Heterogeneity in lattice potentials (like random or quasiperiodic) can localize linear, non-interacting waves and halt their propagation. Nonlinearity induces wave interactions, enabling energy exchange and leading to chaotic dynamics. Understanding the interplay between the two is one of the topical problems of modern wave physics. In particular, one questions whether nonlinearity destroys localization and revives wave propagation, whether thresholds in wave energy/norm exist, and what the resulting wave transport mechanisms and characteristics are. Despite remarkable progress in the field, the answers to these questions remain controversial and no general agreement is currently achieved. This thesis aims at resolving some of the controversies in the framework of nonlinear dynamics and computational physics. Following common practice, basic lattice models (discrete Klein-Gordon and nonlinear Schroedinger equations) were chosen to study the problem analytically and numerically. In the disordered linear case all eigenstates of such lattices are spatially localized manifesting Anderson localization, while nonlinearity couples them, enabling energy exchange and chaotic dynamics. For the first time we present a comprehensive picture of different subdiffusive spreading regimes and self-trapping phenomena, explain the underlying mechanisms and derive precise asymptotics of spreading. Moreover, we have successfully generalized the theory to models with spatially inhomogeneous nonlinearity, quasiperiodic potentials, higher lattice dimensions and arbitrary nonlinearity index. Furthermore, we have revealed a remarkable similarity to the evolution of wave packets in the nonlinear diffusion equation. Finally, we have studied the limits of strong disorder and small nonlinearities to discover the probabilistic nature of Anderson localization in nonlinear disordered systems, demonstrating the finite probability of its destruction for arbitrarily small nonlinearity and exponentially small probability of its survival above a certain threshold in energy. Our findings give a new dimension to the theory of wave packet spreading in localizing environments, explain existing experimental results on matter and light waves dynamics in disordered and quasiperiodic lattice potentials, and offer experimentally testable predictions.

This thesis is concerned with the solution of linear operator equations by projection methods known as minimal residual (MR) and orthogonal residual (OR) methods. We begin with a rather abstract framework of approximation by orthogonal and oblique projection in Hilbert space. When these approximation schemes are applied to sequences of nested spaces, with a simple requirement relating trial and test spaces in case of the OR method, one can derive at this rather general level the basic relations which have been proved for many specific Krylov subspace methods for solving linear systems of equations in the literature. The crucial quantities with which we describe the behavior of these methods are angles between subspaces. By replacing the given inner product with one that is basis-dependent, one can also incorporate methods based on non-orthogonal bases such as those based on the non-Hermitian Lanczos process for solving linear systems. In fact, one can show that any reasonable approximation method based on a nested sequence of approximation spaces can be interpreted as an MR or OR method in this way. When these abstract approximation techniques are applied to the solution of linear operator equations, there are three generic algorithmic formulations, which we identify with some algorithms in the literature. Specializing further to Krylov trial and test spaces, we recover the well known Krylov subspace methods. Moreover, we show that our general framework also covers in a natural way many recent generalizations of Krylov subspace methods, which employ techniques such as augmentation, deflation, restarts and truncation. We conclude with a chapter on error and residual bounds, deriving some old and new results based on the angles framework. This work provides a natural and consistent framework for the sometimes confusing plethora of methods of Krylov subspace type introduced in the last 50 years.

