In this thesis, we analyze an adaptive discontinuous finite element method for symmetric second order linear elliptic operators. Moreover, we obtain a fully computable convergence analysis on the broken energy seminorm in first order symmetric interior penalty discontin- uous Galerkin finite element approximations of this problem. The method is formulated on nonconforming meshes made of triangular elements with first order polynomial in two di- mension. We use an estimator which is completely free of unknown constants and provide a guaranteed numerical bound on the broken energy norm of the error. This estimator is also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and higher order data oscillation terms. Consequently, the estimator yields fully reliable, quantitative error control along with efficiency. As a second problem, explicit expression for constants of the inverse inequality are given in 1D, 2D and 3D. Increasing mathematical analysis of finite element methods is motivating the inclusion of mesh dependent terms in new classes of methods for a variety of applications. Several inequalities of functional analysis are often employed in convergence proofs. Inverse estimates have been used extensively in the analysis of finite element methods. It is char- acterized as tools for the error analysis and practical design of finite element methods with terms that depend on the mesh parameter. Sharp estimates of the constants of this inequality is provided in this thesis.

Many real-life applications such as the shape optimization of technological devices, the identification of parameters in environmental processes and flow control problems lead to optimization problems governed by systems of convection diusion partial dierential equations (PDEs). When convection dominates diusion, the solutions of these PDEs typically exhibit layers on small regions where the solution has large gradients. Hence, it requires special numerical techniques, which take into account the structure of the convection. The integration of discretization and optimization is important for the overall eciency of the solution process. Discontinuous Galerkin (DG) methods became recently as an alternative to the finite dierence, finite volume and continuous finite element methods for solving wave dominated problems like convection diusion equations since they possess higher accuracy. This thesis will focus on analysis and application of DG methods for linear-quadratic convection dominated optimal control problems. Because of the inconsistencies of the standard stabilized methods such as streamline upwind Petrov Galerkin (SUPG) on convection diusion optimal control problems, the discretize-then-optimize and the optimize-then-discretize do not commute. However, the upwind symmetric interior penalty Galerkin (SIPG) method leads to the same discrete optimality systems. The other DG methods such as nonsymmetric interior penalty Galerkin (NIPG) and incomplete interior penalty Galerkin (IIPG) method also yield the same discrete optimality systems when penalization constant is taken large enough. We will study a posteriori error estimates of the upwind SIPG method for the distributed unconstrained and control constrained optimal control problems. In convection dominated optimal control problems with boundary and/or interior layers, the oscillations are propagated downwind and upwind direction in the interior domain, due the opposite sign of convection terms in state and adjoint equations. Hence, we will use residual based a posteriori error estimators to reduce these oscillations around the boundary and/or interior layers. Finally, theoretical analysis will be confirmed by several numerical examples with and without control constraints

Incompressible flows -- flows in which variations in the density of a fluid are negligible -- arise in a wide variety of applications, from hydraulics to aerodynamics. The incompressible Navier-Stokes equations which govern such flows are also of fundamental physical and mathematical interest. They are believed to hold the key to understanding turbulent phenomena; precise conditions for the existence and uniqueness of solutions remain unknown -- and establishing such conditions is the subject of one of the Clay Mathematics Institute's Millennium Prize Problems. Typical solutions of incompressible flow problems involve both fine- and large-scale phenomena, so that a uniform finite element mesh of sufficient granularity will at best be wasteful of computational resources, and at worst be infeasible because of resource limitations. Thus adaptive mesh refinements are required. In industry, the adaptivity schemes used are ad hoc, requiring a domain expert to predict features of the solution. A badly chosen mesh may cause the code to take considerably longer to converge, or fail to converge altogether. Typically, the Navier-Stokes solve will be just one component in an optimization loop, which means that any failure requiring human intervention is costly. Therefore, I pursue technological foundations for a solver of the incompressible Navier-Stokes equations that provides robust adaptivity starting with a coarse mesh. By robust, I mean both that the solver always converges to a solution in predictable time, and that the adaptive scheme is independent of the problem -- no special expertise is required for adaptivity. The cornerstone of my approach is the discontinuous Petrov-Galerkin (DPG) finite element methodology developed by Leszek Demkowicz and Jay Gopalakrishnan. For a large class of problems, DPG can be shown to converge at optimal rates. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. Several approximations to Navier-Stokes are of interest, and I study each of these in turn, culminating in the study of the steady 2D incompressible Navier-Stokes equations. The Stokes equations can be obtained by neglecting the convective term; these are accurate for "creeping" viscous flows. The Oseen equations replace the convective term, which is nonlinear, with a linear approximation. The steady-state incompressible Navier-Stokes equations approximate the transient equations by neglecting time variations. Crucial to this work is Camellia, a toolbox I developed for solving DPG problems which uses the Trilinos numerical libraries. Camellia supports 2D meshes of triangles and quads of variable polynomial order, allows simple specification of variational forms, supports h- and p-refinements, and distributes the computation of the stiffness matrix, among other features. The central contribution of this dissertation is design and development of mathematical techniques and software, based on the DPG method, for solving the 2D incompressible Navier-Stokes equations in the laminar regime (Reynolds numbers up to about 1000). Along the way, I investigate approximations to these equations -- the Stokes equations and the Oseen equations -- followed by the steady-state Navier-Stokes equations. / text

