Neste trabalho, tivemos como objetivo estudar a existência de soluções fracas não triviais para o problema elíptico com não linearidade crítica { - Δu = λu + u2* - 1+ + g(x, u+) + f(x); em Ω u = 0; sobre ∂ Ω , (P) onde Ω é um domínio limitado com fronteira suave em ℝN, com N ≥ 3, 2* = 2N / (N - 2) é o expoente crítico de Sobolev, u+ = max(u; 0), g ∈ C(Ω̄ x ℝ, ℝ+), λ > λ1, λ ∉ σ (- Δ) e f ∈ Lr> (Ω), com r > N. Com o intuito de observar as mudanças que ocorrem do caso subcrítico para o crítico e as diferentes técnicas variacionais para a resolução de problemas elípticos, estudamos, inicialmente, um problema um pouco mais antigo que (P), que, por sua vez, motivou seu estudo. Tal problema é { - Δu = λ u + up+ +f; em Ω u = 0; sobre ∂ Ω(P\') onde consideramos o caso subcrítico, ou seja, quando p ∈ (1; 2* - 1). Com o auxílio do TEOREMA DE ENLACE verificamos que tanto (P) quanto (P\') têm pelo menos duas soluções fracas não triviais. / In this work, we aimed to study the existence of nontrivial weak solutions for the elliptic problem with critical non-linearity { - Δu = λu + u2* - 1+ + g(x, u+) + f(x); in Ω u = 0; on ∂ Ω , (P) where Ω is a bounded domain with smooth boundary in ℝN, with N ≥ 3, 2* = 2N / N -2 is the critical Sobolev exponent, u+ = max(u; 0), g ∈ C(Ω̄ x ℝ, ℝ+), λ > λ1, λ ∉ σ (- Δ) and f ∈ Lr (Ω), with r > N. In order to observe different variational techniques for solving elliptic problems, we studied initially a problem a little older than (P), which, in turn, led to its study. This problem is { - Δu = λ u + up+ +f; inΩ u = 0; on ∂ Ω(P\') where we consider the subcritical case, that is, when p ∈ (1, 2* - 1). With the aid of the LINKING THEOREM we see that both (P) and (P\') have at least two nontrivial weak solutions.

Tato diplomová práce prezentuje problematiku dopravního toku a jeho modelování. Zabývá se především několika LWR modely, které následně rozebírá a hledá řešení pro počáteční úlohy. Ukazuje se, že ne pro všechny počáteční úlohy lze řešení definovat na celém prostoru, ale jen v určitém okolí počáteční křivky. Proto je dále odvozena metoda výpočtu velikosti tohoto okolí a to nejen zcela obecně, ale i pro dané modely. Teoretický rozbor LWR modelů a řešení počátečních úloh jsou demonstrovány několika příklady, které zřetelně ukazují, jak se dopravní tok simulovaný danými modely chová.

Multiphysics systems with interface coupling are used to model a variety of physical phenomena, such as arterial blood flow, air flow around aeroplane wings, or interactions between surface and ground water flows. Numerical methods enable the practical application of these models through computer simulations. Specifically a high level of detail and accuracy is achieved in finite element methods by discretisations which use extremely large numbers of degrees of freedom, rendering the solution process challenging from the computational perspective. In this thesis we address this challenge by developing a twofold strategy for improving the efficiency of standard finite element coupled solvers. First, we propose to solve a monolithic coupled problem using block-preconditioned GMRES with a new Schur complement approximation. This results in a modular and robust method which significantly reduces the computational cost of solving the system. In particular, numerical tests show mesh-independent convergence of the solver for all the considered problems, suggesting that the method is well-suited to solving large-scale coupled systems. Second, we derive an adjoint-based formula for goal-oriented a posteriori error estimation, which leads to a time-space mesh refinement strategy. The strategy produces a mesh tailored to a given problem and quantity of interest. The monolithic formulation of the coupled problem allows us to obtain expressions for the error in the Lagrange multiplier, which often represents a physically relevant quantity, such as the normal stress on the interface between the problem components. This adaptive refinement technique provides an effective tool for controlling the error in the quantity of interest and/or the size of the discrete system, which may be limited by the available computational resources. The solver and the mesh refinement strategy are both successfully employed to solve a coupled Stokes-Darcy-Stokes problem modelling flow through a cartridge filter.

