There are several algorithms for solving the linear system of equations that arise from the finite element method with linear or near-linear computational complexity. One way is to find an approximation of the stiffness matrix that is such that it can be used in a preconditioned conjugate residual method, that is, a preconditioner to the stiffness matrix. We have studied two preconditioners for the conjugate residual method, both based on writing the stiffness matrix in block form, factorising it and then approximating the Schur complement block to get a preconditioner. We have studied the stationary reaction-diffusion-advection equation in two dimensions. The mesh is refined adaptively, giving a hierarchy of meshes. In the first method the Schur complement is approximated by the stiffness matrix at one coarser level of the mesh, in the second method it is approximated as the assembly of local Schur complements corresponding to macro triangles. For two levels the theoretical bound of the condition number is 1/(1-C²) for either method, where C is the Cauchy-Bunyakovsky-Schwarz constant. For multiple levels there is less theory. For the first method it is known that the condition number of the preconditioned stiffness matrix is O(l²), where l is the number of levels of the preconditioner, or, equivalently, the number mesh refinements. For the second method the asymptotic behaviour is not known theoretically. In neither case is the dependency of the condition number of C known. We have tested both methods on several problems and found the first method to always give a better condition number, except for very few levels. For all tested problems, using the first method it seems that the condition number is O(l), in fact it is typically not larger than Cl. For the second method the growth seems to be superlinear.

Sedimentary basins form when water-borne sediments in shallow seas are deposited over periods of millions of years. Sediments compact under their own weight, causing the expulsion of pore water. If this expulsion is sufficiently slow, overpressuring can result, a phenomenon which is of concern in oil drilling operations. The competition between pore water expulsion and burial is complicated by a variety of factors, which include diagenesis (clay dewatering), and different modes (elastic or viscous) of rheological deformation via compaction and pressure solution, which may also include hysteresis in the constitutive behaviours. This thesis is concerned with models which can describe the evolution of porosity and pore pressure in sedimentary basins. We begin by analysing the simplest case of poroelastic compaction which in a 1-D case results in a nonlinear diffusion equation, controlled principally by a dimensionless parameter lambda, which is the ratio of the hydraulic conductivity to the sedimentation rate. We provide analytic and numerical results for both large and small lambda in Chapter 3 and Chapter 4. We then put a more realistic rheological relation with hysteresis into the model and investigate its effects during loading and unloading in Chapter 5. A discontinuous porosity profile may occur if the unloaded system is reloaded. We pursue the model further by considering diagenesis as a dehydration model in Chapter 6, then we extend it to a more realistic dissolution-precipitation reaction-transport model in Chapter 7 by including most of the known physics and chemistry derived from experimental studies. We eventually derive a viscous compaction model for pressure solution in sedimentary basins in Chapter 8, and show how the model suggests radically different behaviours in the distinct limits of slow and fast compaction. When lambda << 1, compaction is limited to a basal boundary layer. When lambda >> 1, compaction occurs throughout the basin, and the basic equilibrium solution near the surface is a near parabolic profile of porosity. But it is only valid to a finite depth where the permeability has decreased sufficiently, and a transition occurs, marking a switch from a normally pressured environment to one with high pore pressures.

The central objective of this thesis is to suggest and develop one simple, unified method for communication over optical fiber networks, valid for all values of dispersion and nonlinearity parameters, and for a single-user channel or a multiple-user network. The method is based on the nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees of freedom in such models, in much the same way that the Fourier transform does for linear systems. In this thesis, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger (NLS) equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear spectrum of the signal. Just as the (ordinary) Fourier transform converts a linear convolutional channel into a number of parallel scalar channels, the nonlinear Fourier transform converts a nonlinear dispersive channel described by a \emph{Lax convolution} into a number of parallel scalar channels. Since, in the spectral coordinates the NLS equation is multiplicative, users of a network can operate in independent nonlinear frequency bands with no deterministic inter-channel interference. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This thesis lays the foundations of such a nonlinear frequency-division multiplexing system.

The central objective of this thesis is to suggest and develop one simple, unified method for communication over optical fiber networks, valid for all values of dispersion and nonlinearity parameters, and for a single-user channel or a multiple-user network. The method is based on the nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees of freedom in such models, in much the same way that the Fourier transform does for linear systems. In this thesis, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger (NLS) equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear spectrum of the signal. Just as the (ordinary) Fourier transform converts a linear convolutional channel into a number of parallel scalar channels, the nonlinear Fourier transform converts a nonlinear dispersive channel described by a \emph{Lax convolution} into a number of parallel scalar channels. Since, in the spectral coordinates the NLS equation is multiplicative, users of a network can operate in independent nonlinear frequency bands with no deterministic inter-channel interference. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This thesis lays the foundations of such a nonlinear frequency-division multiplexing system.

