In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

In an anisotropic adaptive finite element algorithm one usually needs an error estimator that yields the error size but also the stretching directions and stretching ratios of the elements of a (quasi) optimal anisotropic mesh. However the last two ingredients can not be extracted from any of the known anisotropic a posteriori error estimators. Therefore a heuristic approach is pursued here, namely, the desired information is provided by the so-called Hessian strategy. This strategy produces favourable anisotropic meshes which result in a small discretization error. The focus of this paper is on error estimation on anisotropic meshes. It is known that such error estimation is reliable and efficient only if the anisotropic mesh is aligned with the anisotropic solution. The main result here is that the Hessian strategy produces anisotropic meshes that show the required alignment with the anisotropic solution. The corresponding inequalities are proven, and the underlying heuristic assumptions are given in a stringent yet general form. Hence the analysis provides further inside into a particular aspect of anisotropic error estimation.

In an anisotropic adaptive finite element algorithm one usually needs an error estimator that yields the error size but also the stretching directions and stretching ratios of the elements of a (quasi) optimal anisotropic mesh. However the last two ingredients can not be extracted from any of the known anisotropic a posteriori error estimators. Therefore a heuristic approach is pursued here, namely, the desired information is provided by the so-called Hessian strategy. This strategy produces favourable anisotropic meshes which result in a small discretization error. The focus of this paper is on error estimation on anisotropic meshes. It is known that such error estimation is reliable and efficient only if the anisotropic mesh is aligned with the anisotropic solution. The main result here is that the Hessian strategy produces anisotropic meshes that show the required alignment with the anisotropic solution. The corresponding inequalities are proven, and the underlying heuristic assumptions are given in a stringent yet general form. Hence the analysis provides further inside into a particular aspect of anisotropic error estimation.

Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. An estimator that has shown to be one of the most reliable for reaction-diffusion problem is the <i>equilibrated residual method</i> and its modification done by Ainsworth and Babuška for singularly perturbed problem. However, even the modified method is not robust in the case of anisotropic meshes. The present work modifies the equilibrated residual method for anisotropic meshes. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. A numerical example confirms the theory.

City busses are operated on roads where the GPS signal is weak, because of the tall buildings surrounding these roads. The localization of city busses, needs to therefore rely on alternate technique in order to improve the accuracy. Recent standardization of inter vehicular communication has made this a readily available tool which can be used for localization. This thesis presents an approach towards localization of a city bus by means of vehicular ad hoc network. The two main components of localization by this approach is the initialization of location estimate component, and the real time location estimation component. In particular, the thesis develops the use of minimum mean square estimation for initialization and an extended Kalman filtering approach for real time location estimation. The localization method is mathematically described, considering the operating scenarios of a city bus. The accuracy of the proposed method is mathematically evaluated. The developed localization method is implemented in a simulation tool kit for inter vehicular communication. Simulation experiments were performed for operating scenarios of city bus. The result of initialization by minimum mean square error is compared to that of initialization by GPS, in-terms of localization accuracy. Different setups of road side units are compared in-terms of accuracy and update interval. The results show that the proposed method is feasible for localization of a city bus. This thesis was carried out in association with Scania AB, Södertälje. / Stadsbussar åker på vägar som är omgivna av byggnader, vilket försämrar stadsbussarnas GPSmottagning. Lokaliseringen av stadsbussar måste därför förlita sig på alternativ teknik för att förbättra noggrannheten. Nyligen standardiserad kommunikation mellan fordon har blivit till ett lättillgängligt verktyg som kan användas för lokalisering. Den här uppsatsen presenterar en strategi för lokalisering av en stadsbuss med hjälp av fordonets ad hoc-nätverk. Huvudkomponenterna för lokalisering är en initialiseringskomponent och realtidslägesuppskattningskomponent. Speciellt utvecklar arbetet användningen av minsta medelkvadratberäkning för initialisering och en utvidgad kalmanfiltreringsmetod för realtidslägesuppskattning. Lokaliseringsmetoden beskrivs matematiskt med tanke på driftsscenarierna för en stadsbuss. Noggrannheten hos den föreslagna metoden utvärderas matematiskt. Den utvecklade lokaliseringsmetoden implementeras i ett simuleringsverktyg för kommunikation mellan fordon. Simuleringsexperiment utfördes för driftsscenarier för stadsbussar. Resultatet av initialisering med minsta medelkvadratberäkning jämförs med initialiseringen med GPS, i termer av lokaliseringsnoggrannhet. Olika inställningar av vägrensenheter jämförs med avseende på noggrannhet och uppdateringsintervall. Resultaten visar att den föreslagna metoden är möjlig för lokalisering av en stadsbuss. Denna arbetet genomfördes i samarbete med Scania AB, Södertälje.

