Unbounded domains often appear in engineering applications, such as acoustic or elastic wave radiation from a body immersed in an infinite medium. To simulate the unboundedness of the domain special boundary conditions have to be imposed: the Sommerfeld radiation condition. In the present work we focused on steady-state wave propagation. The objective of this research is to obtain accurate prediction of phenomena occurring in exterior acoustics and elastodynamics and ensure the quality of the solutions even for high wavenumbers. To achieve this aim, we develop higher-order domain-based schemes: Spectral Element Method (SEM) coupled to Dirichlet-to-Neumann (DtN ), Perfectly Matched Layer (PML) and Infinite Element (IEM) methods. Spectral elements combine the rapid convergence rates of spectral methods with the geometric flexibility of the classical finite element methods. The interpolation is based on Chebyshev and Legendre polynomials. This work presents an implementation of these techniques and their validation exploiting some benchmark problems. A detailed comparison between the DtN, PML and IEM is made in terms of accuracy and convergence, conditioning and computational cost.

Made available in DSpace on 2015-05-15T11:46:24Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 947145 bytes, checksum: 7294f81daf663930a60afca1aecbee7b (MD5) Previous issue date: 2010-07-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we talk about properties of the Dirichlet-to-Neumann map for the conductivity equation in a smooth manifold with boundary of R2. We use several times the Maximum Principle to conclude a Alternating Property of the Dirichlet-to- Neumann map. Using this property, we and that the Kernel satises a given set of inequalities. Finally, we note that these inequalities imply the Alternating Property of the Kernel of the Dirichlet-to-Neumann map. / Neste trabalho dissertamos sobre Propriedades do Funcional de Dirichlet-Neumann para uma equação de condutividade numa variedade diferenciavel bidimensional com bordo. Utilizamos varias vezes o Principio do Maximo para concluir que esse Funcional tem uma Propriedade Alternada. A partir dessa propriedade, verificamos que o Nucleo do Funcional satisfaz um conjunto especifico de desigualdades. Porém, verificamos que essas desigualdades implicam na Propriedade Alternada do Nucleo do Funcional.

This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation. Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of two such solutions up to the boundary. Paper 2 concerns the solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a Muckenhoupt weight of class 1+2/n. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the parabolic measure, and the Hölder continuity of the quotient of two such solutions up to the boundary. Paper 3 concerns a fractional heat equation. The first main result is that a solution to the fractional heat equation in Euclidean space of dimension n can be extended as a solution to a certain linear degenerate parabolic equation in the upper half space of dimension n+1. The second main result is the Hölder continuity of quotients of two non-negative solutions that vanish continuously on the latteral boundary of a Lipschitz domain. Paper 4 concerns the solutions to uniformly parabolic linear equations with complex coefficients. The first main result is that under certain assumptions on the opperator the bounds for the single layer potentials associated to the opperator are bounded. The second main result is that these bounds always hold if the opperator is realvalued and symmetric.

This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$. The connection is to be determined up to a unitary gauge equivalence equal to the identity at the boundary. In our first approach to the problem, we restrict our attention to conformally transversally anisotropic (cylindrical) manifolds $M \Subset \mathbb{R}\times M_0$. Our strategy can be described as follows: we construct the special Complex Geometric Optics solutions oscillating in the vertical direction, that concentrate near geodesics and use their density in an integral identity to reduce the problem to a suitable $X$-ray transform on $M_0$. The construction is based on our proof of existence of Gaussian Beams on $M_0$, which are a family of smooth approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau \in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure along geodesics when $\tau \to \infty$, whereas the small remainder (that makes the solution exact) can be shown to exist by using suitable Carleman estimates. In the case $m = 1$, we prove the recovery of the connection given the injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For $m > 1$ and $M_0$ simple we reduce the problem to a certain two dimensional $\textit{new non-abelian ray transform}$. In our second approach, we assume that the connection $A$ is a $\textit{Yang-Mills connection}$ and no additional assumption on $M$. We construct a global gauge for $A$ (possibly singular at some points) that ties well with the DN map and in which the Yang-Mills equations become elliptic. By using the unique continuation property for elliptic systems and the fact that the singular set is suitably small, we are able to propagate the gauges globally. For the case $m = 1$ we are able to reconstruct the connection, whereas for $m > 1$ we are forced to make the technical assumption that $(M, g)$ is analytic in order to prove the recovery. Finally, in both approaches we are using the vital fact that is proved in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$ acting on sections of $E|_{\partial M}$, whose full symbol determines the full Taylor expansion of $A$ at the boundary.

Ce mémoire contient deux articles que j’ai rédigés au cours de ma maîtrise. Le premier chapitre sert d’introduction à ces articles. Plusieurs concepts de géométrie spectrale y sont présentés dans le contexte du problème de Steklov, en plus des résultats principaux des chapitres subséquents. Le second chapitre porte sur le problème de Steklov paramétrique sur des surfaces lisses. Un développement asymptotique complet des valeurs propres du problème est obtenu à l’aide de méthodes pseudodifférentielles. Celui-ci généralise l’asymptotique spectrale déjà connue du problème de Steklov classique. Nous en déduisons de nouveaux invariants géométriques déterminés par le spectre. Le troisième chapitre porte sur le problème de ballottement sur des prismes à base triangulaire. Le but est de comprendre comment les angles du prisme affectent le deuxième terme du développement asymptotique de la fonction de compte des valeurs propres. En construisant des quasimodes, nous obtenons une expression de ce terme que nous conjecturons comme étant la bonne pour les vraies valeurs propres. Cette conjecture est alors supportée par des expériences numériques. / This thesis contains two articles that I wrote during my M.Sc. studies. The first chapter serves as an introduction to both articles. Some concepts of spectral geometry in the context of the Steklov problem are presented, as well as the main results of the subsequent chapters. The second chapter concerns the parametric Steklov problem on smooth surfaces. We obtain a complete asymptotic expansion of the eigenvalues of the problem by using pseudodifferential techniques. This generalizes the already known spectral asymptotics of the classical Steklov problem. We deduce new geometric invariants determined by the spectrum. The third chapter concerns the sloshing problem on triangular prisms. The goal is to understand how the angles in the prism affect the second term in the asymptotic expansion of the eigenvalue counting function. By constructing quasimodes, we obtain an expression for this term that we conjecture as being correct for the true eigenvalues. This conjecture is then supported by numerical experiments.

Die Arbeit untersucht einen überraschenden Zusammenhang zwischen Halbflüssen von holomorphen Selbstabbildungen auf einfach zusammenhängenden Gebieten und Halbgruppen, die von Poincaré-Steklov Operatoren erzeugt werden. Mithilfe von Erzeuger von Kompositionshalbgruppen auf Banachräumen von analytischen Funktionen werden insbesondere Dirichlet-zu-Neumann und Dirichlet-zu-Robin Operatoren konstruiert. Dieser Zugang eröffnet einen neuen Ansatz für das Studium partiellen Differentialgleichungen, die mit solchen Operatoren assoziiert sind. / We study a surprising connection between semiflows of holomorphic selfmaps of a simply connected domain and semigroups generated by Poincaré-Steklov operators. In particular, by means of generators of semigroups of composition operators on Banach spaces of analytic functions, we construct Dirichlet-to-Neumann and Dirichlet-to-Robin operators. This approach gives new insights to the theory of partial differential equations associated with such operators.