Numerische Mathematik

The subject of this thesis is the efficient simulation of nonlinear optical phenomena, especially frequency mixing processes. Unlike linear optics, where the response of the material to light remains constant regardless of the light's intensity, in nonlinear optics, this response varies as a function of the light's intensity. Nonlinear optical phenomena have paved the way for significant scientific innovations, e.g. the development of novel optical sources, and have become central to a wide range of technological applications. This work is located in the emerging field of scientific machine learning and contributes to the efficient simulation of nonlinear optical phenomena by combining the rigor of finite element methods with the empirical strength of deep neural networks. Specifically, we focus on frequency mixing processes in the context of THz generation in periodically poled nonlinear crystals through Quasi-Phase-Matching. We develop a mathematical model and numerical simulation techniques in the time domain to simulate complex nonlinear optical processes without the usual slowly varying envelope approximation. The accuracy is illustrated through numerical simulations and comparisons to experimental data. The simulations elucidate the THz generation in periodically poled Lithium Niobate (PPLN), including optical harmonic generation. This thesis introduces a novel approach for solving optimal Dirichlet boundary control problems of nonlinear optics using deep learning. For computing high-resolution approximations of the solution to the nonlinear wave model, we propose higher order space-time finite element methods combined with collocation techniques, ensuring C^l-regularity in time of the global discrete solution. The simulation data trains solution operators that effectively leverage the higher regularity of the training data. The operators, represented by Fourier Neural Operators, are used as the forward solver in the optimal Dirichlet boundary control problem. This approach demonstrates an innovative integration of traditional numerical methods with data-driven techniques. The proposed algorithm is implemented and tested on high-performance computing platforms, with a focus on efficiency and scalability. We showcase its effectiveness on the problem of generating Terahertz radiation in PPLN, optimizing the parametrization of the optical input pulse to maximize the yield of 0.3THz-frequency radiation. By exploiting the periodic layering of the crystal, the networks are trained to learn propagation through one period of the layers. The recursive application of the network onto itself yields an approximation to the full problem. The results indicate a significant speedup compared to classical methods and, when compared to experimental data, demonstrate the potential to revolutionize optimization in nonlinear optics. The proposed approach of data-driven acceleration contributes to a paradigm shift towards more efficient and innovative problem-solving methods in the realm of nonlinear optics and beyond. In addition, we review a promising approach to showing the well-posedness of the nonlinear Maxwell equation and show how it can be applied in the settings we consider throughout this thesis. To this end, the nonlinear wave equations are rewritten as a first-order system in space and time. The well-posedness is ensured by a combination of the abstract theory for evolutionary problems by R. Picard and fixed point arguments.

In dieser Arbeit wird ein zeitliches Multi-Raten Konzept kombiniert mit zielorientierter Fehlerkontrolle basierend auf der Dual Weighted Residual (DWR) Methode für ein gekoppeltes Strömungs- und Transportproblem entwickelt. Das Transportproblem wird durch eine Konvektions-Diffusions-Reaktionsgleichung mit hochdynamischem Zeitverhalten repräsentiert, wohingegen das Strömungsproblem durch zähfließende, zeitabhängige Stokes Gleichungen modelliert wird. Dies erfordert den Einsatz von adaptiven Multi-Raten Methoden, um die unterschiedlichen charakteristischen Zeitskalen der beiden Teilprobleme adäquat auflösen zu können. Zudem wird das Transportproblem als konvektionsdominant angesehen und zu diesem Zweck mithilfe der Streamline Upwind Petrov Galerkin (SUPG) Methode stabilisiert. Für beide Probleme wird als zeitliches Diskretisierungsverfahren eine unstetige Galerkin Methode dG(r) mit beliebigem Polynomgrad r >= 0 und als örtliches Diskretisierungsverfahren eine stetige Galerkin Methode cG(r) mit beliebigem Polynomgrad p>= 1 verwendet. Für eine effiziente numerische Behandlung solch stabilisierter multiphysikalischer Probleme, deren Teilprobleme unterschiedliche Dynamiken hinsichtlich ihrer charakteristische Zeitskalen aufweisen, sind adaptive Gitterverfeinerungsstrategien unumgänglich. Vor diesem Hintergrund werden in dieser Arbeit a posteriori Fehlerschätzer basierend auf der DWR Methode für beide Teilprobleme hergeleitet. Diese Fehlerschätzer, gemessen in physikalisch relevanten Zielgrößen, werden in räumliche und zeitliche Anteile aufgeteilt, sodass ihre lokalisierten Repräsentanten als zellweise Fehlerindikatoren für räumliche und zeitliche Gitterverfeinerung genutzt werden können. Im Hinblick auf eine effiziente numerische Approximation gekoppelter Probleme sind adaptive Verfeinerungsstrategien umso wichtiger, will man hierbei nicht nur wissen, in welcher Region das jeweilige Gitter verfeinert werden soll, sondern auch welches der beiden Teilprobleme stärker zum Gesamtfehler beiträgt und somit mit größerer Genauigkeit zu lösen ist. Um das Zusammenspiel von Stabilisierung und Fehlerkontrolle zu analysieren, werden zunächst stationäre und zeitabhängige konvektionsdominante Transportprobleme mit fest vorgegebener Konvektion behandelt. Hierzu werden zahlreiche Vergleichsstudien im Hinblick auf unterschiedliche Ansätze für das duale Problem sowie unterschiedliche Approximationstechniken für räumliche und zeitliche Gewichtsfaktoren durchgeführt. Die Resultate dienen als Grundlage für das anschließend betrachtete gekoppelte Problem. Für die Implementierung werden sogenannte Raum-Zeit Slabs basiend auf Tensor- Produkten verwendet, welche Galerkin Diskretisierungen mit beliebeigem Polynomgrad in Raum und Zeit ermöglichen. Diese Slabs bilden eine Disketisierung des Raum-Zeit- Gebiets und werden in einem Listen-Objekt gespeichert, was ein einfaches und effizientes Hinzufügen von weiteren Slabs im Sinne einer adaptiven Gitterverfeinerung zulässt. Die Eigenschaften des vorgestellten Konzepts werden anhand von akademischen Testbeispielen, konvektionsdominanten Benchmarks sowie physikalischer Anwendungsbeispiele in zwei und drei Raumdimensionen untersucht. In numerischen Beispielen werden Konvergenzstudien sowie numerische Effiziens- und Stabilisierungsuntersuchungen des zugrundeliegenden Algorithmus durchgeführt.

