In this thesis, we analyze theoretical properties of the discontinuous Galerkin method (DGM) and propose novel approaches to implementation with the aim to increase its efficiency. First, we derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p+1] Pade approximation of exp(-z) when the p-th degree basis functions are used. Then, we extend the analysis to nonuniform meshes where both the size of elements and the composition of the mesh influence the spectrum. We show that the spectrum depends on the ratio of the size of the largest to the smallest cell as well as the number of cells of different types. We find that the spectrum grows linearly as a function of the proportion of small cells present in the mesh when the size of small cells is greater than some critical value. When the smallest cells are smaller than this critical value, the corresponding eigenvalues lie outside of the main spectral curve. Numerical examples on nonuniform meshes are presented to show the improvement on the time step restriction. In particular, this result can be used to improve the time step restriction on Cartesian grids. Finally, we present a discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. Cutting an embedded geometry out of the Cartesian grid creates cut cells, which are difficult to deal with for two reasons. One is the restrictive CFL number and the other is the integration on irregularly shaped cells. We use explicit time integration employing cell merging to avoid restrictively small time steps. We provide an algorithm for splitting complex cells into triangles and use standard quadrature rules on these for numerical integration. To avoid the loss of accuracy due to straight sided grids, we employ the curvature boundary conditions. We show that the proposed method is robust and high-order accurate.

Discontinuous dynamical system defined by impulsive autonomous differential equation is a field that has actually been considered rarely. Also, the properties of such systems have not been discussed thoroughly in the course of mathematical researches so far. This thesis comprises two parts, elaborated with a number of examples. In the first part, some results of the previous studies on the classical dynamical system are exposed. In the second part, the definition of discontinuous dynamical system defined by impulsive autonomous differential equation is formulated, and its properties are investigated, in the view of the known results of the studies on the classical dynamical system and impulsive differential equations.

O principal objetivo desta dissertação é obter um Teorema do Envelope que permita mecanismos não diferenciáveis, preferências arbitrárias e que possa ser aplicado em modelos com múltiplos agentes. Nós alcançamos isto ao expandir a análise de Milgrom e Segal (2002), generalizando seus resultados para espaços de tipos multidimensionais. Dessa forma, continuamos permitindo que a regra de escolha (mecanismo) seja descontínua. Para obter nosso resultado, é necessário o uso do Teorema do Máximo de Berge e, consequentemente, devemos impor compacidade no conjunto de escolha. Inicialmente esta hipótese pode parecer forte, porém argumentamos que em aplicações _e muito improvável termos um conjunto de escolha aberto ou, principalmente, não limitado. Nós também identificamos condições para que a função valor seja absolutamente contínua e mostramos que sua representação integral também é válida para espaços de tipos multidimensionais. Inicialmente propomos uma generalização direta do resultado de Milgrom e Segal (2002), utilizando a hipótese de continuidade absoluta da função de utilidade do agente. Entretanto, esta exigência não possui muito significado econômico e é considerada pouco elegante por parte da literatura. Neste sentido, incorporamos uma hipótese adicional de diferenciabilidade da utilidade em todo o domínio que gera a mesma representação integral e possui uma maior interpretação econômica. Nossos resultados são, em geral, aplicados a modelos com múltiplos agentes, em especial Economia do Setor Público (provisão de bens públicos e taxação ótima) e teoria dos leilões. / The main objective of this dissertation is to obtain an Envelope Theorem that allows non-di erentiable mechanisms, arbitrary preferences, and that can be applied to models with multiple agents. We achieve that by expanding the analysis of Milgrom and Segal (2002) and generalizing their results to multidimensional type spaces. Thus, we continue allowing that the choice rule (mechanism) is discontinuous. For our result, it is necessary to use the Berge's Maximum Theorem and therefore we must impose compactness in the choice set. Initially this assumption may seem strong, but we argue that in applications there is an open or unbounded choice set is very unlikely. We also identify conditions for the value function is absolutely continuous and show that its integral representation is also valid for multidimensional type spaces. Firstly we propose a direct generalization of the Milgrom and Segal (2002)'s result, using the assumption of absolute continuity of the agent's utility function. However, this requirement does not have much economic interpretation and it is considered not very elegant in the literature. In this sense, we incorporate an additional assumption of di erentiability of the utility in all range that generates the same integral representation and it possesses a greater economic interpretation. Our results are generally applied to models with multiple agents, in particular Public Economics (public goods supply and optimal taxation) and auction theory.

