<p dir="ltr">This thesis presents advances in numerical methods for partial differential equations (PDEs) and optimization problems, with a focus on improving efficiency, stability, and accuracy across various applications. We begin by addressing 3D Poisson-type equations, developing a GPU-accelerated spectral-element method that utilizes the tensor product structure to achieve extremely fast performance. This approach enables solving problems with over one billion degrees of freedom in less than one second on modern GPUs, with applications to Schrödinger and Cahn<i>–</i>Hilliard equations demonstrated. Next, we focus on parabolic PDEs, specifically the Cahn<i>–</i>Hilliard equation with dynamical boundary conditions. We propose an efficient energy-stable numerical scheme using a unified framework to handle both Allen<i>–</i>Cahn and Cahn<i>–</i>Hilliard type boundary conditions. The scheme employs a scalar auxiliary variable (SAV) approach to achieve linear, second-order, and unconditionally energy stable properties. Shifting to a machine learning perspective for PDEs, we introduce an unsupervised learning-based numerical method for solving elliptic PDEs. This approach uses deep neural networks to approximate PDE solutions and employs least-squares functionals as loss functions, with a focus on first-order system least-squares formulations. In the realm of optimization, we present an efficient and robust SAV based algorithm for discrete gradient systems. This method modifies the standard SAV approach and incorporates relaxation and adaptive strategies to achieve fast convergence for minimization problems while maintaining unconditional energy stability. Finally, we address optimization in the context of machine learning by developing a structure-guided Gauss<i>–</i>Newton method for shallow ReLU neural network optimization. This approach exploits both the least-squares and neural network structures to create an efficient iterative solver, demonstrating superior performance on challenging function approximation problems. Throughout the thesis, we provide theoretical analysis, efficient numerical implementations, and extensive computational experiments to validate the proposed methods. </p>

Das Thema dieser Dissertation ist die Anwendung von Variationsmethoden auf Evolutionsgleichungen parabolischen und hyperbolischen Typs. Im ersten Teil der Arbeit beschäftigen wir uns mit Reaktions-Diffusions-Systemen, die sich als Gradientensysteme schreiben lassen. Hierbei verstehen wir unter einem Gradientensystem ein Tripel bestehend aus einem Zustandsraum, einem Entropiefunktional und einer Dissipationsmetrik. Wir geben Bedingungen an, die die geodätische Konvexität des Entropiefunktionals sichern. Geodätische Konvexität ist eine wertvolle aber auch starke strukturelle Eigenschaft und schwer zu zeigen. Wir zeigen anhand zahlreicher Beispiele, darunter ein Drift-Diffusions-System, dass dennoch interessante Systeme existieren, die diese Eigenschaft besitzen. Einen weiteren Punkt dieser Arbeit stellt die Anwendung von Gamma-Konvergenz auf Gradientensysteme dar. Wir betrachten hierbei zwei Modellsysteme aus dem Bereich der Mehrskalenprobleme: Erstens, die rigorose Herleitung einer Allen-Cahn-Gleichung mit dynamischen Randbedingungen und zweitens, einer Interface-Bedingung für eine eindimensionale Diffusionsgleichung jeweils aus einem reinen Bulk-System. Im zweiten Teil der Arbeit beschäftigen wir uns mit dem sog. Weighted-Inertia-Dissipation-Energy-Prinzip für Evolutionsgleichungen. Hierbei werden Trajektorien eines Systems als (Grenzwerte von) Minimierer(n) einer parametrisierten Familie von Funktionalen charakterisiert. Dies erlaubt es, Werkzeuge aus der Theorie der Variationsrechung auf Evolutionsprobleme anzuwenden. Wir zeigen, dass Minimierer der WIDE-Funktionale gegen Lösungen des Ausgangsproblems konvergieren. Hierbei betrachten wir getrennt voneinander den Fall des beschränkten und des unbeschränkten Zeitintervalls, die jeweils mit verschiedenen Methoden behandelt werden. / This thesis deals with the application of variational methods to evolution problems governed by partial differential equations. The first part of this work is devoted to systems of reaction-diffusion equations that can be formulated as gradient systems with respect to an entropy functional and a dissipation metric. We provide methods for establishing geodesic convexity of the entropy functional by purely differential methods. Geodesic convexity is beneficial, however, it is a strong structural property of a gradient system that is rather difficult to achieve. Several examples, including a drift-diffusion system, provide a survey on the applicability of the theory. Next, we demonstrate the application of Gamma-convergence, to derive effective limit models for multiscale problems. The crucial point in this investigation is that we rely only on the gradient structure of the systems. We consider two model problems: The rigorous derivation of an Allen-Cahn system with bulk/surface coupling and of an interface condition for a one-dimensional diffusion equation. The second part of this thesis is devoted to the so-called Weighted-Inertia-Dissipation-Energy principle. The WIDE principle is a global-in-time variational principle for evolution equations either of conservative or dissipative type. It relies on the minimization of a specific parameter-dependent family of functionals (WIDE functionals) with minimizers characterizing entire trajectories of the system. We prove that minimizers of the WIDE functional converge, up to subsequences, to weak solutions of the limiting PDE when the parameter tends to zero. The interest for this perspective is that of moving the successful machinery of the Calculus of Variations.