Over the last three decades, CFD simulations have become commonplace as a tool in the engineering and design of high-speed aircraft. Experiments are often complemented by computational simulations, and CFD technologies have proved very useful in both the reduction of aircraft development cycles, and in the simulation of conditions difficult to reproduce experimentally. Great advances have been made in the field since its introduction, especially in areas of meshing, computer architecture, and solution strategies. Despite this, there still exist many computational limitations in existing CFD methods; in particular, reliable higher order and hp-adaptive methods for the Navier-Stokes equations that govern viscous compressible flow. Solutions to the equations of viscous flow can display shocks and boundary layers, which are characterized by localized regions of rapid change and high gradients. The use of adaptive meshes is crucial in such settings -- good resolution for such problems under uniform meshes is computationally prohibitive and impractical for most physical regimes of interest. However, the construction of "good" meshes is a difficult task, usually requiring a-priori knowledge of the form of the solution. An alternative to such is the construction of automatically adaptive schemes; such methods begin with a coarse mesh and refine based on the minimization of error. However, this task is difficult, as the convergence of numerical methods for problems in CFD is notoriously sensitive to mesh quality. Additionally, the use of adaptivity becomes more difficult in the context of higher order and hp methods. Many of the above issues are tied to the notion of robustness, which we define loosely for CFD applications as the degradation of the quality of numerical solutions on a coarse mesh with respect to the Reynolds number, or nondimensional viscosity. For typical physical conditions of interest for the compressible Navier-Stokes equations, the Reynolds number dictates the scale of shock and boundary layer phenomena, and can be extremely high -- on the order of 10⁷ in a unit domain. For an under-resolved mesh, the Galerkin finite element method develops large oscillations which prevent convergence and pollute the solution. The issue of robustness for finite element methods was addressed early on by Brooks and Hughes in the SUPG method, which introduced the idea of residual-based stabilization to combat such oscillations. Residual-based stabilizations can alternatively be viewed as modifying the standard finite element test space, and consequently the norm in which the finite element method converges. Demkowicz and Gopalakrishnan generalized this idea in 2009 by introducing the Discontinous Petrov-Galerkin (DPG) method with optimal test functions, where test functions are determined such that they minimize the discrete linear residual in a dual space. Under the ultra-weak variational formulation, these test functions can be computed locally to yield a symmetric, positive-definite system. The main theoretical thrust of this research is to develop a DPG method that is provably robust for singular perturbation problems in CFD, but does not suffer from discretization error in the approximation of test functions. Such a method is developed for the prototypical singular perturbation problem of convection-diffusion, where it is demonstrated that the method does not suffer from error in the approximation of test functions, and that the L² error is robustly bounded by the energy error in which DPG is optimal -- in other words, as the energy error decreases, the L² error of the solution is guaranteed to decrease as well. The method is then extended to the linearized Navier-Stokes equations, and applied to the solution of the nonlinear compressible Navier-Stokes equations. The numerical work in this dissertation has focused on the development of a 2D compressible flow code under the Camellia library, developed and maintained by Nathan Roberts at ICES. In particular, we have developed a framework allowing for rapid implementation of problems and the easy application of higher order and hp-adaptive schemes based on a natural error representation function that stems from the DPG residual. Finally, the DPG method is applied to several convection diffusion problems which mimic difficult problems in compressible flow simulations, including problems exhibiting both boundary layers and singularities in stresses. A viscous Burgers' equation is solved as an extension of DPG to nonlinear problems, and the effectiveness of DPG as a numerical method for compressible flow is assessed with the application of DPG to two benchmark problems in supersonic flow. In particular, DPG is used to solve the Carter flat plate problem and the Holden compression corner problem over a range of Mach numbers and laminar Reynolds numbers using automatically adaptive schemes, beginning with very under-resolved/coarse initial meshes. / text

This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for some of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.