A look into an implementation of turbulence model into the ADGFEM code for viscous flow. Discretization, theory background and development of the method will be carried during this thesis. Also some numerical examples of the application of the code will be provided. 1

Submitted by Ronan Felipe Jorge null (ronan.jorge@hotmail.com) on 2016-04-07T21:29:57Z No. of bitstreams: 1 RonanFelipeJorgeVersaoFinal.pdf: 3551394 bytes, checksum: 30b15e5d6519d49164a37bcb6b0946c8 (MD5) / Approved for entry into archive by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br) on 2016-04-08T13:07:08Z (GMT) No. of bitstreams: 1 jorge_rf_me_sjrp.pdf: 3551394 bytes, checksum: 30b15e5d6519d49164a37bcb6b0946c8 (MD5) / Made available in DSpace on 2016-04-08T13:07:08Z (GMT). No. of bitstreams: 1 jorge_rf_me_sjrp.pdf: 3551394 bytes, checksum: 30b15e5d6519d49164a37bcb6b0946c8 (MD5) Previous issue date: 2016-03-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O conceito de estabilidade estrutural foi introduzido aos estudos de sistemas dinâmicos contínuos por Andronov e Pontryagin (1937). Em 1988, após investigar osciladores com atrito de Coulomb e apresentar equações diferenciais com lado direito descontínuo, Filippov desenvolveu em sua obra uma nomenclatura para o estudo de sistemas dinâmicos descontínuos. Desde então, estudiosos da área vêm tentando analisar a estabilidade estrutural de sistemas dinâmicos descontínuos por diferentes métodos. Um dos métodos é transformar, sem alterar a estrutura dos campos de vetores, estes sistemas descontínuos em sistemas contínuos onde o estudo da estabilidade estrutural já é conhecido. Esta transformação, também conhecida como regularização, pode ser desenvolvida de diversas formas. Este trabalho tem por objetivo a apresentação de um método de regularização através do uso de uma função de transição de campos de vetores Z do plano com um conjunto de descontinuidade S baseando-se no método de regularização apresentado por Sotomayor e Teixeira (1996) em seu artigo, e realizar um breve estudo sobre estabilidade estrutural em campos de vetores regularizados utilizando tal método. / The concept of structural stability was introduced into the studies of continuous dynamical systems by Andronov and Pontryagin (1937). In 1988, after investigating oscillators with Coulomb friction and differential equations with discontinuous right-hand sides, Filippov developed a nomenclature to the study of discontinuous dynamical systems. Since then, researches have been trying to analyze the structural stability of discontinuous dynamical systems by different methods. One of these methods is to transform, without changing the structure of vector fields, these discontinuous systems in continuous systems where the study of structural stability is already known. This transformation, also known as regularization, can be developed in various ways. This work aims at presenting a regularization method using a transition function of vector fields Z in the plane with a discontinuous set S basing in regularization method presented by Sotomayor and Teixeira (1996) in their article, and, also to do a short study about structural stability in regularized vector fields using this method.