Ce manuscrit de thèse porte sur l'analyse numérique de schémas volumes finis pour la discrétisation de deux systèmes particuliers d'équations. Dans un premier temps nous étudions l'équation de Cahn-Hilliard associée à des conditions aux limites dynamiques dont l'une des principales difficultés est que cette condition aux limites est une équation parabolique, non linéaire, posée sur le bord et couplée avec l'intérieur du domaine. Nous proposons une discrétisation de type volumes finis en espace qui permet de coupler naturellement l'équation dans le domaine et celle sur sa frontière par un terme de flux et qui s'adapte facilement à la géométrie courbe du domaine. Nous montrons l'existence et la convergence des solutions discrètes vers une solution faible du système. Dans un second temps nous étudions la stabilité Inf-Sup du problème de Stokes pour un schéma volumes finis de type dualité discrète (DDFV). Nous donnons une analyse complète de la stabilité Inf-Sup inconditionnelle dans certains cas et de la stabilité de codimension 1 dans le cas de maillages cartésiens. Nous mettons également en place une méthode numérique permettant de calculer la constante Inf-Sup associée à ce schéma pour un maillage donné. On peut ainsi observer le comportement stable ou instable selon les cas en fonction de la géométrie des maillages. Dans une dernière partie nous proposons un schéma DDFV pour un modèle couplé Cahn-Hilliard/Stokes ce qui nécessite l'introduction de nouveaux opérateurs discrets. Nous démontrons la décroissance de l'énergie au niveau discret ainsi que l'existence d'une solution au problème discret. L'ensemble de ces travaux est validé par de nombreux résultats numériques. / This manuscript is devoted to the numerical analysis of finite-volume schemes for the discretization of two particular equations. First, we study the Cahn-Hilliard equation with dynamic boundary conditions whose one of the main difficulties is that this boundary condition is a non-linear parabolic equation on the boundary coupled with the interior of the domain. We propose a spatial finite-volume discretization which is well adapted to the coupling of the dynamics in the domain and those on the boundary by the flux term. Moreover this kind of scheme accounts naturally for the non-flat geometry of the boundary. We prove the existence and the convergence of the discrete solutions towards a weak solution of the system. Second, we study the Inf-Sup stability of the discrete duality finite volume (DDFV) scheme for the Stokes problem. We give a complete analysis of the unconditional Inf-Sup stability in some cases and of codimension 1 Inf-Sup stability for Cartesian meshes. We also implement a numerical method which allows us to compute the Inf-Sup constant associated with this scheme for a given mesh. Thus, we can observe the stable or unstable behaviour that can occur depending on the geometry of the meshes. In a last part we propose a DDFV scheme for a Cahn-Hilliard/Stokes phase field model that required the introduction of new discrete operators. We prove the dissipation of the energy in the discrete case and the existence of a solution to the discrete problem. All these research results are validated by extensive numerical results.

Transient vehicle thermal management simulations have the potential to be an important tool to ensure long component lifetimes in heavy-duty vehicles, as well as save development costs by reducing development time. Time-resolved computational fluid dynamics simulations of complete vehicles are however typically very computationally expensive, and approximation methods must be employed to keep computational costs and turn-around times at a reasonable level. In this thesis, two transient methods are used to simulate two important time-dependent scenarios for complete vehicles; hot shutdowns and long dynamic drive cycles. An approach using a time scaling between fluid solver and thermal solver is evaluated for a short drive cycle and heat soak. A quasi-transient method, utilizing limited steady-state computational fluid dynamics data repeatedly, is used for a long drive cycle. The simulation results are validated and compared with measurements from a climatic wind tunnel. The results indicate that the time-scaling approach is appropriate when boundary conditions are not changing rapidly. Heat-soak simulations show reasonable agreement between three cases with different thermal scale factors. The quasi-transient simulations suggest that complete vehicle simulations for durations of more than one hour are feasible. The quasi-transient results partly agree with measurements, although more component temperature measurements are required to fully validate the method.