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This dissertation mainly consists of three parts. The first part proposes generalized linear minimum mean-square error (GLMMSE) estimation for nonlinear point estimation. The second part proposes a recursive joint decision and estimation (RJDE) algorithm for joint decision and estimation (JDE). The third part analyzes the performance of sequential probability ratio test (SPRT) when the log-likelihood ratios (LLR) are independent but not identically distributed. The linear minimum mean-square error (LMMSE) estimation plays an important role in nonlinear estimation. It searches for the best estimator in the set of all estimators that are linear in the measurement. A GLMMSE estimation framework is proposed in this disser- tation. It employs a vector-valued measurement transform function (MTF) and finds the best estimator among all estimators that are linear in MTF. Several design guidelines for the MTF based on a numerical example were provided. A RJDE algorithm based on a generalized Bayes risk is proposed in this dissertation for dynamic JDE problems. It is computationally efficient for dynamic problems where data are made available sequentially. Further, since existing performance measures for estimation or decision are effective to evaluate JDE algorithms, a joint performance measure is proposed for JDE algorithms for dynamic problems. The RJDE algorithm is demonstrated by applications to joint tracking and classification as well as joint tracking and detection in target tracking. The characteristics and performance of SPRT are characterized by two important functions—operating characteristic (OC) and average sample number (ASN). These two functions have been studied extensively under the assumption of independent and identically distributed (i.i.d.) LLR, which is too stringent for many applications. This dissertation relaxes the requirement of identical distribution. Two inductive equations governing the OC and ASN are developed. Unfortunately, they have non-unique solutions in the general case. They do have unique solutions in two special cases: (a) the LLR sequence converges in distributions and (b) the LLR sequence has periodic distributions. Further, the analysis can be readily extended to evaluate the performance of the truncated SPRT and the cumulative sum test.

The thesis presents a comprehensive study of a posteriori error estimation in the adaptive solution to some classical elliptic partial differential equations. Several new error estimators are proposed for diffusion problems with discontinuous coefficients and for convection-reaction-diffusion problems with dominated convection/reaction. The robustness of the new estimators is justified theoretically. Extensive numerical results demonstrate the robustness of the new estimators for challenging problems and indicate that, compared to the well-known residual-type estimators, the new estimators are much more accurate.

Neste trabalho desenvolveu-se uma formulação adaptativa do método de elementos de contorno (MEC) para a análise de problemas de fratura elástica linear. Foi utilizado o método da colocação para a formulação das equações integrais de deslocamento e de tensão. Para a discretização das equações integrais foram utilizados elementos lineares que possibilitaram a obtenção das expressões exatas das integrais (integração analítica) sobre elementos de contorno e fratura. Para a montagem do sistema de equações algébricas foram utilizadas apenas equações de deslocamento, apenas equações de forças de superfície, ou as duas escritas para nós opostos da fratura levando, portanto ao método dos elementos de contorno dual usualmente empregado na análise de fratura. Para o processo de crescimento da trinca foi desenvolvido um procedimento especial objetivando a correta determinação da direção de crescimento da trinca. Os fatores de intensidade de tensão são calculados por meio da conhecida técnica de correlação de deslocamentos a qual relaciona os deslocamentos atuantes nas faces da fissura. Após a determinação dos fatores de intensidade de tensão é utilizada a teoria da máxima tensão circunferencial para a determinação do ângulo de propagação. O modelo adaptativo empregado é do tipo h onde apenas a sub-divisão dos elementos é feita com base em erros estimados. O erro a ser considerado foi estimado a partir de normas onde se consideraram: a variação aproximada dos deslocamentos, a variação das forças de superfície e a variação da energia de deformação do sistema, calculada com a sua integração sobre o contorno. São apresentados exemplos numéricos para demonstrar a eficiência dos procedimentos propostos. / In this work, an adaptative formulation of the boundary element method is developed to analyze linear elastic fracture problems. The collocation point method was used to formulate the integral equations for the displacements and stresses (or tractions). To discretize the integral equations, linear elements were used to obtain the exact expressions of the integrals over boundary elements and fracture. To construct the linear system of equations were used only displacement equations, traction equations or both of them written for opposite nodes of the fracture, leading to the dual boundary element formulation usually employed in the fracture analyses. For the process of growth of the crack a special procedure was developed aiming at the correct determination of the direction of growth of the crack. The stress intensity factors, to calculate he crack growth angle, are calculated through of correlation displacements technique which relates the displacements actuants in the faces of the crack. The employed adaptative model is the h-type where only the sub-division of the elements is done based on error estimate. The error estimates considered in this work are based on the following norms: displacement, traction and strain energy variations, this last considered from the integration over the boundary. Numerical examples are presented to demonstrate the efficiency of the proposed procedures.

