This thesis studies the approximation of solutions to partial differential equations (PDEs) over domains discretized by the dual of a simplicial mesh. While `primal' methods associate degrees of freedom (DoFs) of the solution with specific geometrical entities of a simplicial mesh (simplex vertices, edges, faces, etc.), a `dual discretization method' associates DoFs with the geometric duals of these objects. In a tetrahedral mesh, for instance, a primal method might assign DoFs to edges of tetrahedra while a dual method for the same problem would assign DoFs to edges connecting circumcenters of adjacent tetrahedra. Dual discretization methods have been proposed for various specific PDE problems, especially in the context of electromagnetics, but have not been analyzed using the full toolkit of modern numerical analysis as is considered here. The recent and still-developing theories of finite element exterior calculus (FEEC) and discrete exterior calculus (DEC) are shown to be essential in understanding the feasibility of dual methods. These theories treat the solutions of continuous PDEs as differential forms which are then discretized as cochains (vectors of DoFs) over a mesh. While the language of DEC is ideal for describing dual methods in a straightforward fashion, the results of FEEC are required for proving convergence results. Our results about dual methods are focused on two types of stability associated with PDE solvers: discretization and numerical. Discretization stability analyzes the convergence of the approximate solution from the discrete method to the continuous solution of the PDE as the maximum size of a mesh element goes to zero. Numerical stability analyzes the potential roundoff errors accrued when computing an approximate solution. We show that dual methods can attain the same approximation power with regard to discretization stability as primal methods and may, in some circumstances, offer improved numerical stability properties. A lengthier exposition of the approach and a detailed description of our results is given in the first chapter of the thesis. / text

Yes / Exterior calculus with its three operations meet, join and hodge star complement, is used for the representation of fermion-hole systems and for fermionic analogues of logical gates. Two different schemes that implement fermionic quantum computation, are proposed. The first scheme compares fermionic gates with Boolean gates, and leads to novel electronic devices that simulate fermionic gates. The second scheme uses a well known map between fermionic and multi-qubit systems, to simulate fermionic gates within multi-qubit systems.

In dieser Arbeit geben wir eine Einführung in das Diskrete Äußere Kalkül (engl.: Discrete Exterior Calculus, kurz: DEC), das sich mit der Diskretisierung von Differentialformen und -operatoren beschäftigt. Wir beschränken uns hierbei auf zweidimensionalen orientierten kompakten Riemannschen Mannigfaltigkeiten und zeigen auf, wie diese als wohlzentrierte Simplizialkomplexe zu approximieren sind. Dabei beschreiben wir die Implementierung der Methode und testen diese an Beispielen, wie Helmholtz-artige PDEs und die Berechnung von in- und extrinsischen Krümmungsgrößen.

In dieser Arbeit geben wir eine Einführung in das Diskrete Äußere Kalkül (engl.: Discrete Exterior Calculus, kurz: DEC), das sich mit der Diskretisierung von Differentialformen und -operatoren beschäftigt. Wir beschränken uns hierbei auf zweidimensionalen orientierten kompakten Riemannschen Mannigfaltigkeiten und zeigen auf, wie diese als wohlzentrierte Simplizialkomplexe zu approximieren sind. Dabei beschreiben wir die Implementierung der Methode und testen diese an Beispielen, wie Helmholtz-artige PDEs und die Berechnung von in- und extrinsischen Krümmungsgrößen.:0 Einführung 1 Diskrete Mannigfaltigkeiten 1.1 Primär- und Dualgitter 1.2 Kettenkomplexe 1.3 Gittergenerierung für Oberflächen 1.4 Implizit gegebene Oberflächen 2 Diskretes Äußeres Kalkül (DEC) 2.1 Diskrete Differentialformen 2.2 Äußere Ableitung 2.3 Hodge-Stern-Operator 2.4 Laplace-Operator 2.5 Primär-Dual-Gradient im Mittel 3 Anwendung: Oberflächenkrümmung 3.1 Weingartenabbildung 3.2 Krümmungsvektor 3.3 Gauß-Bonnet-Operator 3.4 Numerisches Experiment 4 Fazit und Ausblicke 5 Appendix 5.1 Häufige Bezeichner 5.2 Algorithmen 5.3 Krümmungen für impliziten Oberflächen 5.4 Ausgewählte Oberflächen Literaturverzeichnis

