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.

Variations in total and differential milk somatic cell counts and plasmin levels and their role in proteolysis and quality of milk and cheese

Increased plasmin and plasminogen levels and elevated somatic cell counts (SCC) and polymorphonuclear leucocyte levels (PMN) were evident in late lactation milk. Compositional changes in these milks were associated with increased SCC. The quality of late lactation milks was related to nutritional status of herds, with milks from herds on a high plane of nutrition having composition and clotting properties similar to, or superior to, early-mid lactation milks. Nutritionally-deficient cows had elevated numbers of polymorphonuclear leucocytes (PMNs) in their milk, elevated plasmin levels and increased overall proteolytic activity. The dominant effect of plasmin on proteolysis in milks of low SCC was established. When present in elevated numbers, somatic cells and PMNs in particular had a more significant influence on the proteolysis of both raw and pasteurised milks than plasmin. PMN protease action on the caseins showed proteolysis products of two specific enzymes, cathepsin B and elastase, which were also shown in high SCC milk. Crude extracts of somatic cells had a high specificity on αs1-casein. Cheeses made from late lactation milks had increased breakdown of αs1-casein, suggestive of the action of somatic cell proteinases, which may be linked to textural defects in cheese. Late lactation cheeses also showed decreased production of small peptides and amino acids, the reason for which is unknown. Plasmin, which is elevated in activity in late lactation milk, accelerated the ripening of Gouda-type cheese, but was not associated with defects of texture or flavour. The retention of somatic cell enzymes in cheese curd was confirmed, and a potential role in production of bitter peptides identified. Cheeses made from milks containing high levels of PMNs had accelerated αs1-casein breakdown relative to cheeses made from low PMN milk of the same total SCC, consistent with the demonstrated action of PMN proteinases. The two types of cheese were determined significantly different by blind triangle testing.

Differential and numerical models of hysteretic systems with stochastic and deterministic inputs

Many deterministic models with hysteresis have been developed in the areas of economics, finance, terrestrial hydrology and biology. These models lack any stochastic element which can often have a strong effect in these areas. In this work stochastically driven closed loop systems with hysteresis type memory are studied. This type of system is presented as a possible stochastic counterpart to deterministic models in the areas of economics, finance, terrestrial hydrology and biology. Some price dynamics models are presented as a motivation for the development of this type of model. Numerical schemes for solving this class of stochastic differential equation are developed in order to examine the prototype models presented. As a means of further testing the developed numerical schemes, numerical examination is made of the behaviour near equilibrium of coupled ordinary differential equations where the time derivative of the Preisach operator is included in one of the equations. A model of two phenotype bacteria is also presented. This model is examined to explore memory effects and related hysteresis effects in the area of biology. The memory effects found in this model are similar to that found in the non-ideal relay. This non-ideal relay type behaviour is used to model a colony of bacteria with multiple switching thresholds. This model contains a Preisach type memory with a variable Preisach weight function. Shown numerically for this multi-threshold model is a pattern formation for the distribution of the phenotypes among the available thresholds.

Leadership, fairness and gender: a practitioner's perspective

This Portfolio of Exploration outlines the course of learning by a practitioner on a personal leadership journey. It constructs a functional examination of ideas around leadership. It explores the belief system which underpins leadership practice and it also explores to what extent the increased participation of women can lead to a qualitatively different type of leadership. Different models of leadership are explored to find the model which can best fit with the author’s perception of leadership. The Women’s Intercultural Leadership Model is discussed. This model conceptualises leadership as the envisioning of oneself and others as potential leaders. It seeks to promote that recognition whilst at the same time encouraging personal transformation. The promotion of a values based leadership is examined. The value of fairness is examined as an element of functional leadership. The extent to which increasing the number of women in leadership can lead to a qualitatively different type of leadership is explored. Drucker’s assertion that every practice rests on theory is used to unearth the underlying assumptions of this reflective practitioner doctorate. Kegan’s Theory of Adult Mental Development and Subject-Object separation is used to explicitly examine knowledge construction. Myers Briggs Type Indicator (MBTI) and Assumption/Commitment/Behaviour/Outcome (ACBO) template are used to promote practice change.