This thesis investigates two a posteriori error estimators, based on gradient recovery, aiming to fill the gap of the error estimations for the Generalized FEM (GFEM) and, mainly, its modified versions called Corrected XFEM (C-XFEM) and Stable GFEM (SGFEM). In order to reach this purpose, firstly, brief reviews regarding the GFEM and its modified versions are presented, where the main advantages attributed to each numerical method are highlighted. Then, some important concepts related to the error study are presented. Furthermore, some contributions involving a posteriori error estimations for the GFEM are shortly described. Afterwards, the two error estimators hereby proposed are addressed focusing on linear elastic fracture mechanics problems. The first estimator was originally proposed for the C-XFEM and is hereby extended to the SGFEM framework. The second one is based on a splitting of the recovered stress field into two distinct parts: singular and smooth. The singular part is computed with the help of the J integral, whereas the smooth one is calculated from a combination between the Superconvergent Patch Recovery (SPR) and Singular Value Decomposition (SVD) techniques. Finally, various numerical examples are selected to assess the robustness of the error estimators considering different enrichment types, versions of the GFEM, solicitant modes and element types. Relevant aspects such as effectivity indexes, error distribution and convergence rates are used for describing the error estimators. The main contributions of this thesis are: the development of two efficient a posteriori error estimators for the GFEM and its modified versions; a comparison between the GFEM and its modified versions; the identification of the positive features of each error estimator and a detailed study concerning the blending element issues. / Esta tese investiga dois estimadores de erro a posteriori, baseados na recuperação do gradiente, visando preencher o hiato das estimativas de erro para o Generalized FEM (GFEM) e, sobretudo, suas versões modificadas denominadas Corrected XFEM (C-XFEM) e Stable GFEM (SGFEM). De modo a alcançar este objetivo, primeiramente, breves revisões a respeito do GFEM e suas versões modificadas são apresentadas, onde as principais vantagens atribuídas a cada método são destacadas. Em seguida, alguns importantes conceitos relacionados ao estudo do erro são apresentados. Além disso, algumas contribuições envolvendo estimativas de erro a posteriori para o GFEM são brevemente descritas. Posteriormente, os dois estimadores de erro propostos neste trabalho são abordados focando em problemas da mecânica da fratura elástico linear. O primeiro estimador foi originalmente proposto para o C-XFEM e por este meio é estendido para o âmbito do SGFEM. O segundo é baseado em uma divisão do campo de tensões recuperadas em duas partes distintas: singular e suave. A parte singular é calculada com o auxílio da integral J, enquanto que a suave é calculada a partir da combinação entre as técnicas Superconvergent Patch Recovery (SPR) e Singular Value Decomposition (SVD). Finalmente, vários exemplos numéricos são selecionados para avaliar a robustez dos estimadores de erro considerando diferentes tipos de enriquecimento, versões do GFEM, modos solicitantes e tipos de elemento. Aspectos relevantes tais como índices de efetividade, distribuição do erro e taxas de convergência são usados para descrever os estimadores de erro. As principais contribuições desta tese são: o desenvolvimento de dois eficientes estimadores de erro a posteriori para o GFEM e suas versões modificadas; uma comparação entre o GFEM e suas versões modificadas; a identificação das características positivas de cada estimador de erro e um estudo detalhado sobre a questão dos elementos de mistura.