In this thesis, we investigate the regularity and approximation of a hyperbolic-elliptic coupled problem. In particular, we consider the Poisson and the transport equation where both are assigned nonhomogeneous Dirichlet boundary conditions. The coupling of the two problems is executed as follows. The right hand side function of the Poisson equation is the solution ρ of the transport equation whereas the gradient field E = −∇u, with u being solution of the Poisson problem, is the convective field for the transport equation. The analysis is done throughout on a nonconvex, not simply connected domain that is supposed to be homeomorph to an annular domain. In the first part of this thesis, we will focus on the existence and uniqueness of a classical solution to this highly nonlinear problem using the framework of Hölder continuous functions. Herein, we distinguish between a time dependent and time independent formulation. In both cases, we investigate the streamline functions defined by the convective field E. These are used in the time dependent case to derive an operator equation whose fixed point is the streamline function to the gradient of the classical solution u. In the time independent setting, we formulate explicitly the solution operators L for the Poisson and T for the transport equation and show with a fixed point argument the existence and uniqueness of a classical solution (u,ρ). The second part of this thesis deals with the approximation of the coupled problem in Sobolev spaces. First, we show that the nonlinear transport equation can be formulated equivalently as variational inequality and analyse its Galerkin finite element discretization. Due to the nonlinearity of the coupled problem, it is necessary to use iterative solvers. We will introduce the staggered algorithm which is an iterative method solving alternating the Poisson and transport equation until convergence is obtained. Assuming that LοT is a contraction in the Sobolev space H1(Ω), we will investigate the convergence of the discrete staggered algorithm and obtain an error estimate. Subsequently, we present numerical results in two and three dimensions. Beside the staggered algorithm, we will introduce other iterative solvers that are based on linearizing the coupled problem by Newton’s method. We illustrate that all iterative solvers converge satisfactorily to the solution (u, ρ).

We present a method for Poisson’s equation that computes guaranteed upper and lower bounds for the values of linear functional outputs of the exact weak solution of the infinite dimensional continuum problem using traditional finite element approximations. The guarantee holds uniformly for any level of refinement, not just in the asymptotic limit of refinement. Given a finite element solution and its output adjoint solution, the method can be used to provide a certificate of precision for the output with an asymptotic complexity which is linear in the number of elements in the finite element discretization. / Singapore-MIT Alliance (SMA)

A new multiresolution scheme based on interpolating scaling function(ISF) on adaptive gridding(AG) shows promising in the first principle calculation. We also use ISFs on solving Poisson equation(PE), and find good approximations on the expansions of the second derivatives of ISFs. It is simpler than the wavelet scheme and fully implements the fast wavelet transformation so that the method is very suitable to problems with frequently updating charge density such as the first-principle calculation in electronic structures in atoms, molecules, and solids. Although the scheme is similar to the AG scheme on real space, the ISFs can represent fields more effectively and it needs less grids than the scheme of real space does. This simple and effective method provides an alternative to both the real space and the wavelet methods in the first principle calculation. Also, The method can be easily parallelized due to the block structure of the grid layout.

No description available.