The Levenberg-Marquardt algorithm is a classical method for solving nonlinear systems of equations that can come from various applications in engineering and economics. Recently, Levenberg-Marquardt methods turned out to be a valuable principle for obtaining fast convergence to a solution of the nonlinear system if the classical nonsingularity assumption is replaced by a weaker error bound condition. In this way also problems with nonisolated solutions can be treated successfully. Such problems increasingly arise in engineering applications and in mathematical programming. In this thesis we use Levenberg-Marquardt algorithms to deal with nonlinear equations, multi-objective optimization and complementarity problems. We develop new algorithms for solving these problems and investigate their convergence properties. For sufficiently smooth nonlinear equations we provide convergence results for inexact Levenberg-Marquardt type algorithms. In particular, a sharp bound on the maximal level of inexactness that is sufficient for a quadratic (or a superlinear) rate of convergence is derived. Moreover, the theory developed is used to show quadratic convergence of a robust projected Levenberg-Marquardt algorithm. The use of Levenberg-Marquardt type algorithms for unconstrained multi-objective optimization problems is investigated in detail. In particular, two globally and locally quadratically convergent algorithms for these problems are developed. Moreover, assumptions under which the error bound condition for a Pareto-critical system is fulfilled are derived. We also treat nonsmooth equations arising from reformulating complementarity problems by means of NCP functions. For these reformulations, we show that existing smoothness conditions are not satisfied at degenerate solutions. Moreover, we derive new results for positively homogeneous functions. The latter results are used to show that appropriate weaker smoothness conditions (enabling a local Q-quadratic rate of convergence) hold for certain reformulations. / Der Levenberg-Marquardt-Algorithmus ist ein klassisches Verfahren zur Lösung von nichtlinearen Gleichungssystemen, welches in verschiedenen Anwendungen der Ingenieur-und Wirtschaftswissenschaften vorkommen kann. Kürzlich, erwies sich das Verfahren als ein wertvolles Instrument für die Gewährleistung einer schnelleren Konvergenz für eine Lösung des nichtlinearen Systems, wenn die klassische nichtsinguläre Annahme durch eine schwächere Fehlerschranke der eingebundenen Bedingung ersetzt wird. Auf diese Weise, lassen sich ebenfalls Probleme mit nicht isolierten Lösungen erfolgreich behandeln. Solche Probleme ergeben sich zunehmend in den praktischen, ingenieurwissenschaftlichen Anwendungen und in der mathematischen Programmierung. In dieser Arbeit verwenden wir Levenberg-Marquardt- Algorithmus für nichtlinearere Gleichungen, multikriterielle Optimierung - und nichtlineare Komplementaritätsprobleme. Wir entwickeln neue Algorithmen zur Lösung dieser Probleme und untersuchen ihre Konvergenzeigenschaften. Für ausreichend differenzierbare nichtlineare Gleichungen, analysieren und bieten wir Konvergenzergebnisse für ungenaue Levenberg-Marquardt-Algorithmen Typen. Insbesondere, bieten wir eine strenge Schranke für die maximale Höhe der Ungenauigkeit, die ausreichend ist für eine quadratische (oder eine superlineare) Rate der Konvergenz. Darüber hinaus, die entwickelte Theorie wird verwendet, um quadratische Konvergenz eines robusten projizierten Levenberg-Marquardt-Algorithmus zu zeigen. Die Verwendung von Levenberg-Marquardt-Algorithmen Typen für unbeschränkte multikriterielle Optimierungsprobleme im Detail zu untersucht. Insbesondere sind zwei globale und lokale quadratische konvergente Algorithmen für multikriterielle Optimierungsprobleme entwickelt worden. Die Annahmen wurden hergeleitet, unter welche die Fehlerschranke der eingebundenen Bedingung für ein Pareto-kritisches System erfüllt ist. Wir behandeln auch nicht differenzierbare nichtlineare Gleichungen aus Umformulierung der nichtlinearen Komplementaritätsprobleme durch NCP-Funktionen. Wir zeigen für diese Umformulierungen, dass die bestehenden differenzierbaren Bedingungen nicht zufrieden mit degenerierten Lösungen sind. Außerdem, leiten wir neue Ergebnisse für positiv homogene NCP-Funktionen. Letztere Ergebnisse werden verwendet um zu zeigen, dass geeignete schwächeren differenzierbare Bedingungen (so dass eine lokale Q-quadratische Konvergenzgeschwindigkeit ermöglichen) für bestimmte Umformulierungen gelten.

Die Dissertation behandelt methodische und algorithmische Aspekte der Spektralelementemethode zur räumlichen Diskretisierung partieller Differentialgleichungen. Die Weiterentwicklung einer symmetriebasierten Faktorisierung ermöglicht effiziente Operatoren für Tetraederelemente. Auf Grundlage einer umfassenden Leistungsanalyse werden Engpässe in der Implementierung der Operatoren identifiziert und durch algorithmische Modifikationen der Methode eliminiert.

