In this theses, an iterativ linear solver for the non-linear p-Poisson problem is introduced. After the theoretical convergence results some numerical examples of a fully adaptive solver are presented.

This is a book on numerical methods for singular perturbation problems - in particular stationary convection-dominated convection-diffusion problems. More precisely it is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. An early important contribution towards the optimization of numerical methods by means of special meshes was made by N.S. Bakhvalov in 1969. His paper spawned a lively discussion in the literature with a number of further meshes being proposed and applied to various singular perturbation problems. However, in the mid 1980s this development stalled, but was enlivend again by G.I. Shishkin's proposal of piecewise- equidistant meshes in the early 1990s. Because of their very simple structure they are often much easier to analyse than other meshes, although they give numerical approximations that are inferior to solutions on competing meshes. Shishkin meshes for numerous problems and numerical methods have been studied since and they are still very much in vogue. With this contribution we try to counter this development and lay the emphasis on more general meshes that - apart from performing better than piecewise-uniform meshes - provide a much deeper insight in the course of their analysis. In this monograph a classification and a survey are given of layer-adapted meshes for convection-diffusion problems. It tries to give a comprehensive review of state-of-the art techniques used in the convergence analysis for various numerical methods: finite differences, finite elements and finite volumes. While for finite difference schemes applied to one-dimensional problems a rather complete convergence theory for arbitrary meshes is developed, the theory is more fragmentary for other methods and problems and still requires the restriction to certain classes of meshes.

Be it in the weather forecast or while swimming in the Baltic Sea, in almost every aspect of every day life we are confronted with flow phenomena. A common model to describe the motion of viscous incompressible fluids are the Navier-Stokes equations. These equations are not only relevant in the field of physics, but they are also of great interest in a purely mathematical sense. One of the difficulties of the Navier-Stokes equations originates from a non-linear term. In this thesis, we consider the Oseen equations as a linearisation of the Navier-Stokes equations. We restrict ourselves to the two-dimensional case. Our domain will be the unit square. The aim of this thesis is to find a suitable numerical method to overcome known instabilities in discretising these equations. One instability arises due to layers of the analytical solution. Another instability comes from a divergence constraint, where one gets poor numerical accuracy when the irrotational part of the right-hand side of the equations is large. For the first cause, we investigate the layer behaviour of the analytical solution of the corresponding stream function of the problem. Assuming a solution decomposition into a smooth part and layer parts, we create layer-adapted meshes in Chapter 3. Using these meshes, we introduce a numerical method for equations whose solutions are of the assumed structure in Chapter 4. To reduce the instability caused by the divergence constraint, we add a grad-div stabilisation term to the standard Galerkin formulation. We consider Taylor-Hood elements and elements with a discontinous pressure space. We can show that there exists an error bound which is independent of our perturbation parameter and get information about the convergence rate of the method. Numerical experiments in Chapter 5 confirm our theoretical results.

What is flagellar swimming? Cilia and flagella are whip-like cell appendages that can exhibit regular bending waves. This active process emerges from the non-equilibrium dynamics of molecular motors distributed along the length of cilia and flagella. Eukaryotic cells can possess many cilia and flagella that beat in a coordinated fashion, thus transporting fluids, as in mammalian airways or the ventricular system inside the brain. Many unicellular organisms posses just one or two flagella, rendering them microswimmers that are propelled through fluids by the flagellar beat including sperm cells and the biflagellate green alga Chlamydomonas. Objectives. In this thesis in theoretical biological physics, we seek to understand the nonlinear dynamics of flagellar swimming and synchronization. We investigate the flow fields induced by beating flagella and how in turn external hydrodynamic flows change speed and shape of the flagellar beat. This flagellar load-response is a prerequisite for flagellar synchronization. We want to find the physical principals underlying stable synchronization of the two flagella of Chlamydomonas cells. Results. First, we employed realistic hydrodynamic simulations of flagellar swimming based on experimentally measured beat patterns. For this, we developed analysis tools to extract flagellar shapes from high-speed videoscopy data. Flow-signatures of flagellated swimmers are analysed and their effect on a neighboring swimmer is compared to the effect of active noise of the flagellar beat. We were able to estimate a chemomechanical energy efficiency of the flagellar beat and determine its waveform compliance by comparing findings from experiments, in which a clamped Chlamydomonas is exposed to external flow, to predictions from an effective theory that we designed. These mechanical properties have interesting consequences for the synchronization dynamics of Chlamydomonas, which are revealed by computer simulations. We propose that direct elastic coupling between the two flagella of Chlamydomonas, as suggested by recent experiments, in combination with waveform compliance is crucial to facilitate in-phase synchronization of the two flagella of Chlamydomonas.