Problemas de interface do tipo elípticos são frequentemente encontrados em dinâmicas de fluidos, ciências dos materiais, mecânica e outros campos de estudo. Em particular, o clássico Método de Interface Imersa (IIM) figura como uma das abordagens numéricas mais robustas para resolver problemas dessa categoria, o qual tem sido empregado recorrentemente para simular o comportamento de fluxos sobre corpos imersos em malhas cartesianas. Embora esse método seja eficiente e robusto, técnicas construídas com base no IIM impõem como restrições matemáticas diversos tipos de condições de salto na interface a fim de serem passíveis de utilização na prática. Nesta tese, introduzimos um novo método de Interface Imersa para resolver problemas elípticos com coeficientes descontínuos em malhas cartesianas. Diferentemente da maioria das formulações existentes que dependem de vários tipos de condições de salto para produzirem uma solução para o problema elíptico, o esquema aqui proposto reduz significativamente o número de restrições ao solucionar a EDP estudada, isto é, apenas os saltos de ordem zero das incógnitas devem ser fornecidos. A técnica apresentada combina esquemas de Diferenças Finitas, abordagem do Ponto Fantasma, modelos de correções e regras de interpolação em uma metodologia única e concisa. Além disso, o método proposto é capaz de produzir soluções de alta ordem, incluindo cenários onde há poucos dados disponíveis onde o quesito alta precisão é indispensável. A robustez e a precisão do método proposto são verificadas através de uma variedade de experimentos numéricos envolvendo diversos problemas elípticos com interfaces arbitrárias. Finalmente, a partir dos testes numéricos conduzidos, é possível concluir que o método projetado produz aproximações de alta ordem a partir de um número muito condensado de restrições matemáticas. / Elliptic interface problems are often encountered in fluid dynamics, material sciences, mechanics and other relevant fields of study. In particular, the well-known Immersed Interface Method (IIM) figures among the most effective approaches for solving non-trivial problems, where the method is traditionally used to simulate the flow behavior over complex bodies immersed in a cartesian mesh. Although their powerfulness and versatility, techniques that are built in light of the IIM impose as constraints different types of jump conditions at the interface in order to be properly managed and applicable for specific purposes. In this thesis, we introduce a novel Immersed Interface Method for solving Elliptic problems with discontinuous coefficients on cartesian grids. Different from most existing formulations that rely on various jump conditions types to get a valid solution, the present scheme reduces significatively the number of constraints when solving the PDE problem, i.e., only the ordinary jumps of the unknowns are required to be given, a priori. Our technique combines Finite Difference schemes, Ghost node strategy, correction models, and interpolation rules into a unified and concise methodology. Moreover, the method is capable of producing high-order solutions, succeeding in many practical scenarios with little available data wherein high precision is indispensable. We attest the robustness and the accuracy of the proposed method through a variety of numerical experiments involving several Elliptic problems with arbitrary interfaces. Finally, from the conducted numerical tests, we verify that the designed method produces high-order approximations from a very limited number of valid jump constraints.

O objetivo deste trabalho é o estudo de métodos multigrid para a solução de equações elípticas na esfera, discretizadas em malhas esféricas geodésicas icosaédricas. Malhas esféricas geradas a partir de sólidos platônicos receberam crescente atenção ao longo da última década, por serem razoavelmente uniformes e não apresentarem concentração de pontos em torno dos pólos como as tradicionais malhas latitude-longitude. Em especial, as malhas geodésicas icosaédricas (geradas a partir de um icosaedro inscrito na esfera com suas faces projetadas na superfície) têm sido adotadas no desenvolvimento de diversos modelos atmosféricos. Nestes é comum a necessidade de resolução de equações do tipo Poisson como parte do método de integração, motivando o nosso trabalho. Adotamos uma discretização do operador de Laplace baseada em volumes finitos. Para tal escrevemos o laplaciano como o divergente do gradiente. O divergente é discretizado com base nos fluxos nos pontos médios das arestas das células computacionais (com o auxílio do teorema da divergência de Gauss) e no uso de diferenças centradas para aproximar as derivadas nesses pontos médios. Validamos a discretização para o operador de Laplace resolvendo uma equação de Poisson através dos métodos iterativos de Jacobi e Gauss-Seidel. Estes sabidamente não são eficientes computacionalmente, devido ao grande e crescente número de iterações necessárias para atingir a convergência ao refinar a malha. Uma alternativa muito eficiente para a resolução de equações elípticas é a métodologia multigrid. Investigamos alguns métodos multigrid propostos na literatura para a solução destas equações na malha esférica geodésica icosaédrica. A partir desse estudo, utilizando também como referência a Análise Local de Fourier para a equação de Poisson em malhas hexagonais uniformes, como uma aproximação para malhas geodésicas icosaédricas, escolhemos um algoritmo multigrid para implementação. Testamos algumas opções para as componentes do esquema multigrid. Obtivemos taxas de convergência muito boas com V(1,1) ciclos com relaxação por Gauss-Seidel, restrição full weighting e interpolação linear. / This work is dedicated to the numerical solution of elliptic equations on the sphere, discretized on geodesic icosahedral grids. Spherical meshes generated from projections of platonic solids received considerable attention in the last decade, once they are almost isotropic and do not present a concentration of grid points around the poles, as traditional latitude-longitude grids. In particular, the geodesic icosahedral spherical grids have been adopted in the development of several atmospheric models. In these models, the necessity to solve Poisson type equations is very common, providing a motivation for our present work. We have employed a discretization of the Laplace operator based on finite volumes. We write the Laplacian as the divergent of the gradient operator and use Gauss theorem to derive the discretization of the operator. We integrate the fluxes along the cell borders and approximate them through finite-differences. We first validated the discretization solving Poisson\'s equation with a simple (and very innefficient) Jacobi-Relaxation and Gauss-Seidel. We then investigated the use of multigrid type schemes for the solution of this equation. We have analysed some schemes proposed in the literature, also using an idealized Local Fourier Analysis on hexagonal (planar) grids to estimate the behaviour of the schemes on the icosaedral grids. We have implemented and tested a multigrid method, comparing the performance with different relaxation schemes and transfer operators. We have obtained a very efficient method employing V(1,1) cycles with Gauss-Seidel relaxation, and full-weighting and linear interpolation as transfer-operators.