In this thesis singularly perturbed convection-diffusion equations in the unit square are considered. Due to the presence of a small perturbation parameter the solutions of those problems exhibit an exponential layer near the outflow boundary and two parabolic layers near the characteristic boundaries. Discretisation of such problems on standard meshes and with standard methods leads to numerical solutions with unphysical oscillations, unless the mesh size is of order of the perturbation parameter which is impracticable. Instead we aim at uniformly convergent methods using layer-adapted meshes combined with standard methods. The meshes considered here are S-type meshes--generalisations of the standard Shishkin mesh. The domain is dissected in a non-layer part and layer parts. Inside the layer parts, the mesh might be anisotropic and non-uniform, depending on a mesh-generating function. We show, that the unstabilised Galerkin finite element method with bilinear elements on an S-type mesh is uniformly convergent in the energy norm of order (almost) one. Moreover, the numerical solution shows a supercloseness property, i.e. the numerical solution is closer to the nodal bilinear interpolant than to the exact solution in the given norm. Unfortunately, the Galerkin method lacks stability resulting in linear systems that are hard to solve. To overcome this drawback, stabilisation methods are used. We analyse different stabilisation techniques with respect to the supercloseness property. For the residual-based methods Streamline Diffusion FEM and Galerkin Least Squares FEM, the choice of parameters is addressed additionally. The modern stabilisation technique Continuous Interior Penalty FEM--penalisation of jumps of derivatives--is considered too. All those methods are proved to possess convergence and supercloseness properties similar to the standard Galerkin FEM. With a suitable postprocessing operator, the supercloseness property can be used to enhance the accuracy of the numerical solution and superconvergence of order (almost) two can be proved. We compare different postprocessing methods and prove superconvergence of above numerical methods on S-type meshes. To recover the exact solution, we apply continuous biquadratic interpolation on a macro mesh, a discontinuous biquadratic projection on a macro mesh and two methods to recover the gradient of the exact solution. Special attentions is payed to the effects of non-uniformity due to the S-type meshes. Numerical simulations illustrate the theoretical results.

Wir untersuchen ein Defektkorrekturverfahren, das ein einfaches Upwind-Differenzenverfahren erster Ordnung mit einem zentralen Differenzenverfahren kombiniert, für ein- und zweidimensionale singulär gestörte Konvektions-Diffusions-Probleme auf einer Klasse von Shishkin-Typ-Gittern. Im eindimensionalen Fall wird nachgewiesen, dass das Verfahren von (fast) zweiter Ordnung, gleichmäßig bezüglich des Diffusionsparameters $\epsilon$ konvergiert. Zur Konvergenzanalyse für das zweidimensionale Modellproblem werden verschiedene Techniken diskutiert. In einem Spezialfall kann auf einem stückweise uniformen Shishkin-Gitter die $\epsilon$-gleichmäßige Konvergenz des Verfahrens von fast zweiter Ordnung gezeigt werden. Ferner sind die bisher bekannten Stabilitätsaussagen und ihre Verwendung zur Konvergenzanalysis der betrachteten Differenzenverfahren sowie Methoden zur Analyse von Defektkorrekturverfahren zusammengestellt. Einige Bemerkungen zu Defektkorrekturverfahren und Finite-Elemente-Methoden schließen die Arbeit ab. Numerische Experimente untermauern die theoretischen Resultate. / We consider a defect correction method that combines a first-order upwinded difference scheme with a second-order central difference scheme for model singularly perturbed convection-diffusion problems in one and two dimensions on a class of Shishkin-Type meshes. In one dimension, the method is shown to be convergent uniformly in the diffusion parameter $\epsilon$ of second order in the discrete maximum norm. To analyze the two-dimensional case, we discuss several proof techniques for defect correction methods. For a special problem with constant coefficients on a piecewise uniform Shishkin-mesh we can show the second order convergence of the considered scheme, uniformly with respect to the diffusion parameter. Moreover the known stability properties and their impact on the convergence analysis of the considered differnce schemes are compiled. Some remarks on defect correction and finite elements conclude the theses. Numerical experiments support our theoretical results.