O principal objetivo desta dissertação é obter um Teorema do Envelope que permita mecanismos não diferenciáveis, preferências arbitrárias e que possa ser aplicado em modelos com múltiplos agentes. Nós alcançamos isto ao expandir a análise de Milgrom e Segal (2002), generalizando seus resultados para espaços de tipos multidimensionais. Dessa forma, continuamos permitindo que a regra de escolha (mecanismo) seja descontínua. Para obter nosso resultado, é necessário o uso do Teorema do Máximo de Berge e, consequentemente, devemos impor compacidade no conjunto de escolha. Inicialmente esta hipótese pode parecer forte, porém argumentamos que em aplicações _e muito improvável termos um conjunto de escolha aberto ou, principalmente, não limitado. Nós também identificamos condições para que a função valor seja absolutamente contínua e mostramos que sua representação integral também é válida para espaços de tipos multidimensionais. Inicialmente propomos uma generalização direta do resultado de Milgrom e Segal (2002), utilizando a hipótese de continuidade absoluta da função de utilidade do agente. Entretanto, esta exigência não possui muito significado econômico e é considerada pouco elegante por parte da literatura. Neste sentido, incorporamos uma hipótese adicional de diferenciabilidade da utilidade em todo o domínio que gera a mesma representação integral e possui uma maior interpretação econômica. Nossos resultados são, em geral, aplicados a modelos com múltiplos agentes, em especial Economia do Setor Público (provisão de bens públicos e taxação ótima) e teoria dos leilões. / The main objective of this dissertation is to obtain an Envelope Theorem that allows non-di erentiable mechanisms, arbitrary preferences, and that can be applied to models with multiple agents. We achieve that by expanding the analysis of Milgrom and Segal (2002) and generalizing their results to multidimensional type spaces. Thus, we continue allowing that the choice rule (mechanism) is discontinuous. For our result, it is necessary to use the Berge's Maximum Theorem and therefore we must impose compactness in the choice set. Initially this assumption may seem strong, but we argue that in applications there is an open or unbounded choice set is very unlikely. We also identify conditions for the value function is absolutely continuous and show that its integral representation is also valid for multidimensional type spaces. Firstly we propose a direct generalization of the Milgrom and Segal (2002)'s result, using the assumption of absolute continuity of the agent's utility function. However, this requirement does not have much economic interpretation and it is considered not very elegant in the literature. In this sense, we incorporate an additional assumption of di erentiability of the utility in all range that generates the same integral representation and it possesses a greater economic interpretation. Our results are generally applied to models with multiple agents, in particular Public Economics (public goods supply and optimal taxation) and auction theory.

abstract: This dissertation describes a process for interface capturing via an arbitrary-order, nearly quadrature free, discontinuous Galerkin (DG) scheme for the conservative level set method (Olsson et al., 2005, 2008). The DG numerical method is utilized to solve both advection and reinitialization, and executed on a refined level set grid (Herrmann, 2008) for effective use of processing power. Computation is executed in parallel utilizing both CPU and GPU architectures to make the method feasible at high order. Finally, a sparse data structure is implemented to take full advantage of parallelism on the GPU, where performance relies on well-managed memory operations. With solution variables projected into a kth order polynomial basis, a k+1 order convergence rate is found for both advection and reinitialization tests using the method of manufactured solutions. Other standard test cases, such as Zalesak's disk and deformation of columns and spheres in periodic vortices are also performed, showing several orders of magnitude improvement over traditional WENO level set methods. These tests also show the impact of reinitialization, which often increases shape and volume errors as a result of level set scalar trapping by normal vectors calculated from the local level set field. Accelerating advection via GPU hardware is found to provide a 30x speedup factor comparing a 2.0GHz Intel Xeon E5-2620 CPU in serial vs. a Nvidia Tesla K20 GPU, with speedup factors increasing with polynomial degree until shared memory is filled. A similar algorithm is implemented for reinitialization, which relies on heavier use of shared and global memory and as a result fills them more quickly and produces smaller speedups of 18x. / Dissertation/Thesis / Doctoral Dissertation Aerospace Engineering 2015

This thesis studies the numerical solution of non-linear convection-diffusion problems using the space- time discontinuous Galerkin method, which perfectly suits the space as well as time local adaptation. We aim to develop a posteriori error estimates reflecting the spatial, temporal, and algebraic errors. These estimates are based on the measurement of the residuals in dual norms. We derive these estimates and numerically verify their properties. Finally, we derive an adaptive algorithm and apply it to the numerical simulation of non-stationary viscous compressible flows. Powered by TCPDF (www.tcpdf.org)

As traditional communication technology we use in our day-to-day life reaches its limitations, the international community searches for new methods to communicate information. One such novel approach is the so-called molecular communication system. During the last few decades, molecular communication systems become more and more popular. The main difference between traditional communication and molecular communication systems is that in the latter, information transfer occurs through chemical means, most often between microorganisms. This process already happens all around us naturally, for example, in the human body. Even though the molecular communication topic is attractive to researchers, and a lot of theoretical results are available - one cannot claim the same about the practical use of molecular communication. As for experimental results, a few studies have been done on the macroscale, but investigations at the micro- and nanoscale ranges are still lacking because they are a challenging task. In this work, a self-contained introduction of the underlying theory of molecular communication is provided, which includes knowledge from different areas such as biology, chemistry, communication theory, and applied mathematics. Two numerical methods are implemented for three well-studied partial differential equations of the MC field where advection, diffusion, and the reaction are taken into account. Numerical results for test cases in one and three dimensions are presented and discussed in detail. Conclusions and essential analytical and numerical future directions are then drawn.

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