<p dir="ltr">A novel computational electromagnetic (CEM) solver using potential-based formulations and discrete exterior calculus (DEC) is proposed. The proposed solver consists of two parts: the DEC A-Phi solver and the DEC F-Psi solver. A and Phi are the magnetic vector potential and electric scalar potential of the electromagnetic (EM) field, respectively; F and Psi are the electric vector potential and magnetic scalar potential, respectively. The two solvers are dual to each other, and most research is carried out with respect to the DEC A-Phi solver.</p><p dir="ltr">Systematical approach for constructing the DEC A-Phi matrix equations is provided in this thesis, including the construction of incidence matrices, Hodge star operators and different boundary conditions. The DEC A-Phi solver is proved to be broadband stable from DC to optics, while classical CEM solvers suffer from stability issues at low frequencies (also known as the low-frequency breakdown). The proposed solver is ideal for broadband and multi-scale analysis, which is of great importance in modern industry.</p><p dir="ltr">To empower the proposed solver with the ability to solve industry problems with large number of unknowns, iterative solvers are preferred. The error-minimization mechanism buried in iterative solvers allows user to control the effect of numerical error accumulation to the solution vector. Proper preconditioners are almost always needed to accelerate the convergence of iterative solvers in large scale problems. In this thesis, preconditioning schemes for the proposed solver are studied.</p><p dir="ltr">In the DEC A-Phi solver, current sources can be applied easily, but it is difficult to implement voltage sources. To incorporate voltage sources in the potential-based solver, the DEC F-Psi solver is proposed. The DEC A-Phi and F-Psi solvers are dual formulations to each other, and the construction of the F-Psi solver can be generalized from the A-Phi solver straightforward.</p>

Les bases de fonctions sont des outils indispensables de la géométrie numérique puisqu'ils permettent de représenter des fonctions comme des vecteurs, c'est à dire d'appliquer les outils de l'algèbre linéaire à l'analyse fonctionnelle. Dans cette thèse, nous présentons plusieurs constructions de bases de fonctions sur des surfaces pour la géométrie numérique. Nous commençons par présenter les bases de fonctions usuelles des éléments finis et du calcul extérieur discret, leur théorie et leurs limites. Nous étudions ensuite le Laplacien et sa discrétisation, ce qui nous permettra de construire une base de fonctions particulière~: les fonctions propres de l'opérateur de Laplace-Beltrami, ou harmoniques variétés. Celles-ci permettent de généraliser la transformée de Fourier et le filtrage spectral aux fonctions définies sur des surfaces. Nous présentons ensuite des applications de cette base de fonction à la géométrie numérique. En particulier, nous montrons qu'une fois calculée, cette base de fonction permet de filtrer la géométrie en temps interactif. Pour pouvoir définir des bases de fonctions de façon plus indépendante du maillage de la surface, nous nous intéressons ensuite aux paramétrisations globales, et en particulier aux champs de directions à symétries qui permettent de les définir. Ainsi, dans la dernière partie, nous étudions ces champs de directions à symétries, et en particulier leur géométrie et leur topologie. Nous donnons alors des outils pour les construire, les manipuler et les visualiser / Function bases are fundamental objects in geometry processing as they allow to represent functions as vectors, that is to apply tools from linear algebra to functional analysis. In this thesis, we present various constructions of useful functions bases for geometry processing. We start by presenting usual function bases, their theory and limits. We then study the Laplacian operator and its discretization, and use it to define a particular function basis: Laplacian eigenfunctions or Manifold harmonics. The Manifold Hamonics form a function basis that allows to generalize the Fourier transform and spectral filtering on a surface. We present some applications and extensions of this basis for geometry processing. To define function bases in a mesh-independant manner, we need to build a global parameterization, and especially the direction fields required to define them. Thus, in the last part of this thesis we study N-symmetry direction fields on surfaces, and in particular their geometry and topology. We then give tools to build, edit, control and visualize them

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