Be it in the weather forecast or while swimming in the Baltic Sea, in almost every aspect of every day life we are confronted with flow phenomena. A common model to describe the motion of viscous incompressible fluids are the Navier-Stokes equations. These equations are not only relevant in the field of physics, but they are also of great interest in a purely mathematical sense. One of the difficulties of the Navier-Stokes equations originates from a non-linear term. In this thesis, we consider the Oseen equations as a linearisation of the Navier-Stokes equations. We restrict ourselves to the two-dimensional case. Our domain will be the unit square. The aim of this thesis is to find a suitable numerical method to overcome known instabilities in discretising these equations. One instability arises due to layers of the analytical solution. Another instability comes from a divergence constraint, where one gets poor numerical accuracy when the irrotational part of the right-hand side of the equations is large. For the first cause, we investigate the layer behaviour of the analytical solution of the corresponding stream function of the problem. Assuming a solution decomposition into a smooth part and layer parts, we create layer-adapted meshes in Chapter 3. Using these meshes, we introduce a numerical method for equations whose solutions are of the assumed structure in Chapter 4. To reduce the instability caused by the divergence constraint, we add a grad-div stabilisation term to the standard Galerkin formulation. We consider Taylor-Hood elements and elements with a discontinous pressure space. We can show that there exists an error bound which is independent of our perturbation parameter and get information about the convergence rate of the method. Numerical experiments in Chapter 5 confirm our theoretical results.:Acknowledgement III Notation IV 1 Introduction 1 1.1 Existence of solutions 2 1.2 Transformation into a fourth-order problem 4 2 Asymptotic analysis 6 2.1 A fourth-order problem in 1D 6 2.2 A fourth-order problem in 2D 14 2.2.1 Asymptotic expansion 19 2.2.2 Estimation of the residual 26 2.2.3 Asymptotic expansion without compatibility conditions 30 3 Solution decomposition and layer-adapted meshes 32 3.1 Solution decomposition 32 3.2 Layer-adapted meshes 33 3.3 Interpolation errors on layer-adapted meshes 36 4 Galerkin method and stabilisation 41 4.1 Discrete problem and stabilised formulation 41 4.2 A priori error estimates 44 5 Numerical results 48 5.1 Numerical evaluation of inf-sup constants 48 5.1.1 Theoretical aspects 48 5.1.2 Numerical results for β0 and B0 50 5.2 Convergence studies 53 5.2.1 Uniformity in ε 54 5.2.2 Convergence order 55 5.2.3 Necessity of stabilisation 56 5.2.4 Further experiments without known exact solution 56 6 Conclusions and outlook 60 A Numerical study of the stability estimate (2.35) 62 Bibliography 67

