic length scales. In view of greater structural insight and less computational effort, it is our aim to rigorously derive effective equations as the period tends to zero such that solutions of the original model converge to solutions of the effective model. To account for the periodic microstructure as well as for the different diffusion length scales, we employ the method of two-scale convergence via periodic unfolding. In the first part of the thesis, we consider reaction-diffusion systems, where for some species the diffusion length scale is of order of the macroscopic length scale and for other species it is of order of the microscopic one. Based on the notion of strong two-scale convergence, we prove that the effective model is a two-scale reaction-diffusion system depending on the macroscopic and the microscopic scale. Our approach supplies explicit rates for the convergence of the solution. In the second part, we consider Cahn-Hilliard-type equations with position-dependent mobilities and general potentials. It is well-known that the classical Cahn-Hilliard equation admits a gradient structure. Based on the Gamma-convergence of the energies and the dissipation potentials, we prove evolutionary Gamma-convergence, for the associated gradient system such that we obtain in the limit of vanishing periods a Cahn-Hilliard equation with homogenized coefficients.

Title: Numerical analysis of approximation of nonpolygonal domains for discon- tinuous Galerkin method Author: Filip Klouda Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vít Dolejší, Ph.D., DSc., KNM MFF UK Abstract: In this work we use the discontinuous Galerkin finite element method for the semidiscretization of a nonlinear nonstationary convection-diffusion pro- blem defined on a nonpolygonal two-dimensional domain. Using so called appro- ximating curved elements we define a piecewise polynomial approximation of the boundary of the domain and a space on which we search for a solution. We study the convergence of the method considering a symmetric as well as nonsymmetric discretization of diffusion terms and with the interior and boundary penalty. The obtained results allow us to derive an error estimate for the Discontinuous Galer- kin method employing the approximating curved elements. This estimate depends on the order of the approximation of the solution and also on the order of the approximation of the boundary. We describe one possibility of the construction of the approximating curved elements with the aid of a polynomial mapping given by an interpolation of points on the boundary. We present numerical experiments. Keywords: nonlinear convection-diffusion equation, discontinuous...

The thesis consists of four papers on numerical complexityanalysis of weak approximation of ordinary and partialstochastic differential equations, including illustrativenumerical examples. Here by numerical complexity we mean thecomputational work needed by a numerical method to solve aproblem with a given accuracy. This notion offers a way tounderstand the efficiency of different numerical methods. The first paper develops new expansions of the weakcomputational error for Itˆo stochastic differentialequations using Malliavin calculus. These expansions have acomputable leading order term in a posteriori form, and arebased on stochastic flows and discrete dual backward problems.Beside this, these expansions lead to efficient and accuratecomputation of error estimates and give the basis for adaptivealgorithms with either deterministic or stochastic time steps.The second paper proves convergence rates of adaptivealgorithms for Itˆo stochastic differential equations. Twoalgorithms based either on stochastic or deterministic timesteps are studied. The analysis of their numerical complexitycombines the error expansions from the first paper and anextension of the convergence results for adaptive algorithmsapproximating deterministic ordinary differential equations.Both adaptive algorithms are proven to stop with an optimalnumber of time steps up to a problem independent factor definedin the algorithm. The third paper extends the techniques to theframework of Itˆo stochastic differential equations ininfinite dimensional spaces, arising in the Heath Jarrow Mortonterm structure model for financial applications in bondmarkets. Error expansions are derived to identify differenterror contributions arising from time and maturitydiscretization, as well as the classical statistical error dueto finite sampling. The last paper studies the approximation of linear ellipticstochastic partial differential equations, describing andanalyzing two numerical methods. The first method generates iidMonte Carlo approximations of the solution by sampling thecoefficients of the equation and using a standard Galerkinfinite elements variational formulation. The second method isbased on a finite dimensional Karhunen- Lo`eve approximation ofthe stochastic coefficients, turning the original stochasticproblem into a high dimensional deterministic parametricelliptic problem. Then, adeterministic Galerkin finite elementmethod, of either h or p version, approximates the stochasticpartial differential equation. The paper concludes by comparingthe numerical complexity of the Monte Carlo method with theparametric finite element method, suggesting intuitiveconditions for an optimal selection of these methods. 2000Mathematics Subject Classification. Primary 65C05, 60H10,60H35, 65C30, 65C20; Secondary 91B28, 91B70. / QC 20100825

