Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.

The role of the IL-1 type 1 receptor in the life, death and differentiation of adult hippocampal neural precursor cells

Neurogenesis occurs in two distinct regions of the adult brain; the subgranular zone (SGZ) of the dentate gyrus (DG) in the hippocampus, and the subventricular zone (SVZ) lining the lateral ventricles. It is now well-known that adult hippocampal neurogenesis can be modulated by a number of intrinsic and extrinsic factors e.g. local signalling molecules, exercise, environmental enrichment and learning. Moreover, levels of adult hippocampal neurogenesis decrease with age, at least in rodents, and alterations in hippocampal neurogenesis have been reported in animal models and human studies of neuropsychiatric and neurodegenerative conditions. Neuroinflammation is a common pathological feature of these conditions and is also a potent modulator of adult hippocampal neurogenesis. Recently, the orphan nuclear receptor TLX has been identified as an important regulator of adult hippocampal neurogenesis as its expression is necessary to maintain the neural precursor cell (NPC) pool in the adult DG. Likewise, exposure of animals to voluntary exercise has been consistently demonstrated to promote adult hippocampal neurogenesis. Lentivirus (LV)- mediated gene transfer is a useful tool to elucidate gene function and to explore potential therapeutic candidates across an array of conditions as it facilitates sustained gene expression in both dividing and post-mitotic cell populations. Both intrinsic and extrinsic factors are important regulators of adult hippocampal neurogenesis. Examining how these factors are affected by an inflammatory stimulus, and the subsequent effects on adult hippocampal neurogenesis provides important information for the development of novel treatment strategies for neuropsychiatric and neurodegenerative conditions in which adult hippocampal neurogenesis is impaired. The aims of the series of experiments presented in this thesis were to examine the effect of the pro-inflammatory cytokine interleukin-1β (IL-1β) on adult hippocampal NPCs both in vitro and in vivo. In vitro, we have shown that IL-1β reduces proliferation of adult hippocampal NPCs in a dose and time-dependent manner. In addition, we have demonstrated that TLX expression is reduced by IL-1β. Blockade of IL-1β signalling prevented both the IL-1β-induced reduction in cell proliferation and TLX expression. In vivo, we examined the effect of short term and long term exposure to LV-IL-1β in sedentary mice and in mice exposed to voluntary running. We demonstrated that impaired hippocampal neurogenesis is only evident after long term exposure to IL-1β. In mice exposed to voluntary running, hippocampal neurogenesis is significantly increased following short-term but not long-term exposure to running. Moreover, short-term running effectively prevents any IL-1β-induced effects on hippocampal neurogenesis; however, no such effects are seen following long-term exposure to running.

Theory of elasticity and electric polarization effects in the group-III nitrides

In this work, the properties of strained tetrahedrally bonded materials are explored theoretically, with special focus on group-III nitrides. In order to do so, a multiscale approach is taken: accurate quantitative calculations of material properties are carried out in a quantum first-principles frame, for small systems. These properties are then extrapolated and empirical methods are employed to make predictions for larger systems, such as alloys or nanostructures. We focus our attention on elasticity and electric polarization in semiconductors. These quantities serve as input for the calculation of the optoelectronic properties of these systems. Regarding the methods employed, our first-principles calculations use highly- accurate density functional theory (DFT) within both standard Kohn-Sham and generalized (hybrid functional) Kohn-Sham approaches. We have developed our own empirical methods, including valence force field (VFF) and a point-dipole model for the calculation of local polarization and local polarization potential. Our local polarization model gives insight for the first time to local fluctuations of the electric polarization at an atomistic level. At the continuum level, we have studied composition-engineering optimization of nitride nanostructures for built-in electrostatic field reduction, and have developed a highly efficient hybrid analytical-numerical staggered-grid computational implementation of continuum elasticity theory, that is used to treat larger systems, such as quantum dots.

Pitting and porous layer formation on n-InP anodes

Surface pitting occurs when InP electrodes are anodized in KOH electrolytes at concentrations in the range 2 - 5 mol dm-3. The process has been investigated using atomic force microscopy (AFM) and the results correlated with cross-sectional transmission electron microscopy (TEM) and electroanalytical measurements. AFM measurements show that pitting of the surface occurs and the density of pits is observed to increase with time under both potentiodynamic and potentiostatic conditions. This indicates a progressive pit nucleation process and implies that the development of porous domains beneath the surface is also progressive in nature. Evidence for this is seen in plan view TEM images in which individual domains are seen to be at different stages of development. Analysis of the cyclic voltammograms of InP electrodes in 5 mol dm-3 KOH indicates that, above a critical potential for pit formation, the anodic current is predominantly time dependent and there is little differential dependence of the current on potential. Thus, pores continue to grow with time when the potential is high enough to maintain depletion layer breakdown conditions.