What is flagellar swimming? Cilia and flagella are whip-like cell appendages that can exhibit regular bending waves. This active process emerges from the non-equilibrium dynamics of molecular motors distributed along the length of cilia and flagella. Eukaryotic cells can possess many cilia and flagella that beat in a coordinated fashion, thus transporting fluids, as in mammalian airways or the ventricular system inside the brain. Many unicellular organisms posses just one or two flagella, rendering them microswimmers that are propelled through fluids by the flagellar beat including sperm cells and the biflagellate green alga Chlamydomonas. Objectives. In this thesis in theoretical biological physics, we seek to understand the nonlinear dynamics of flagellar swimming and synchronization. We investigate the flow fields induced by beating flagella and how in turn external hydrodynamic flows change speed and shape of the flagellar beat. This flagellar load-response is a prerequisite for flagellar synchronization. We want to find the physical principals underlying stable synchronization of the two flagella of Chlamydomonas cells. Results. First, we employed realistic hydrodynamic simulations of flagellar swimming based on experimentally measured beat patterns. For this, we developed analysis tools to extract flagellar shapes from high-speed videoscopy data. Flow-signatures of flagellated swimmers are analysed and their effect on a neighboring swimmer is compared to the effect of active noise of the flagellar beat. We were able to estimate a chemomechanical energy efficiency of the flagellar beat and determine its waveform compliance by comparing findings from experiments, in which a clamped Chlamydomonas is exposed to external flow, to predictions from an effective theory that we designed. These mechanical properties have interesting consequences for the synchronization dynamics of Chlamydomonas, which are revealed by computer simulations. We propose that direct elastic coupling between the two flagella of Chlamydomonas, as suggested by recent experiments, in combination with waveform compliance is crucial to facilitate in-phase synchronization of the two flagella of Chlamydomonas.:1 Introduction 1.1 Physics of cell motility: flagellated swimmers as model system 2 1.1.1 Tissue cells and unicellular eukaryotic organisms have cilia and flagella 2 1.1.2 The conserved architecture of flagella 3 1.1.3 Synchronization in collections of flagella 5 1.2 Hydrodynamics at the microscale 9 1.2.1 Navier-Stokes equation 10 1.2.2 The limit of low Reynolds number 10 1.2.3 Multipole expansion of flow fields 11 1.3 Self-propulsion by viscous forces 13 1.3.1 Self propulsion requires broken symmetries 13 1.3.2 Signatures of flowfields: pusher & puller 15 1.4 Overview of the thesis 16 2 Flow signatures of flagellar swimming 2.1 Self-propulsion of flagellated swimmers 20 2.1.1 Representation of flagellar shapes 20 2.1.2 Computation of hydrodynamic friction forces 21 2.1.3 Material frame and rigid-body transformations 22 2.1.4 The grand friction matrix 23 2.1.5 Dynamics of swimming 23 2.2 The hydrodynamic far field: pusher and puller 26 2.2.1 The flow generated by a swimmer 26 2.2.2 Force dipole characterization 27 2.2.3 Flagellated swimmers alternate between pusher and puller 29 2.2.4 Implications for two interacting Chlamydomonas cells 31 2.3 Inertial screening of oscillatory flows 32 2.3.1 Convection and oscillatory acceleration 33 2.3.2 The oscilet: fundamental solution of unsteady flow 35 2.3.3 Screening length of oscillatory flows 35 2.4 Energetics of flagellar self-propulsion 36 2.4.1 Impact of inertial screening on hydrodynamic dissipation 37 2.4.2 Case study: the green alga Chlamydomonas 38 2.4.3 Discussion: evolutionary optimization and the number of molecular motors 38 2.5 Summary 39 3 The load-response of the flagellar beat 3.1 Experimental collaboration: flagellated swimmers exposed to flows 41 3.1.1 Description of the experimental setup 42 3.1.2 Computed flow profile in the micro-fluidic device 43 3.1.3 Image processing and flagellar tracking 43 3.1.4 Mode decomposition and limit-cycle reconstruction 47 3.1.5 Changes of limit-cycle dynamics: deformation, translation, acceleration 49 3.2 An effective theory of flagellar oscillations 50 3.2.1 A balance of generalized forces 50 3.2.2 Hydrodynamic friction in generalized coordinates 51 3.2.3 Intra-flagellar friction 52 3.2.4 Calibration of active flagellar driving forces 52 3.2.5 Stability of the limit cycle of the flagellar beat 53 3.2.6 Equations of motion 55 3.3 Comparison of theory and experiment 56 3.3.1 Flagellar mean curvature 57 3.3.2 Susceptibilities of phase speed and amplitude 57 3.3.3 Higher modes and stalling of the flagellar beat at high external load 59 3.3.4 Non-isochrony of flagellar oscillations 63 3.4 Summary 63 4 Flagellar load-response facilitates synchronization 4.1 Synchronization to external driving 65 4.2 Inter-flagellar synchronization in the green alga Chlamydomonas 67 4.2.1 Equations of motion for inter-flagellar synchronization 68 4.2.2 Synchronization strength for free-swimming and clamped cells 70 4.2.3 The synchronization strength depends on energy efficiency and waveform compliance 73 4.2.4 The case of an elastically clamped cell 74 4.2.5 Basal body coupling facilitates in-phase synchronization 75 4.2.6 Predictions for experiments 78 4.3 Summary 80 5 Active flagellar fluctuations 5.1 Effective description of flagellar oscillations 84 5.2 Measuring flagellar noise 84 5.2.1 Active phase fluctuations are much larger than thermal noise 84 5.2.2 Amplitude fluctuations are correlated 85 5.3 Active flagellar fluctuations result in noisy swimming paths 86 5.3.1 Effective diffusion of swimming circles of sperm cell 86 5.3.2 Comparison of the effect of noise and hydrodynamic interactions 87 5.4 Summary 88 6 Summary and outlook 6.1 Summary of our results 89 6.2 Outlook on future work 90 A Solving the Stokes equation A.1 Multipole expansion 95 A.2 Resistive-force theory 96 A.3 Fast multipole boundary element method 97 B Linearized Navier-Stokes equation B.1 Linearized Navier-Stokes equation 101 B.2 The case of an oscillating sphere 102 B.3 The small radius limit 103 B.4 Greens function 104 C Hydrodynamic friction C.1 A passive particle 107 C.2 Multiple Particles 107 C.3 Generalized coordinates 108 D Data analysis methods D.1 Nematic filter 111 D.1.1 Nemat 111 D.1.2 Nematic correlation 111 D.2 Principal-component analysis 112 D.3 The quality of the limit-cycle projections of experimental data 113 E Adler equation F Sensitivity analysis for computational results F.1 The distance function of basal coupling 117 F.2 Computed synchronization strength for alternative waveform 118 F.3 Insensitivity of computed load-response to amplitude correlation time 118 List of Symbols List of Figures Bibliography