It is well-known that the solution of a so-called singularly perturbed differential equation exhibits layers. These are small regions in the domain where the solution changes drastically. These layers deteriorate the convergence of standard numerical algorithms, such as the finite element method on a uniform mesh. In the past many approaches were developed to overcome this difficulty. In this context it was very helpful to understand the structure of the solution - especially to know where the layers can occur. Therefore, we have a lot of analysis in the literature concerning the properties of solutions of such problems. Nevertheless, this field is far from being understood conclusively. More recently, there is an increasing interest in the numerics of optimal control problems subject to a singularly perturbed convection-diffusion equation and box constraints for the control. However, it is not much known about the solutions of such optimal control problems. The proposed solution methods are based on the experience one has from scalar singularly perturbed differential equations, but so far, the analysis presented does not use the structure of the solution and in fact, the provided bounds are rather meaningless for solutions which exhibit boundary layers, since these bounds scale like epsilon^(-1.5) as epsilon converges to 0. In this thesis we strive to prove bounds for the solution and its derivatives of the optimal control problem. These bounds show that there is an additional layer that is weaker than the layers one expects knowing the results for scalar differential equation problems, but that weak layer deteriorates the convergence of the proposed methods. In Chapter 1 and 2 we discuss the optimal control problem for the one-dimensional case. We consider the case without control constraints and the case with control constraints separately. For the case without control constraints we develop a method to prove bounds for arbitrary derivatives of the solution, given the data is smooth enough. For the latter case we prove bounds for the derivatives up to the second order. Subsequently, we discuss several discretization methods. In this context we use special Shishkin meshes. These meshes are piecewise equidistant, but have a very fine subdivision in the region of the layers. Additionally, we consider different ways of discretizing the control constraints. The first one enforces the compliance of the constraints everywhere and the other one enforces it only in the mesh nodes. For each proposed algorithm we prove convergence estimates that are independent of the parameter epsilon. Hence, they are meaningful even for small values of epsilon. As a next step we turn to the two-dimensional case. To be able to adapt the proofs of Chapter 2 to this case we require bounds for the solution of the scalar differential equation problem for a right hand side f only in W^(1,infty). Although, a lot of results for this problem can be found in the literature but we can not apply any of them, because they require a smooth right hand side f in C^(2,alpha) for some alpha in (0,1). Therefore, we dedicate Chapter 3 to the analysis of the scalar differential equations problem only using a right hand side f that is not very smooth. In Chapter 4 we strive to prove bounds for the solution of the optimal control problem in the two dimensional case. The analysis for this problem is not complete. Especially, the characteristic layers induce subproblems that are not understood completely. Hence, we can not prove sharp bounds for all terms in the solution decomposition we construct. Nevertheless, we propose a solution method. Numerical results indicate an epsilon-independent convergence for the considered examples - although we are not able to prove this.