New numerical methods are proposed for computing with smooth scalar and vector valued functions of two variables defined on rectangular domains. Functions are approximated to essentially machine precision by an iterative variant of Gaussian elimination that constructs near-optimal low rank approximations. Operations such as integration, differentiation, and function evaluation are particularly efficient. Explicit convergence rates are shown for the singular values of differentiable and separately analytic functions, and examples are given to demonstrate some paradoxical features of low rank approximation theory. Analogues of QR, LU, and Cholesky factorizations are introduced for matrices that are continuous in one or both directions, deriving a continuous linear algebra. New notions of triangular structures are proposed and the convergence of the infinite series associated with these factorizations is proved under certain smoothness assumptions. A robust numerical bivariate rootfinder is developed for computing the common zeros of two smooth functions via a resultant method. Using several specialized techniques the algorithm can accurately find the simple common zeros of two functions with polynomial approximants of high degree (≥ 1,000). Lastly, low rank ideas are extended to linear partial differential equations (PDEs) with variable coefficients defined on rectangles. When these ideas are used in conjunction with a new one-dimensional spectral method the resulting solver is spectrally accurate and efficient, requiring O(n²) operations for rank $1$ partial differential operators, O(n³) for rank 2, and O(n⁴) for rank &geq,3 to compute an n x n matrix of bivariate Chebyshev expansion coefficients for the PDE solution. The algorithms in this thesis are realized in a software package called Chebfun2, which is an integrated two-dimensional component of Chebfun.

Scientific research in areas of physics, chemistry, and biology traditionally depends purely on experimental and theoretical methods. Recently numerical simulation is emerging as the third way of science discovery beyond the experimental and theoretical approaches. This work describes some general procedures in numerical computation, and presents several applications of numerical modeling in bioheat transfer and biomechanics, jet diffusion flame, and bio-molecular interactions of proteins in blood circulation. A three-dimensional (3D) multilayer model based on the skin physical structure is developed to investigate the transient thermal response of human skin subject to external heating. The temperature distribution of the skin is modeled by a bioheat transfer equation. Different from existing models, the current model includes water evaporation and diffusion, where the rate of water evaporation is determined based on the theory of laminar boundary layer. The time-dependent equation is discretized using the Crank-Nicolson scheme. The large sparse linear system resulted from discretizing the governing partial differential equation is solved by GMRES solver. The jet diffusion flame is simulated by fluid flow and chemical reaction. The second-order backward Euler scheme is applied for the time dependent Navier-Stokes equation. Central difference is used for diffusion terms to achieve better accuracy, and a monotonicity-preserving upwind difference is used for convective ones. The coupled nonlinear system is solved via the damped Newton's method. The Newton Jacobian matrix is formed numerically, and resulting linear system is ill-conditioned and is solved by Bi-CGSTAB with the Gauss-Seidel preconditioner. A novel convection-diffusion-reaction model is introduced to simulate fibroblast growth factor (FGF-2) binding to cell surface molecules of receptor and heparan sulfate proteoglycan and MAP kinase signaling under flow condition. The model includes three parts: the flow of media using compressible Navier-Stokes equation, the transport of FGF-2 using convection-diffusion transport equation, and the local binding and signaling by chemical kinetics. The whole model consists of a set of coupled nonlinear partial differential equations (PDEs) and a set of coupled nonlinear ordinary differential equations (ODEs). To solve the time-dependent PDE system we use second order implicit Euler method by finite volume discretization. The ODE system is stiff and is solved by an ODE solver VODE using backward differencing formulation (BDF). Findings from this study have implications with regard to regulation of heparin-binding growth factors in circulation.

This thesis concerns the modelling of sound propagation in an urban environment. For most of the thesis a point source of sound source is assumed, and both 2D and 3D geometries are considered. Buildings are modelled as rigid blocks, with the effects of surface inhomogeneities neglected. In the time-harmonic case, assuming that the wavelength is short compared to typical lengthscales of the domain (street widths and lengths), ray theory is used to derive estimates for the time-averaged acoustic power flows through a network of interconnecting streets in the form of integrals over ray angles. In the impulsive case, the propagation of wave-field singularities in the presence of obstacles is considered, and a general principle concerning the weakening of singularities when they are diffracted by edges and vertices is proposed. The problem of switching on a time-harmonic source is also studied, and an exact solution for the diffraction of a switched on plane wave by a rigid half-line is obtained and analysed. The pulse diffraction theory is then applied in a study of the inverse problem for an impulsive source, where the aim is to locate an unknown source using Time Differences Of Arrival (TDOA) at multiple receivers. By using reflected and diffracted pulse arrivals, the standard free-space TDOA method is extended to urban environments. In particular, approximate source localisation is found to be possible even when the exact building distribution is unknown.