O objetivo deste trabalho é o estudo de métodos multigrid para a solução de equações elípticas na esfera, discretizadas em malhas esféricas geodésicas icosaédricas. Malhas esféricas geradas a partir de sólidos platônicos receberam crescente atenção ao longo da última década, por serem razoavelmente uniformes e não apresentarem concentração de pontos em torno dos pólos como as tradicionais malhas latitude-longitude. Em especial, as malhas geodésicas icosaédricas (geradas a partir de um icosaedro inscrito na esfera com suas faces projetadas na superfície) têm sido adotadas no desenvolvimento de diversos modelos atmosféricos. Nestes é comum a necessidade de resolução de equações do tipo Poisson como parte do método de integração, motivando o nosso trabalho. Adotamos uma discretização do operador de Laplace baseada em volumes finitos. Para tal escrevemos o laplaciano como o divergente do gradiente. O divergente é discretizado com base nos fluxos nos pontos médios das arestas das células computacionais (com o auxílio do teorema da divergência de Gauss) e no uso de diferenças centradas para aproximar as derivadas nesses pontos médios. Validamos a discretização para o operador de Laplace resolvendo uma equação de Poisson através dos métodos iterativos de Jacobi e Gauss-Seidel. Estes sabidamente não são eficientes computacionalmente, devido ao grande e crescente número de iterações necessárias para atingir a convergência ao refinar a malha. Uma alternativa muito eficiente para a resolução de equações elípticas é a métodologia multigrid. Investigamos alguns métodos multigrid propostos na literatura para a solução destas equações na malha esférica geodésica icosaédrica. A partir desse estudo, utilizando também como referência a Análise Local de Fourier para a equação de Poisson em malhas hexagonais uniformes, como uma aproximação para malhas geodésicas icosaédricas, escolhemos um algoritmo multigrid para implementação. Testamos algumas opções para as componentes do esquema multigrid. Obtivemos taxas de convergência muito boas com V(1,1) ciclos com relaxação por Gauss-Seidel, restrição full weighting e interpolação linear. / This work is dedicated to the numerical solution of elliptic equations on the sphere, discretized on geodesic icosahedral grids. Spherical meshes generated from projections of platonic solids received considerable attention in the last decade, once they are almost isotropic and do not present a concentration of grid points around the poles, as traditional latitude-longitude grids. In particular, the geodesic icosahedral spherical grids have been adopted in the development of several atmospheric models. In these models, the necessity to solve Poisson type equations is very common, providing a motivation for our present work. We have employed a discretization of the Laplace operator based on finite volumes. We write the Laplacian as the divergent of the gradient operator and use Gauss theorem to derive the discretization of the operator. We integrate the fluxes along the cell borders and approximate them through finite-differences. We first validated the discretization solving Poisson\'s equation with a simple (and very innefficient) Jacobi-Relaxation and Gauss-Seidel. We then investigated the use of multigrid type schemes for the solution of this equation. We have analysed some schemes proposed in the literature, also using an idealized Local Fourier Analysis on hexagonal (planar) grids to estimate the behaviour of the schemes on the icosaedral grids. We have implemented and tested a multigrid method, comparing the performance with different relaxation schemes and transfer operators. We have obtained a very efficient method employing V(1,1) cycles with Gauss-Seidel relaxation, and full-weighting and linear interpolation as transfer-operators.