In dieser Arbeit untersuchen wir mehrere Anwendungen von photonischen und plasmonischen Nanostrukturen unter Verwendung zweier verschiedener numerischer Methoden: die Fourier-Moden-Methode (FMM) und ein unstetiges Galerkin-Zeitraumverfahren (discontinuous Galerkin time-domain method, DGTD method). Die Methoden werden für vier verschiedene Anwendungen eingesetzt, die alle eine Materialmodellerweiterung in der Implementierung der Methoden erfordern. Diese Anwendungen beinhalten die Untersuchung von dünnen, freistehenden, periodisch perforierten Goldfilmen. Wir charakterisieren die auftretenden Oberflächenplasmonenpolaritonen durch die Berechnung von Transmissions- und Elektronenenergieverlustspektren, die mit experimentellen Messungen verglichen werden. Dazu stellen wir eine Erweiterung der DGTD-Methode zur Verfügung, die sowohl absorbierende, impedanzangepasste Randschichten als auch Anregung mit geglätteter Ladungsverteilung für materialdurchdringende Elektronenstrahlen beinhaltet. Darüber hinaus wird eine Erweiterung auf nicht-dispersive anisotrope Materialien für eine Formoptimierung einer volldielektrischen magneto-optischen Metaoberfläche verwendet. Diese Optimierung ermöglicht eine verstärkte Faraday-Rotation zusammen mit einer hohen Transmission. Zusätzlich untersuchen wir abstimmbare hyperbolische Metamaterialresonatoren im nahen Infrarot mit Hilfe der FMM. Wir berechnen deren Resonanzen und vergleichen sie mit dem Experiment. Zum Schluss wird die Implementierung eines nichtlinearen Vier-Niveau-System-Materialmodells in der DGTD-Methode verwendet, um die Laserschwellen eines Mikroresonators mit Bragg-Spiegeln zu berechnen. Bei Einführung eines Silbergitters mit variablen Spaltgrößen wird eine defektinduzierte Kontrolle der Laserschwellen ermöglicht. Die Berechnung der vollständigen, zeitaufgelösten Felddynamik innerhalb des Resonator gibt dabei Aufschluss über die beteiligten Lasermoden. / In this thesis, we study several applications of photonic and plasmonic nanostructures by employing two different numerical methods: the Fourier modal method (FMM) and discontinuous Galerkin time-domain (DGTD) method. The methods are used for four different applications, all of which require a material model extension for the implementation of the methods. These applications include the investigation of thin, free-standing periodically perforated gold films. We characterize the emerging surface plasmon polaritons by computing both transmittance and electron energy loss spectra, which are compared to experimental measurements. To this end, we provide an extension of the DGTD method, including absorbing stretched coordinate perfectly matched layers as well as excitations with smoothed charge distribution for material-penetrating electron beams. Furthermore, an extension to non-dispersive anisotropic materials is used for shape optimization of an all-dielectric magneto-optic metasurface. This optimization enables an enhanced Faraday rotation along with high transmittance. Additionally, we study tuneable near-infrared hyperbolic metamaterial cavities with the help of the FMM. We compute the cavity resonances and compare them to the experiment. Finally, the implementation of a non-linear four-level system material model in the DGTD method is used to compute lasing thresholds of a distributed Bragg reflector microcavity. Introducing a silver grating with variable gap sizes allows for a defect-induced lasing threshold control. The computation of the full time-resolved field dynamics of the cavity provides information on the involved lasing modes.