Scientific computing is used frequently in an increasing number of disciplines to accelerate scientific discovery. Many such computing problems involve the numerical solution of partial differential equations (PDE). In this thesis we explore and develop methodology for high-performance implementations of PDE solvers for shared-memory multiprocessor architectures. We consider three realistic PDE settings: solution of the Maxwell equations in 3D using an unstructured grid and the method of conjugate gradients, solution of the Poisson equation in 3D using a geometric multigrid method, and solution of an advection equation in 2D using structured adaptive mesh refinement. We apply software optimization techniques to increase both parallel efficiency and the degree of data locality. In our evaluation we use several different shared-memory architectures ranging from symmetric multiprocessors and distributed shared-memory architectures to chip-multiprocessors. For distributed shared-memory systems we explore methods of data distribution to increase the amount of geographical locality. We evaluate automatic and transparent page migration based on runtime sampling, user-initiated page migration using a directive with an affinity-on-next-touch semantic, and algorithmic optimizations for page-placement policies. Our results show that page migration increases the amount of geographical locality and that the parallel overhead related to page migration can be amortized over the iterations needed to reach convergence. This is especially true for the affinity-on-next-touch methodology whereby page migration can be initiated at an early stage in the algorithms. We also develop and explore methodology for other forms of data locality and conclude that the effect on performance is significant and that this effect will increase for future shared-memory architectures. Our overall conclusion is that, if the involved locality issues are addressed, the shared-memory programming model provides an efficient and productive environment for solving many important PDE problems.

This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.

This thesis consists of three parts: The first and second parts focus on long-time asymptotics of macroscopic and kinetic models respectively, while in the third part we connect these regimes using different scaling approaches. (1) Keller–Segel-type aggregation-diffusion equations: We study a Keller–Segel-type model with non-linear power-law diffusion and non-local particle interaction: Does the system admit equilibria? If yes, are they unique? Which solutions converge to them? Can we determine an explicit rate of convergence? To answer these questions, we make use of the special gradient flow structure of the equation and its associated free energy functional for which the overall convexity properties are not known. Special cases of this family of models have been investigated in previous works, and this part of the thesis represents a contribution towards a complete characterisation of the asymptotic behaviour of solutions. (2) Hypocoercivity techniques for a fibre lay-down model: We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modelling the fibre lay-down process in non-woven textile production. Further, we prove convergence to equilibrium with an explicit rate. This part of the thesis is an extension of previous work which considered the case of a stationary conveyor belt. Adding the movement of the belt, the global equilibrium state is not known explicitly and a more general hypocoercivity estimate is needed. Although we focus here on a particular application, this approach can be used for any equation with a similar structure as long as it can be understood as a certain perturbation of a system for which the global Gibbs state is known. (3) Scaling approaches for collective animal behaviour models: We study the multi-scale aspects of self-organised biological aggregations using various scaling techniques. Not many previous studies investigate how the dynamics of the initial models are preserved via these scalings. Firstly, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local kinetic 1D and 2D models to simpler models existing in the literature. Secondly, we investigate how some of the kinetic spatio-temporal patterns are preserved via these scalings using asymptotic preserving numerical methods.