Cette thèse est composée de deux parties dans lesquelles nous étudions d'une part des estimations d'erreurs pour des schémas numériques associés à des équations de Hamilton-Jacobi du premier ordre. D'autre part, nous nous intéressons a l'étude de la stabilité et de la contrôlabilité exacte frontière indirecte des équations d'onde couplées.Dans un premier temps, en utilisant la technique de Crandall-Lions, nous établissons une estimation d'erreur d'un schéma numérique monotone aux différences finies pour des conditions de jonction dites a flux limité, pour une équation de Hamilton-Jacobi du premier ordre. Ensuite, nous montrons que ce schéma numérique peut être généralisé à des conditions de jonction générales. Nous établissons alors la convergence de la solution discrétisée vers la solution de viscosité du problème continu. Enfin, nous proposons une nouvelle approche, à la Crandall-Lions, pour améliorer les estimations d'erreur déjà obtenues, pour une classe des Hamiltoniens bien choisis. Cette approche repose sur l'interprétation du type contrôle optimal de l'équation de Hamilton-Jacobi considérée.Dans un second temps, nous étudions la stabilisation et la contrôlabilité exacte frontière indirecte d'un système monodimensionnel d’équations d'ondes couplées. D'abord, nous considérons le cas d'un couplage avec termes de vitesses, et par une méthode spectrale, nous montrons que le système est exactement contrôlable moyennant un seul contrôle à la frontière. Les résultats dépendent de la nature arithmétique du quotient des vitesses de propagation et de la nature algébrique du terme de couplage. De plus, ils sont optimaux. Ensuite, nous considérons le cas d'un couplage d'ordre zéro et nous établissons un taux polynômial optimal de la décroissance de l'énergie. Enfin, nous montrons que le système est exactement contrôlable moyennant un seul contrôle à la frontière / The aim of this work is mainly to study on the one hand a numerical approximation of a first order Hamilton-Jacobi equation posed on a junction. On the other hand, we are concerned with the stability and the exact indirect boundary controllability of coupled wave equations in a one-dimensional setting.Firstly, using the Crandall-Lions technique, we establish an error estimate of a finite difference scheme for flux-limited junction conditions, associated to a first order Hamilton-Jacobi equation. We prove afterwards that the scheme can generally be extended to general junction conditions. We prove then the convergence of the numerical solution towards the viscosity solution of the continuous problem. We adopt afterwards a new approach, using the Crandall-Lions technique, in order to improve the error estimates for the finite difference scheme already introduced, for a class of well chosen Hamiltonians. This approach relies on the optimal control interpretation of the Hamilton-Jacobi equation under consideration.Secondly, we study the stabilization and the indirect exact boundary controllability of a system of weakly coupled wave equations in a one-dimensional setting. First, we consider the case of coupling by terms of velocities, and by a spectral method, we show that the system is exactly controllable through one single boundary control. The results depend on the arithmetic property of the ratio of the propagating speeds and on the algebraic property of the coupling parameter. Furthermore, we consider the case of zero coupling parameter and we establish an optimal polynomial energy decay rate. Finally, we prove that the system is exactly controllable through one single boundary control

Le but de cette thèse est d’étudier certaines équations aux dérivées partielles hyperboliques-dispersives. Une part importante est consacrée à l’analyse numérique et plus particulièrement à la convergence de schémas aux différences finies pour l’équation de Korteweg-de Vries et les systèmes abcd de Boussinesq. L’étude numérique suit les étapes classiques de consistance et de stabilité. Nous transposons au niveau discret la propriété de stabilité fort-faible des lois de conservations hyperboliques. Nous déterminons l’ordre de convergence des schémas et le quantifions en fonction de la régularité de Sobolev de la donnée initiale. Si nécessaire, nous régularisons la donnée initiale afin de toujours assurer les estimations de consistance. Une étape d’optimisation est alors nécessaire entre cette régularisation et l’ordre de convergence du schéma. Une seconde partie est consacrée à l’existence d’ondes progressives pour l’équation de Korteweg de Vries-Kuramoto-Sivashinsky. Par des méthodes classiques de systèmes dynamiques : système augmenté, fonction de Lyapunov, intégrale de Melnikov, par exemple, nous démontrons l’existence d’ondes oscillantes de petite amplitude. / The aim of this thesis is to study some hyperbolic-dispersive partial differential equations. A significant part is devoted to the numerical analysis and more precisely to the convergence of some finite difference schemes for the Korteweg-de Vries equation and abcd systems of Boussinesq. The numerical study follows the classical steps of consistency and stability. The main idea is to transpose at the discrete level the weak-strong stability property for hyperbolic conservation laws. We determine the convergence rate and we quantify it according to the Sobolev regularity of the initial datum. If necessary, we regularize the initial datum for the consistency estimates to be always valid. An optimization step is thus necessary between this regularization and the convergence rate of the scheme. A second part is devoted to the existence of traveling waves for the Korteweg-de Vries-Kuramoto-Sivashinsky equation. By classical methods of dynamical systems : extended systems, Lyapunov function, Melnikov integral, for instance, we prove the existence of oscillating small amplitude traveling waves.