The discontinuous Galerkin finite element time-domain (DGFETD) method has been successfully applied to the solution of the coupled curl Maxwell’s equations. In this dissertation, important extensions to the DGFETD method are provided, including the ability to model lumped circuit elements and the ability to model thin-wire structures within a discrete DGFETD solution. To this end, a hybrid DGFETD/SPICE formulation is proposed for high-frequency circuit simulation, and a hybrid DGFETD/Thin-wire formulation is proposed for modeling thin-wire structures within a three-dimensional problem space. To aid in the efficient modeling of open-region structures, a Complex Frequency Shifted-Perfectly Matched Layer (CFS-PML) absorbing medium is applied to the DGFETD method for the first time. An efficient CFS-PML method that reduces the computational complexity and improves accuracy as compared to previous PML formulations is proposed. The methods have been successfully implemented, and a number of test cases are provided that validate the proposed methods. The proposed hybrid formulations and the new CFS-PML formulation dramatically enhances the ability of the DGFETD method to be efficiently applied to simulate complex, state of the art radio frequency systems.

Nowadays there is an ever increasing demand to obtain more accurate numericalsimulation results while at the same time using fewer computations. One area withsuch a demand is oil reservoir simulations, which builds upon Poisson's equation withvariable coefficients (PEWVC). This thesis focuses on applying and testing a high ordernumerical scheme to solve the PEWVC, namely Summation By Parts - SimultaneousApproximation Term (SBP-SAT). The thesis opens with proving that the method isconvergent at arbitrary high orders given sufficiently smooth coefficients. Theconvergence is furthermore verified in practice by test cases on the Poisson'sequation with smoothly variable permeability coefficients. To balance observed lowerboundary flux convergence, the SBP-SAT method was modified with additionalpenalty terms that were subsequently shown to work as expected. Finally theSBP-SAT method was tested on a semi-realistic model of an oil reservoir withdiscontinuous permeability. The correctness of the resulting pressure distributionvaried and it was shown that flux leakage was the probable cause. Hence theproposed SBP-SAT method performs, as expected, very well in continuous settingsbut typically allows undesirable leakage in discontinuous settings. There are possiblefixes, but these are outside the scope of this thesis.

The groundwater flow in aquifers contain evaporite rocks can cause problems such as geo-mechanical subsidence or collapse. In this work, we focus on the development of numerical models to simulate the flow in porous and non-porous domains in order to study the dissolution phenomenon and fractures evolution over time. The first part of this thesis is devoted to developing new solutions for the validation of numerical models to simulate density driven flow in porous media. The new procedure consist of solving simultaneously the flow and the transport equations using the Levenberg-Marquardt algorithm. The use of this technique allowed us to develop, for the first time, semi-analytical solutions of saltwater intrusion in the case of small diffusion and in the case of a large density contrast. In the second part of this work, we studied the flow in evaporitic rocks. A numerical code was developed to solve the nonlinear system using advanced numerical methods. To validate this new model, we have developed a semi-analytical solution for the density Stokes flow. The third part of this work is devoted to transport with dissolution of rock salt. As a first step, we studied the influence of various parameters on the dissolution of salt in Adlertunnel located at a depth of 160 m in the region of Basel in Switzerland. In a second step, we are interested in the simulation of the fracture's evolution as a result of the dissolution. The numerical model takes into account the Stokes flow and mass transport effects and dissolution of the fracture walls.

The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint.

Made available in DSpace on 2014-06-11T19:27:07Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-02-26Bitstream added on 2014-06-13T19:06:47Z : No. of bitstreams: 1 santos_ild_me_sjrp.pdf: 434711 bytes, checksum: 230caec3d969a14efac9b1700fd1dd97 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho introduzimos uma classe de sistemas dinâmicos descontínuos com espaço tempo contínuo e analisamos Teoremas que asseguram condições suficientes para a estabilidade de Lyapunov utilizando funções de Lyapunov. Além disso, consideramos também Teoremas de Recíproca, que sob algumas condições garantem uma determinada necessidade para esses Teoremas de estabilidade de Lyapunov. / In this work we introduce a class of discontinuous dynamical systems with time space continuous and we analyze Theorems that ensure sufficient conditions for the Lyapunov stability using Lyapunov functions. Moreover, we also consider Converse Theorems, which under some conditions guarantee a determined necessity for those Theorems of Lyapunov stability.