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.

A numerical treatment of non-linear higher-order geometric evolution equations with the level set and the finite element method is presented. The isotropic, weak anisotropic and strong anisotropic situation is discussed. Most of the equations considered in this work arise from the field of thin film growth. A short introduction to the subject is given. Four different models are discussed: mean curvature flow, surface diffusion, a kinetic model, which combines the effects of mean curvature flow and surface diffusion and includes a further kinetic component, and an adatom model, which incorporates in addition free adatoms. As an introduction to the numerical schemes, first the isotropic and weak anisotropic situation is considered. Then strong anisotropies (non-convex anisotropies) are used to simulate the phenomena of faceting and coarsening. The experimentally observed effect of corner and edge roundings is reached in the simulation through the regularization of the strong anisotropy with a higher-order curvature term. The curvature regularization leads to an increase by two in the order of the equations, which results in highly non-linear equations of up to 6th order. For the numerical solution, the equations are transformed into systems of second order equations, which are solved with a Schur complement approach. The adatom model constitutes a diffusion equation on a moving surface. An operator splitting approach is used for the numerical solution. In difference to other works, which restrict to the isotropic situation, also the anisotropic situation is discussed and solved numerically. Furthermore, a treatment of geometric evolution equations on implicitly given curved surfaces with the level set method is given. In particular, the numerical solution of surface diffusion on curved surfaces is presented. The equations are discretized in space by standard linear finite elements. For the time discretization a semi-implicit discretization scheme is employed. The derivation of the numerical schemes is presented in detail, and numerous computational results are given for the 2D and 3D situation. To keep computational costs low, the finite element grid is adaptively refined near the moving curves and surfaces resp. A redistancing algorithm based on a local Hopf-Lax formula is used. The algorithm has been extended by the authors to the 3D case. A detailed description of the algorithm in 3D is presented in this work. / In der Arbeit geht es um die numerische Behandlung nicht-linearer geometrischer Evolutionsgleichungen höherer Ordnung mit Levelset- und Finite-Elemente-Verfahren. Der isotrope, schwach anisotrope und stark anisotrope Fall wird diskutiert. Die meisten in dieser Arbeit betrachteten Gleichungen entstammen dem Gebiet des Dünnschicht-Wachstums. Eine kurze Einführung in dieses Gebiet wird gegeben. Es werden vier verschiedene Modelle diskutiert: mittlerer Krümmungsfluss, Oberflächendiffusion, ein kinetisches Modell, welches die Effekte des mittleren Krümmungsflusses und der Oberflächendiffusion kombiniert und zusätzlich eine kinetische Komponente beinhaltet, und ein Adatom-Modell, welches außerdem freie Adatome berücksichtigt. Als Einführung in die numerischen Schemata, wird zuerst der isotrope und schwach anisotrope Fall betrachtet. Anschließend werden starke Anisotropien (nicht-konvexe Anisotropien) benutzt, um Facettierungs- und Vergröberungsphänomene zu simulieren. Der in Experimenten beobachtete Effekt der Ecken- und Kanten-Abrundung wird in der Simulation durch die Regularisierung der starken Anisotropie durch einen Krümmungsterm höherer Ordnung erreicht. Die Krümmungsregularisierung führt zu einer Erhöhung der Ordnung der Gleichung um zwei, was hochgradig nicht-lineare Gleichungen von bis zu sechster Ordnung ergibt. Für die numerische Lösung werden die Gleichungen auf Systeme zweiter Ordnungsgleichungen transformiert, welche mit einem Schurkomplement-Ansatz gelöst werden. Das Adatom-Modell bildet eine Diffusionsgleichung auf einer bewegten Fläche. Zur numerischen Lösung wird ein Operatorsplitting-Ansatz verwendet. Im Unterschied zu anderen Arbeiten, die sich auf den isotropen Fall beschränken, wird auch der anisotrope Fall diskutiert und numerisch gelöst. Außerdem werden geometrische Evolutionsgleichungen auf implizit gegebenen gekrümmten Flächen mit Levelset-Verfahren behandelt. Insbesondere wird die numerische Lösung von Oberflächendiffusion auf gekrümmten Flächen dargestellt. Die Gleichungen werden im Ort mit linearen Standard-Finiten-Elementen diskretisiert. Als Zeitdiskretisierung wird ein semi-implizites Diskretisierungsschema verwendet. Die Herleitung der numerischen Schemata wird detailliert dargestellt, und zahlreiche numerische Ergebnisse für den 2D und 3D Fall sind gegeben. Um den Rechenaufwand gering zu halten, wird das Finite-Elemente-Gitter adaptiv an den bewegten Kurven bzw. den bewegten Flächen verfeinert. Es wird ein Redistancing-Algorithmus basierend auf einer lokalen Hopf-Lax Formel benutzt. Der Algorithmus wurde von den Autoren auf den 3D Fall erweitert. In dieser Arbeit wird der Algorithmus für den 3D Fall detailliert beschrieben.