This thesis is concerned with analysis and implementation of Time discontinuous Galerkin method. Important part of it is constructing of algorithm for solving nonlinear convection-diffusion equations, which combines Discontinuous Galerkin method in space (DGFEM) with Time discontinuous Galerkin method (TDG). Nonlinearity of the problem is overcome by damped Newton-like method. This approach provides easy adaptivity manipulation as well as high order approximation with respect to both space and time variables. The second part of the thesis is focused on Time discontinuous Galerkin method, applied to ordinary differential equations. It is shown that the solution of Time discontinuous Galerkin equals the solution obtained by Radau IIA implicit Runge-Kutta method in the roots of right Radau Quadrature. By virtue of this relation, error estimates of the order higher by one than the standard order can be obtained in these points. Furthermore, almost two times higher order can be achieved in the endpoints of the intervals of time discretization. Finally, the thesis deals with the phenomenon of stiffness, which may dramatically decrease the order of the applied method. The theoretical results are verified by numerical experiments. Powered by TCPDF (www.tcpdf.org)

In the first part of this thesis we are concerned with the constitutive the- ory for incompressible fluids characterized by a continuous monotone rela- tion between the velocity gradient and the Cauchy stress. We, in particular, investigate a class of activated fluids that behave as the Euler fluid prior activation, and as the Navier-Stokes or power-law fluid once the activation takes place. We develop a large-data existence analysis for both steady and unsteady three-dimensional flows of such fluids subject either to the no-slip boundary condition or to a range of slip-type boundary conditions, including free-slip, Navier's slip, and stick-slip. In the second part we show that the W−1,q norm is localizable provided that the functional in question vanishes on locally supported functions which constitute a partition of unity. This represents a key tool for establishing local a posteriori efficiency for partial differential equations in divergence form with residuals in W−1,q . In the third part we provide a novel analysis for the pressure convection- diffusion (PCD) preconditioner. We first develop a theory for the precon- ditioner considered as an operator in infinite-dimensional spaces. We then provide a methodology for constructing discrete PCD operators for a broad class of pressure discretizations. The...

