Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

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.

Genetic analysis of bacteriophages from clinical and environmental samples

Bacteriophages, viruses infecting bacteria, are uniformly present in any location where there are high numbers of bacteria, both in the external environment and the human body. Knowledge of their diversity is limited by the difficulty to culture the host species and by the lack of the universal marker gene present in all viruses. Metagenomics is a powerful tool that can be used to analyse viral communities in their natural environments. The aim of this study was to investigate diverse populations of uncultured viruses from clinical (a sputum of patient with cystic fibrosis, CF) and environmental samples (a sludge from a dairy food wastewater treatment plant) containing rich bacterial populations using genetic and metagenomic analyses. Metagenomic sequencing of viruses obtained from these samples revealed that the majority of the metagenomic reads (97-99%) were novel when compared to the NCBI protein database using BLAST. A large proportion of assembled contigs were assignable as novel phages or uncharacterised prophages, the next largest assignable group being single-stranded eukaryotic virus genomes. Sputum from a cystic fibrosis patient contained DNA typical of phages of bacteria that are traditionally involved in CF lung infections and other bacteria that are part of the normal oral flora. The only eukaryotic virus detected in the CF sputum was Torque Teno virus (TTV). A substantial number of assigned sequences from dairy wastewater could be affiliated with phages of bacteria that are typically found in the soil and aquatic environments, including wastewater. Eukaryotic viral sequences were dominated by plant pathogens from the Geminiviridae and Nanoviridae families, and animal pathogens from the Circoviridae family. Antibiotic resistance genes were detected in both metagenomes suggesting phages could be a source for transmissible antimicrobial resistance. Overall, diversity of viruses in the CF sputum was low, with 89 distinct viral genotypes predicted, and higher (409 genotypes) in the wastewater. Function-based screening of a metagenomic library constructed from DNA extracted from dairy food wastewater viruses revealed candidate promoter sequences that have ability to drive expression of GFP in a promoter-trap vector in Escherichia coli. The majority of the cloned DNA sequences selected by the assay were related to ssDNA circular eukaryotic viruses and phages which formed a minority of the metagenome assembly, and many lacked any significant homology to known database sequences. Natural diversity of bacteriophages in wastewater samples was also examined by PCR amplification of the major capsid protein sequences, conserved within T4-type bacteriophages from Myoviridae family. Phylogenetic analysis of capsid sequences revealed that dairy wastewater contained mainly diverse and uncharacterized phages, while some showed a high level of similarity with phages from geographically distant environments.

Fabrication and characterization of doped and porous silicon nanowires as anodes for lithium ion batteries

By using Si(100) with different dopant type (n++-type (As) or p-type (B)), it is shown how metal-assisted chemically (MAC) etched silicon nanowires (Si NWs) can form with rough outer surfaces around a solid NW core for p-type NWs, and a unique, defined mesoporous structure for highly doped n-type NWs. High resolution electron microscopy techniques were used to define the characteristic roughening and mesoporous structure within the NWs and how such structures can form due to a judicious choice of carrier concentration and dopant type. Control of roughness and internal mesoporosity is demonstrated during the formation of Si NWs from highly doped n-type Si(100) during electroless etching through a systematic investigation of etching parameters (etching time, AgNO3 concentration, %HF and temperature). Raman scattering measurements of the transverse optical phonon confirm quantum size effects and phonon scattering in mesoporous wires associated with the etching condition, including quantum confinement effects for the nanocrystallites of Si comprising the internal structure of the mesoporous NWs. Laser power heating of NWs confirms phonon confinement and scattering from internal mesoporosity causing reduced thermal conductivity. The Li+ insertion and extraction characteristics at n-type and p-type Si(100) electrodes with different carrier density and doping type are investigated by cyclic voltammetry and constant current measurements. The insertion and extraction potentials are demonstrated to vary with cycling and the occurrence of an activation effect is shown in n-type electrodes where the charge capacity and voltammetric currents are found to be much higher than p-type electrodes. X-ray photo-electron spectroscopy (XPS) and Raman scattering demonstrate that highly doped n-type Si(100) retains Li as a silicide and converts to an amorphous phase as a two-step phase conversion process. The findings show the succinct dependence of Li insertion and extraction processes for uniformly doped Si(100) single crystals and how the doping type and its effect on the semiconductor-solution interface dominate Li insertion and extraction, composition, crystallinity changes and charge capacity. The effect of dopant, doping density and porosity of MAC etched Si NWs are investigated. The CV response is shown to change in area (current density) with increasing NW length and in profile shape with a changing porosity of the Si NWs. The CV response also changes with scan rate indicative of a transition from intercalation or alloying reactions, to pseudocapactive charge storage at higher scan rates and for p-type NWs. SEM and TEM show a change in structure of the NWs after Li insertion and extraction due to expansion and contraction of the Si NWs. Galvanostatic measurements show the cycling behavior and the Coulombic efficiency of the Si NWs in comparison to their bulk counterparts.

Redox biology of myeloid leukaemias

Internal tandem duplication of FMS-like receptor tyrosine kinase (FLT3-ITD) has been associated with an aggressive AML phenotype. FLT3-ITD expressing cell lines have been shown to generate increased levels of reactive oxygen species (ROS) and DNA double strand breaks (dsbs). However, the molecular basis of how FLT3-ITD-driven ROS leads to the aggressive form of AML is not clearly understood. Herein, we observe that the majority of H2O2 in FLT3-ITD-expressing MV4-11 cells colocalises to the endoplasmic reticulum (ER). Furthermore, ER localisation of ROS in MV4-11 cells corresponds to the localisation of p22phox, a small membrane-bound subunit of NOX complex. Furthermore, we show that 32D cells, a myeloblast-like cell line transfected with FLT3-ITD, possess higher steady protein levels of p22phox than their wild type FLT3 (FLT3-WT)-expressing counterparts. Moreover, the inhibition of FLT3-ITD, using various FLT3 tyrosine kinase inhibitors, uniformly results in a posttranslational downregulation of p22phox. We also show that depletion of NOX2 and NOX4 and p22phox, but not NOX1 proteins causes a reduction in endogenous H2O2 levels. We show that genomic instability induced by FLT3-ITD leads to an increase in nuclear levels of H2O2. The presence of H2O2 in the nucleus is largely reduced by inhibition of FLT3-ITD or NOX. Furthermore, similar results are also observed following siRNA knockdowns of p22phox or NOX4. We demonstrate that 32D cells transfected with FLT3-ITD have a higher level of DNA damage than 32D cells transfected with FLT3-WT. Additionally, inhibition of FLT3-ITD, p22phox and NOX knockdowns decrease the number of DNA dsbs. In summary, this study presents a novel mechanism of genomic instability generation in FLT3-ITD-expressing AML cells, whereby FLT3-ITD activates NOX complexes by stabilising p22phox. This in turn leads to elevated generation of ROS and DNA damage in these cells.