In this thesis we consider constrained systems of equations. The focus is on local Newton-type methods for the solution of constrained systems which converge locally quadratically under mild assumptions implying neither local uniqueness of solutions nor differentiability of the equation function at solutions. The first aim of this thesis is to improve existing local convergence results of the constrained Levenberg-Marquardt method. To this end, we describe a general Newton-type algorithm. Then we prove local quadratic convergence of this general algorithm under the same four assumptions which were recently used for the local convergence analysis of the LP-Newton method. Afterwards, we show that, besides the LP-Newton method, the constrained Levenberg-Marquardt method can be regarded as a special realization of the general Newton-type algorithm and therefore enjoys the same local convergence properties. Thus, local quadratic convergence of a nonsmooth constrained Levenberg-Marquardt method is proved without requiring conditions implying the local uniqueness of solutions. As already mentioned, we use four assumptions for the local convergence analysis of the general Newton-type algorithm. The second aim of this thesis is a detailed discussion of these convergence assumptions for the case that the equation function of the constrained system is piecewise continuously differentiable. Some of the convergence assumptions seem quite technical and difficult to check. Therefore, we look for sufficient conditions which are still mild but which seem to be more familiar. We will particularly prove that the whole set of the convergence assumptions holds if some set of local error bound conditions is satisfied and in addition the feasible set of the constrained system excludes those zeros of the selection functions which are not zeros of the equation function itself, at least in a sufficiently small neighborhood of some fixed solution. We apply our results to constrained systems arising from complementarity systems, i.e., systems of equations and inequalities which contain complementarity constraints. Our new conditions are discussed for a suitable reformulation of the complementarity system as constrained system of equations by means of the minimum function. In particular, it will turn out that the whole set of the convergence assumptions is actually implied by some set of local error bound conditions. In addition, we provide a new constant rank condition implying the whole set of the convergence assumptions. Particularly, we provide adapted formulations of our new conditions for special classes of complementarity systems. We consider Karush-Kuhn-Tucker (KKT) systems arising from optimization problems, variational inequalities, or generalized Nash equilibrium problems (GNEPs) and Fritz-John (FJ) systems arising from GNEPs. Thus, we obtain for each problem class conditions which guarantee local quadratic convergence of the general Newton-type algorithm and its special realizations to a solution of the particular problem. Moreover, we prove for FJ systems of GNEPs that generically some full row rank condition is satisfied at any solution of the FJ system of a GNEP. The latter condition implies the whole set of the convergence assumptions if the functions which characterize the GNEP are sufficiently smooth. Finally, we describe an idea for a possible globalization of our Newton-type methods, at least for the case that the constrained system arises from a certain smooth reformulation of the KKT system of a GNEP. More precisely, a hybrid method is presented whose local part is the LP-Newton method. The hybrid method turns out to be, under appropriate conditions, both globally and locally quadratically convergent.