To simulate real world problems modeled by differential equations, it is often not sufficient to  consider and tackle a single equation. Rather, complex phenomena are modeled by several partial dierential equations that are coupled to each other. For example, a heart beat involve electric activity, mechanics of the movement of the walls and valves, as well as blood fow - a true multiphysics problem. There may also be ordinary differential equations modeling the reactions on a cellular level, and these may act on a much finer scale in both space and time. Determining efficient and accurate simulation tools for such multiscalar multiphysics problems is a challenge. The five scientific papers constituting this thesis investigate and present solutions to issues regarding accurate and efficient simulation using adaptive finite element methods. These include handling local accuracy through submodeling, analyzing error propagation in time-dependent  multiphysics problems, developing efficient algorithms for adaptivity in time and space, and deriving error analysis for coupled PDE-ODE systems. In all these examples, the error is analyzed and controlled using the framework of dual-weighted residuals, and the spatial meshes are handled using octree based data structures. However, few realistic geometries fit such grid and to address this issue a discontinuous Galerkin Nitsche method is presented and analyzed.

Accuracy of the finite difference sensitivity calculations are improved by calculating the optimum finite difference interval sizes. In an aerodynamic inverse design algorithm, a compressor cascade geometry is perturbed by shape functions and finite differences sensitivity derivatives of the flow variables are calculated with respect to the base geometry flow variables. Sensitivity derivatives are used in an optimization code and a new airfoil is designed verifying given design characteristics. Accurate sensitivities are needed for optimization process. In order to find the optimum finite difference interval size, a method is investigated. Convergence error estimation techniques in iterative solutions and second derivative estimations are investigated to facilitate this method. For validation of the method, analytical sensitivity calculations of Euler equations are used and several applications are performed. Efficiency of the finite difference sensitivity calculations is improved by parallel computing. Finite difference sensitivity calculations are independent tasks in an inverse aerodynamic design algorithm and can be computed separately. Sensitivity calculations are performed on parallel processors and computing time is decreased.

Use of imaging systems for performing various situational awareness tasks in militaryand commercial settings has a long history. There is increasing recognition,however, that a much better job can be done by developing non-traditional opticalsystems that exploit the task-specific system aspects within the imager itself. Insome cases, a direct consequence of this approach can be real-time data compressionalong with increased measurement fidelity of the task-specific features. In others,compression can potentially allow us to perform high-level tasks such as direct trackingusing the compressed measurements without reconstructing the scene of interest.In this dissertation we present novel advancements in feature-specific (FS) imagersfor large field-of-view surveillence, and estimation of temporal object-scene changesutilizing the compressive imaging paradigm. We develop these two ideas in parallel.In the first case we show a feature-specific (FS) imager that optically multiplexesmultiple, encoded sub-fields of view onto a common focal plane. Sub-field encodingenables target tracking by creating a unique connection between target characteristicsin superposition space and the target's true position in real space. This isaccomplished without reconstructing a conventional image of the large field of view.System performance is evaluated in terms of two criteria: average decoding time andprobability of decoding error. We study these performance criteria as a functionof resolution in the encoding scheme and signal-to-noise ratio. We also includesimulation and experimental results demonstrating our novel tracking method. Inthe second case we present a FS imager for estimating temporal changes in the objectscene over time by quantifying these changes through a sequence of differenceimages. The difference images are estimated by taking compressive measurementsof the scene. Our goals are twofold. First, to design the optimal sensing matrixfor taking compressive measurements. In scenarios where such sensing matrices arenot tractable, we consider plausible candidate sensing matrices that either use theavailable <italic>a priori</italic> information or are non-adaptive. Second, we develop closed-form and iterative techniques for estimating the difference images. We present results to show the efficacy of these techniques and discuss the advantages of each.

In this thesis we focus on multiscale and multiphysics problems. We derive a posteriori error estimates for a one way coupled multiphysics problem, using the dual weighted residual method. Such estimates can be used to drive local mesh refinement in adaptive algorithms, in order to efficiently obtain good accuracy in a desired goal quantity, which we demonstrate numerically. Furthermore we prove existence and uniqueness of finite element solutions for a two way coupled multiphysics problem. The possibility of deriving dual weighted a posteriori error estimates for two way coupled problems is also addressed. For a two way coupled linear problem, we show numerically that unless the coupling of the equations is to strong the propagation of errors between the solvers goes to zero. We also apply a variational multiscale method to both an elliptic and a hyperbolic problem that exhibits multiscale features. The method is based on numerical solutions of decoupled local fine scale problems on patches. For the elliptic problem we derive an a posteriori error estimate and use an adaptive algorithm to automatically tune the resolution and patch size of the local problems. For the hyperbolic problem we demonstrate the importance of how to construct the patches of the local problems, by numerically comparing the results obtained for symmetric and directed patches.

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 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.