In this work, various aspects of space-time finite element discretizations for the Navier-Stokes equations and wave problems are investigated. First, an overview of the current state of the science is given in Chapter 1 and the notation is introduced in Chapter 2. Then, in Chapter 3, flow problems on time-dependent domains are studied. These play a major role in engineering practice, for example in the simulation of fluid-structure interaction problems, such as the interaction of a fluid with a rotor. In this work, rigid bodies moving in a viscous fluid described by the time-dependent Navier-Stokes equations are considered as an example problem for this purpose. This means that the flow area changes depending on the position of the rigid body. A classical way to simulate such problems is the Arbitrary-Lagrangian-Eulerian method (ALE method), which is based on the transformation of the flow domain to a reference mesh. This transformation has the disadvantage that it is computationally intensive, can induce nonlinearities, and also, for larger deformations of the flow domain, an update of the underlying computational grid is inevitable. Further, such a remeshing is again computationally expensive and usually means that all matrices and vectors of the underlying simulation have to be rebuilt. In this work, a method is presented in which all computations are performed on a fixed background grid that can be arbitrarily cut by the rigid body. This method is called the Cut Finite Elmenent Method (CutFEM). A prerequisite for an implementation is to be able to integrate functions over arbitrarily cut cells in order to be able to assemble the underlying system matrices and vectors. To accomplish this, a flexible method was developed, based on the iterated application of 1d integrals. In the case of cells that become very small, if they are cut by the rigid body, unphysical steep gradients of the solution arise in these cells. These result in poor conditioning of the system matrix and numerical instabilities. To circumvent the problem, a so-called "ghost penalty" stabilization was implemented, which constrains the gradients and makes the method robust. Furthermore, the stabilization operator implicitly propagates the flow solution into the entire rigid-body domain. This allows all calculations to be performed on a fixed, time-independent background grid. All aspects mentioned so far have been verified in numerical convergence tests and simulations. Then, in Chapter 4, a geometric multigrid method for higher order finite element space-time discretizations of the Navier-Stokes equations is then implemented for time-independent domains. The method is based on a Vanka smoother, which dampens high-frequency error components over all space-time degrees of freedom in each time step. All major parts of our code are implemented in parallel using the Message Passing Interface (MPI). The multigrid method, used in this work, is verified by simulating the 3d DFG benchmark "flow around a cylinder", with more than 96 million space-time degrees of freedom per time step. Adapting the method to problems with time-dependent domains in conjunction with CutFEM results in increased iteration counts, depending on the size of the rigid body. The causes of this are analyzed and investigated numerically in this work. In Chapter 5 classical continuous finite element methods for time discretization are combined with collocation methods to obtain efficient higher order methods in time, which also result in a discrete solution of higher regularity. Such Galerkin-Collocation methods are developed and applied first for the wave equation and then for the Navier-Stokes equations. Compared to traditional continuous finite element time discretizations, we could significantly increase the time step size with these methods, while maintaining the same quality of the solution. Especially in coupled problems, methods of this type could be advantageous if coefficients of the derivative of one problem are included in the problem description of the other problem.