Les méthodes d’optimisation de forme basées sur le gradient reposent sur le calcul de la dérivée de forme. Dans beaucoup d’applications, la fonctionnelle coût dépend de la solution d’une EDP. Il s’en suit qu’elle ne peut être résolue exactement et que seule une approximation de celle-ci peut être calculée, par exemple par la méthode des éléments finis. Il en est de même pour la dérivée de forme. Ainsi, les méthodes de gradient en optimisation de forme - basées sur des approximations du gradient - ne garantissent pas a priori que la direction calculée à chaque itération soit effectivement une direction de descente pour la fonctionnelle coût. Cette thèse est consacrée à la construction d’une procédure de certification de la direction de descente dans des algorithmes de gradient en optimisation de forme grâce à des estimations a posteriori de l’erreur introduite par l’approximation de la dérivée de forme par la méthode des éléments finis. On présente une procédure pour estimer l’erreur dans une Quantité d’Intérêt et on obtient une borne supérieure certifiée et explicitement calculable. L’Algorithme de Descente Certifiée (CDA) pour l’optimisation de forme identifie une véritable direction de descente à chaque itération et permet d’établir un critère d’arrêt fiable basé sur la norme de la dérivée de forme. Deux applications principales sont abordées dans la thèse. Premièrement, on considère le problème scalaire d’identification de forme en tomographie d’impédance électrique et on étudie différentes estimations d’erreur. Une première approche est basée sur le principe de l’énergie complémentaire et nécessite la résolution de problèmes globaux additionnels. Afin de réduire le coût de calcul de la procédure de certification, une estimation qui dépend seulement de quantités locales est dérivée par la reconstruction des flux équilibrés. Après avoir validé les estimations de l’erreur pour un cas bidimensionnel, des résultats numériques sont présentés pour tester les méthodes discutées. Une deuxième application est centrée sur le problème vectoriel de la conception optimale des structures élastiques. Dans ce cadre figure, on calcule l’expression volumique de la dérivée de forme de la compliance à partir de la formulation primale en déplacements et de la formulation duale mixte pour l’équation de l’élasticité linéaire. Quelques résultats numériques préliminaires pour la minimisation de la compliance sous une contrainte de volume en 2D sont obtenus à l’aide de l’Algorithme de Variation de Frontière et une estimation a posteriori de l’erreur de la dérivée de forme basée sur le principe de l’énergie complémentaire est calculée. / Gradient-based shape optimization strategies rely on the computation of the so-called shape gradient. In many applications, the objective functional depends both on the shape of the domain and on the solution of a PDE which can only be solved approximately (e.g. via the Finite Element Method). Hence, the direction computed using the discretized shape gradient may not be a genuine descent direction for the objective functional. This Ph.D. thesis is devoted to the construction of a certification procedure to validate the descent direction in gradient-based shape optimization methods using a posteriori estimators of the error due to the Finite Element approximation of the shape gradient.By means of a goal-oriented procedure, we derive a fully computable certified upper bound of the aforementioned error. The resulting Certified Descent Algorithm (CDA) for shape optimization is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion basedon the norm of the shape gradient.Two main applications are tackled in the thesis. First, we consider the scalar inverse identification problem of Electrical Impedance Tomography and we investigate several a posteriori estimators. A first procedure is inspired by the complementary energy principle and involves the solution of additionalglobal problems. In order to reduce the computational cost of the certification step, an estimator which depends solely on local quantities is derived via an equilibrated fluxes approach. The estimators are validated for a two-dimensional case and some numerical simulations are presented to test the discussed methods. A second application focuses on the vectorial problem of optimal design of elastic structures. Within this framework, we derive the volumetric expression of the shape gradient of the compliance using both H 1 -based and dual mixed variational formulations of the linear elasticity equation. Some preliminary numerical tests are performed to minimize the compliance under a volume constraint in 2D using the Boundary Variation Algorithm and an a posteriori estimator of the error in the shape gradient is obtained via the complementary energy principle.

Title: Fluid-structure interaction of compressible flow Author: RNDr. Jaroslava Hasnedlová Department: Department of Numerical Mathematics, Institute of Applied Mathematics Supervisors: Prof. RNDr. Miloslav Feistauer, DrSc., Dr. h. c., Prof. Dr. Dr. h. c. Rolf Rannacher Supervisors' e-mail addresses: feist@karlin.mff.cuni.cz, rannacher@iwr.uni-heidelberg.de Abstract: The presented work is split into two parts. The first part is devoted to the theory of the discontinuous Galerkin finite element (DGFE) method for the space-time discretization of a nonstationary convection-diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The DGFE method is applied sep- arately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time discretization. The main result is the proof of error estimates in L2 (L2 )-norm and in DG-norm formed by the L2 (H1 )-seminorm and penalty terms. The second part of the thesis deals with the realization of fluid-structure interaction problem of the compressible viscous flow with the elastic structure. The time-dependence of the domain occupied by the fluid is treated by the ALE (Arbitrary Lagrangian-Eulerian) method, when the compress- ible Navier-Stokes equations are formulated in...