In this thesis, the problem of well-posedness of nonlinear viscoelasticity under the assumptions allowing for phase transformations in solids is considered. In one space dimension we prove existence and uniqueness of the solutions for the quasistatic version of the model using approximating sequences corresponding to the case when initial data takes finitely many values. This special case also provides upper and lower bounds for the solutions which are interesting in their own rights. We also show equivalence of the existence theory we develop with that of gradient flows when the stored-energy function is assumed to be -convex. Asymptotic behaviour of the solutions as time goes to infinity is then investigated and stabilization results are obtained by means of a new argument. Finally, we look at the problem from the viewpoint of curves of maximal slope and follow a time-discretization approach. We introduce a three-dimensional method based on composition of time-increments, as a result of which we are able to deal with the physical requirement of frame-indifference. In order to test this method and distinguish the difficulties for possible generalizations, we look at the problem in a convex setting. At the end we are able to obtain convergence of the minimization scheme as time step goes to zero.

This thesis concerns the analytical and practical aspects of applying the Closest Point Method to solve elliptic partial differential equations (PDEs) on smooth surfaces and domains with smooth boundaries. A new numerical scheme is proposed to solve surface elliptic PDEs and a novel geometric multigrid solver is constructed to solve the resulting linear system. The method is also applied to coupled bulk-surface problems. A new embedding equation in a narrow band surrounding the surface is formulated so that it agrees with the original surface PDE on the surface and has a unique solution which is constant along the normals to the surface. The embedding equation is then discretized using standard finite difference scheme and barycentric Lagrange interpolation. The resulting scheme has 2nd-order accuracy in practice and is provably 2nd-order convergent for curves without boundary embedded in ℝ². To apply the method to solve elliptic equations on surfaces and domains with boundaries, the "ghost" point approach is adopted to handle Dirichlet, Neumann and Robin boundary conditions. A systematic method is proposed to represent values of ghost points by values of interior points according to boundary conditions. A novel geometric multigrid method based on the closest point representation of the surface is constructed to solve the resulting large sparse linear systems. Multigrid solvers are designed for surfaces with or without boundaries and domains with smooth boundaries. Numerical results indicate that the convergence rate of the multigrid solver stays roughly the same as we refine the mesh, as is desired of a multigrid algorithm. Finally the above methods are combined to solve coupled bulk-surface PDEs with some applications to biology.

The main purpose of this thesis is to define the stability of a system of linear difference equations of the form, ∇y(t) = Ay(t), and to analyze the stability theory for such a system using the eigenvalues of the corresponding matrix A in nabla discrete calculus and nabla fractional discrete calculus. Discrete exponential functions and the Putzer algorithms are studied to examine the stability theorem. This thesis consists of five chapters and is organized as follows. In the first chapter, the Gamma function and its properties are studied. Additionally, basic definitions, properties and some main theorem of discrete calculus are discussed by using particular example. In the second chapter, we focus on solving the linear difference equations by using the undetermined coefficient method and the variation of constants formula. Moreover, we establish the matrix exponential function which is the solution of the initial value problems (IVP) by the Putzer algorithm.

Dislocation channel-veins and persistent slip bands (PSBs) are characteristic dislocation configurations that are of interest to both industry and academia. However, existing mathematical models are not adequate to describe the mechanism of the transition between these two states. In this thesis, a series of models are proposed to give a quantitative description to such a transition. The full problem has been considered from two angles. Firstly, the general motion and instabilities of arbitrary curved dislocations have been studied both analytically and numerically. Then the law of motion and local expansions are used to track the shapes of screw segments moving through channels, which are believed to induce dislocation multiplication by cross-slip. The second approach has been to investigate the collective behavior of a large number of dislocations, both geometrically necessary and otherwise. The traditional method of multiple scales does not apply well to describe the pile-up of two arrays of dislocations of opposite signs on a pair of neighbouring glide planes in two dimensional space. Certain quantities have to be more accurately defined under the multiple-scale coordinates to capture the much more localised resultant stress caused by these dislocation pairs. Through detailed calculations, one-dimensional dipoles can be homogenised to obtain some insightful results both on a local scale where the dipole pattern is the key diagnostic and on a macroscopic scale on which density variations are of most interest. Equilibria of dislocation dipoles in a two-dimensional regular lattice have been also studied. Some natural transitions between different patterns can be found as a result of geometrical instabilities.