Traditional methods in optical design like ray tracing suffer from slow convergence and are not constructive, i.e., each minimal perturbation of input parameters might lead to “chaotic” changes in the output. However, so-called inverse methods can be helpful in designing optical systems of reflectors and lenses. The equations in R2 become ordinary differential equations, while in R3 the equations become partial differential equations. These equations are then used to transform source distributions into target distributions, where the distributions are arbitrary, though assumed to be positive and integrable. In this project, we derive the governing equations and solve them numerically, for the systems presented by our instructor Martijn Anthonissen [Anthonissen et al. 2021]. Additionally, we show how point sources can be derived as a special case of a interval source with di- rected source interval, i.e., with each point in the source interval there is also an associated unit direction vector which could be derived from a system of two interval sources in R2. This way, it is shown that connecting source distributions with target distributions can be classified into two instead of three categories. The resulting description of point sources as a source along an interval with directed rays could potentially be extended to three dimensions, leading to interpretations of point sources as directed sources on convex or star-shaped sets.:1 Abstract 4 2 Notation And Conventions 4 3 Introduction 5 4 ECMI Modeling Week Challenges 5 4.1 Problem 1 - Parallel to Near-Field Target 5 4.1.1 Description 5 4.1.2 Deriving The Equations 5 4.2 Problem 2 - Parallel Source To Two Targets 8 4.3 Problem 3 - Point Source To Near-Field Target 9 4.3.1 Deriving The Equations 9 4.4 Problem 4 - Point Source To Two Targets 11 5 Validation - Ray tracing 13 5.1 Splines 13 5.1.1 Piece-Wise Affine Reflectors 13 5.1.2 Piece-Wise Cubic Reflectors 14 5.2 Error Estimates For Spline Reflectors 14 5.2.1 Lemma: A Priori Feasibility Of Starting Values For Near-Field Problems 15 5.2.2 Estimates for single reflector, near-field targets 16 5.3 Ray Tracing Errors - Illumination Errors 17 5.3.1 Definition: Axioms For Errors 18 5.3.2 Extrapolated Ray Tracing Error (ERTE) 18 5.3.3 Definition: Minimal Distance Ray Tracing Error (MIRTE) 19 5.3.4 Lemma: Continuity Of The Ray Traced Reflection Projection Of Smooth Reflectors 19 5.3.5 Theorem: Convergence Of The MIRTE 20 5.3.6 Convergence Of The ERTE 21 5.3.7 Application 21 6 Numerical Implementation 21 6.1 The DOPTICS Library 21 6.2 Pseudocode Of The Implementation 21 6.2.1 Solutions Of The Problems 22 6.2.2 Ray Tracing And Ray Tracing Error 22 6.3 ERTE Implementation 25 7 Results 26 7.1 Problem 1: Results 26 7.2 Problem 2: Results 26 7.3 Problem 3: Results 27 7.4 Problem 4: Results 27 8 Generalizations In Two Dimensions 29 8.1 Directed Densities 29 8.2 Generalized, Orthogonally Emitting Sources in R2 30 8.2.1 Point Light Sources As Orthogonally Emitting Sources 30 9 Conclusion and Future Research 32 10 Group Dynamic 32 References 32

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.

This is a book on numerical methods for singular perturbation problems - in particular stationary convection-dominated convection-diffusion problems. More precisely it is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. An early important contribution towards the optimization of numerical methods by means of special meshes was made by N.S. Bakhvalov in 1969. His paper spawned a lively discussion in the literature with a number of further meshes being proposed and applied to various singular perturbation problems. However, in the mid 1980s this development stalled, but was enlivend again by G.I. Shishkin's proposal of piecewise- equidistant meshes in the early 1990s. Because of their very simple structure they are often much easier to analyse than other meshes, although they give numerical approximations that are inferior to solutions on competing meshes. Shishkin meshes for numerous problems and numerical methods have been studied since and they are still very much in vogue. With this contribution we try to counter this development and lay the emphasis on more general meshes that - apart from performing better than piecewise-uniform meshes - provide a much deeper insight in the course of their analysis. In this monograph a classification and a survey are given of layer-adapted meshes for convection-diffusion problems. It tries to give a comprehensive review of state-of-the art techniques used in the convergence analysis for various numerical methods: finite differences, finite elements and finite volumes. While for finite difference schemes applied to one-dimensional problems a rather complete convergence theory for arbitrary meshes is developed, the theory is more fragmentary for other methods and problems and still requires the restriction to certain classes of meshes.