The hyperbolic conservation laws model the phenomena of nonlinear waves including discontinuities. The coupled nonlinear equations representing such conservation laws may lead to discontinuous solutions even for smooth initial data. To solve such equations, developing numerical methods which are accurate, robust, and resolve all the wave structures appearing in the solutions is a challenging task. Among several discretization techniques developed for solving hyperbolic conservation laws numerically, Finite Volume Method (FVM) is the most popular. Numerical algorithms, in the framework of FVM, are broadly classified as upwind and central discretization methods. Upwind methods mimic the features of hyperbolic conservation laws very well. However, most of the popular upwind schemes are known to suffer from the shock instabilities. Many upwind methods are heavily dependent on eigen-structure, therefore methods developed for one system of conservation laws are not straightforwardly extended to other systems. On the contrary, central discretization methods are simple, independent of eigen-structure, and therefore, are easily extended to other systems. In the first part of the thesis, a hybrid central discretization method is introduced for Euler equations of gas dynamics. This hybrid scheme is then extended to other hyperbolic conservation laws namely, shallow water equations of oceanography and ideal magnetohydrodynamics equations. The baseline solver for the new hybrid scheme, Method of Optimal Viscosity for Enhanced Resolution of Shocks (MOVERS), is an accurate scheme capable of capturing grid aligned steady discontinuities exactly. This central scheme is free from complicated Riemann solvers and therefore is easy to implement. This low diffusive algorithm produces sonic glitches at the expansion regions involving sonic points and is prone to shock instabilities. Therefore it requires an entropy fix to avoid these problems. With the use of entropy fix the exact discontinuity capturing property of the scheme is lost, although sonic glitches and shock instabilities are avoided. The motivation for this work is to develop a numerical method which exactly preserves the steady contacts, is accurate, free of multi-dimensional shock instabilities and yet avoids the entropy fix. This is achieved by constructing a coefficient of numerical diffusion based on pressure gradient sensor. The pressure gradients are known to detect shocks and they vanish across contact discontinuities. This property of pressure sensor is utilized in constructing the coefficient of numerical diffusion. In addition to the numerical diffusion of the baseline solver, a numerical diffusion based on the pressure sensor, scaled by the maximum of eigen-spectrum, is used to avoid shock instabilities. At contact discontinuities, pressure gradients vanish and coefficient of numerical diffusion of MOVERS is automatically retained to capture steady contact discontinuities exactly. This simple hybrid central solver is accurate, captures steady contact discontinuities exactly and is free of multi-dimensional shock instabilities. This novel method is extended to shallow water and ideal magnetohydrodynamics equations in a similar way. In the second part of the thesis, an entropy stable central discretization method for hyperbolic conservation laws is introduced. In a quest for optimal numerical viscosity, development of entropy stable schemes gained importance in recent times. In this work, the entropy conservation equation is used as a guideline to fix the coefficient of numerical diffusion for smooth regions of the flow. At the large gradients, coefficient of numerical diffusion of baseline solver is used. Switch over between smooth and large gradients of the flow is done using limiter functions which are known to distinguish between smooth and high gradient regions of the flow. This simple and stable central scheme termed MOVERS-LE captures grid aligned steady discontinuities exactly and is free of shock instabilities in multi-dimensions. Both the above algorithms are tested on various well established benchmark test problems.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-10T18:13:32Z No. of bitstreams: 1 ROMERO ALVES DE MELO - DISSERTAÇÃO PPGMAT 2006..pdf: 1038740 bytes, checksum: d9fd10d289c6cf822fe688e743b58356 (MD5) / Made available in DSpace on 2018-07-10T18:13:32Z (GMT). No. of bitstreams: 1 ROMERO ALVES DE MELO - DISSERTAÇÃO PPGMAT 2006..pdf: 1038740 bytes, checksum: d9fd10d289c6cf822fe688e743b58356 (MD5) Previous issue date: 2006-12 / Capes / Neste trabalho usaremos a Teoria de Semigrupos para demonstrar resultados de existência e unicidade de solução para Equações Diferenciais Ordinárias, em espaços de Banach. Usando esta teoria resolvemos problemas de valor inicial, com relação a equação do calor e a equação da onda. (Para visualizar a equação ou fórmula deste resumo recomendamos o download do arquivo). / In this work we use semigroup theory to prove some results of existence and unicity for a class Ordinary Differential Equation, on Banach spaces. Using this tool, we show the existence of solutions for wave and heat equations. (To visualize the equation or formula of this summary we recommend downloading the file).

Cette thèse présente quelques contributions à l'étude qualitative de solutions d'équations aux dérivées partielles non linéaires dans des modèles issus de l'optique et de la mécanique des fluides. Nous nous intéressons plus précisément à l'existence de solutions et à leur stabilité temporelle. Le Chapitre 1 est consacré à l'équation de Lugiato-Lefever, qui est une variante de l'équation de Schrödinger non linéaire et qui a été dérivée dans plusieurs contextes en optique. En utilisant des outils de la théorie des bifurcations et des formes normales, nous procédons à une étude systématique des solutions stationnaires de cette équation, et prouvons l'existence de solutions périodiques et localisées. Dans le Chapitre 2, nous présentons un critère simple d'instabilité linéaire pour des ondes non linéaires. Nous appliquons ce résultat aux équations de Lugiato-Lefever, de Kadomtsev-Petviashvili-I et de Davey-Stewartson. Ces deux dernières équations sont des équations modèles dérivées en mécanique des fluides. Dans le Chapitre 3, nous montrons un critère d'instabilité linéaire pour des solutions périodiques de petite amplitude, par rapport à certaines perturbations quasipériodiques. Ce résultat est ensuite appliqué à l'équation de Lugiato-Lefever. / In this thesis we present several contributions to qualitative study of solutions of nonlinear partial differential equations in optics and fluid mechanics models. More precisely, we focus on the existence of solutions and their stability properties. In Chapter 1, we study the Lugiato-lefever equation, which is a variant of the nonlinear Schrödinger equation arising in sereval contexts in nonlinear optics. Using tools from bifurcation and normal forms theory, we perfom a systematic analysis of stationary solutions of this equation and prove the existence of periodic and localized solutions. In Chapter 2, we present a simple criterion for linear instability of nonlinear waves. We then apply this result to the Lugiato-Lefever equation, to the Kadomtsev-Petviashvili-I equation and the Davey-Stewartson equations. These last two equations are model equations arising in fluid mechanics. In Chapter 3, we prove a criterion for linear instability of periodic solutions with small amplitude, with respect to certain quasiperiodic perturbations. This result is then applied to the Lugiato-Lefever equation.