In this thesis, the applications of image stitching and object insertion are considered and two gradient based approaches offering solutions are proposed. An essential part of the proposed methods is obtaining an image from a given gradient data set. This is done using an existing Haar wavelet based reconstruction technique, which consists of two main steps. First, the Haar wavelet decomposition of the image to be reconstructed is obtained directly from a given gradient. Second, the image is obtained using Haar wavelet synthesis. In both stitching and object insertion applications considered, the gradient from which the image must be reconstructed is a non-conservative vector field and this requires adding an iterative Poisson solver at each resolution level, during the synthesis step of the reconstruction technique. The performance of the reconstruction algorithm is evaluated by comparing it with other existing techniques, in terms of solution accuracy and computation speed. The proposed image stitching technique consists of three main parts: registering the images to be combined, blending their gradients over a region of interest and obtaining a composite image from a gradient. The object insertion technique considers the images registered and has two main stages: gradient blending of images in a region of interest and recovering an image from the gradient. The performance of the stitching algorithm is evaluated visually, by presenting the results produced to combine images with varying orientation, scales, illumination, and color conditions. Experimental results illustrate both the stitching and the insertion techniques proposed, and indicate that they yield seamless composite images. / Graduate

This thesis identifies the obstacles in efficient modelling of interaction of plasma and solid surfaces. It presents an enhanced method of solving the Poisson equation optimized to meet the requirements of the Particle-in-Cell modelling approach. A number of applications are discussed, including models of the plasma-facing wall of the nuclear fusion device, tokamak, and its interaction with particle fluxes driven by the intrinsic magnetic field. Another area of applications covers the modelling of plasma probes deployed to diagnose properties of plasma in various experiments. The thesis also includes the computer library code and instructions enabling a rapid use of the Poisson solver method in a third party computer code which implements the PIC approach in a compatible manner.

Le développement d’une couche limite turbulente sur des structures entraîne des vibrations et des nuisances sonores. Celles-ci sont estimées par des calculs vibro-acoustiques qui nécessitent le spectre de pression pariétale turbulente en fréquence-nombre d’onde. Ce spectre est généralement calculé via des modèles empiriques. Or ces modèles ont un domaine de validité très restreint et ne sont pas adaptés pour des écoulements complexes, avec notamment des gradients de pression. Dans ces travaux, une méthode est proposée pour calculer les corrélations spatio-temporelles de pression pariétale à partir d’une solution sous forme intégrale de l’équation de Poisson. Le spectre de pression est obtenu à partir de la transformation de Fourier de ces corrélations. L’expression retenue pour ces dernières fait intervenir les dérivées d’une fonction de Green ainsi que les champs de la vitesse moyenne et des tensions de Reynolds qui sont obtenus par simulation RANS. Elle fait aussi intervenir des coefficients de corrélation de vitesse spatio-temporelle qui doivent être modélisés. Pour cela, un nouveau modèle de coefficient de corrélation spatiale a été développé : l’Extended Anisotropic Model. Le calcul des corrélations et du spectre de pression est réalisé en utilisant une méthode numérique basée sur une stratégie d’échantillonnage adaptatif combinée à du krigeage. Elle permet de réduire le nombre de valeurs de corrélation de pression nécessaires pour obtenir le spectre de pression pariétale et donc de réduire le temps de calcul. La méthode est appliquée à des écoulements de couche limite turbulente sur une plaque plane et sur un profil NACA-0012 avec un gradient de pression adverse. / Turbulent boundary layer flows over structures induce vibrations and noise. The latter are estimated by vibro-acoustic studies which require the wavenumber-frequency turbulent wall-pressure spectrum. This spectrum is generally computed via empirical models. However, these models have a very narrow domain of validity and are not adapted for complex flows, in particular with pressure gradients. In this work, a method is proposed to compute space-time wall-pressure correlations from an integral solution of the Poisson equation. The pressure spectrum is obtained by the Fourier transform of these correlations. The expression retained for the pressure correlations involves the derivatives of a Green function as well as the mean velocity field and the Reynolds stresses which are obtained by RANS solutions. It also involves space-time velocity correlation coefficients that have to be modelled. To achieve this, a new model was developed for the spatial correlation coefficients: the Extended Anisotropic Model. To compute the wall-pressure correlations and spectrum, a numerical method based on a self adaptive sampling strategy combined with Kriging is used. It reduces the number of pressure correlation values required to compute the wall-pressure spectrum and thus reduces the computation time. The method is applied to turbulent boundary layer flows over a flat plate and over a NACA-0012 profile with an adverse pressure gradient.