An investigation into mechanisms of visceral pain in rodents

Visceral pain is a debilitating symptom of irritable bowel syndrome (IBS), a disorder affecting up to 30% of adults. A better understanding of the mechanisms underlying visceral hypersensitivity may facilitate development of more targeted therapies, improving the quality of life of these individuals. The studies performed in this thesis were designed to investigate important factors of visceral pain, including early-life manipulations, genetic predisposition and sex hormones. Maternal separation (MS) consistently reproduces visceral hypersensitivity and altered anxiety-like behaviours in rats, symptoms associated with IBS. It has been found that 5-HT2B receptor antagonism blocks visceral pain but no difference in relative 5-HT2B receptor mRNA expression was found in hippocampus, amygdala and colon. The neuronal activation patterns of prefrontal cortex and amygdala of MS rats were then investigated. MS animals are characterised by differential activation of the prefrontal cortex (anterior cingulate cortex (ACC), infralibic cortex, prelimbic cortex) as well as the central nucleus of the amygdala (CeA). Genetic factors also contribute to pain syndromes such as IBS. We utilised the Wistar Kyoto (WKY) rat, a stress-sensitive strain, as an animal model of brain-gut axis dysfunction. WKY rats have a lower expression of the glutamate transporter EAAT2 and mGlu4 receptor in the ACC. Another early-life factor that can increase susceptibility to functional gastrointestinal symptoms later life is disruption of the gut microbiota, thus early-life antibiotic treatment was used to assess this effect. Antibiotic treatment induced visceral hypersensitivity in adulthood and may be related to observed reductions in spinal cord alpha-2A adrenoreceptor (adra2A) mRNA. Lastly, we investigated sex differences in visceral sensitivity. EAAT1 & 2 mRNA levels are lower in females, potentially increasing glutamatergic concentration at the symaptic level. Moreover, NR1 and NR2B subunits mRNA of NMDA receptor were increased in caudal ACC of females. These findings may account for sex differences in visceral sensitivity.

The Shawlies: A study of early neoliberal ‘gender-fication’. The Street Trading Act, 1926, modern gender-fication, and the implications for Cork’s women street traders

For centuries Cork’s Shawlies, working-class women, survived by trading on public streets. My study explores how the first Irish Free State government, and Cork’s local authority, limited the rights of poor women to earn by subsistence trading with The Street Trading Act, 1926. The government insisted this would regulate street trading. In practice it further marginalised the women economically and socially, containing them outside the privileged, commercial city centre. In Cork the legislation facilitated the gradual disappearance of the Shawlies amid entrenched social processes and relations, contingencies that allowed for the abuse of their rights in the service of amalgamated business interests. This study address the role of discourses in deepening this marginalisation. My theoretical framework is designed to demonstrate how a seemingly innocuous piece of legislation would, in practice, do this. I set out the concepts of ‘Thriving State’, ‘Prosperous State’, and state of ‘Best Intentions’ that uses gentrification to meet these goals. The existing knowledge on women in trade is then examined, highlighting the gaps in what is known about the Shawlies. Chapter 3 details the theory behind my genealogical method. The legislation, debate, and other data produced at the national level is then examined, before moving to the local data. Chapter 6 is devoted to the Shawlies, setting their stories in the larger context of the debates. An examination of studies of contemporary women street traders in poor nations follows, along with a brief history of the decline of street trading in New York city under gentrification. Points of convergence between that process and the one in Cork are identified, along with convergences between contemporary traders and the Shawlies. The conclusion sets out my methodological, theoretical and substantive discoveries, and comments on current nostalgic renderings of the Shawlies in Cork’s newly gentrified Corn Market Street.