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.

Not thinking straight? A critical discourse analysis of the 2006 Irish High Court ruling in Zappone and Gilligan v. Revenue Commissioners and Attorney General

The recognition and protection of constitutional rights is a fundamental precept. In Ireland, the right to marry is provided for in the equality provisions of Article 40 of the Irish Constitution (1937). However, lesbians and gay men are denied the right to marry in Ireland. The ‘last word’ on this issue came into being in the High Court in 2006, when Katherine Zappone and Ann Louise Gilligan sought, but failed, to have their Canadian marriage recognised in Ireland. My thesis centres on this constitutional court ruling. So as to contextualise the pursuit of marriage equality in Ireland, I provide details of the Irish trajectory vis-à-vis relationship and family recognition for same-sex couples. In Chapter One, I discuss the methodological orientation of my research, which derives from a critical perspective. Chapter Two denotes my theorisation of the principle of equality and the concept of difference. In Chapter Three, I discuss the history of the institution of marriage in the West with its legislative underpinning. Marriage also has a constitutional underpinning in Ireland, which derives from Article 41 of our Constitution. In Chapter Four, I discuss ways in which marriage and family were conceptualised in Ireland, by looking at historical controversies surrounding the legalisation of contraception and divorce. Chapter Five denotes a Critical Discourse Analysis of the High Court ruling in Zappone and Gilligan. In Chapter Six, I critique text from three genres of discourse, i.e. ‘Letters to the Editor’ regarding same-sex marriage in Ireland, communication from legislators vis-à-vis the 2004 legislative impediment to same-sex marriage in Ireland, and parliamentary debates surrounding the 2010 enactment of civil partnership legislation in Ireland. I conclude my research by reflecting on my methodological and theoretical considerations with a view to answering my research questions. Author’s Update: Following the outcome of the 2015 constitutional referendum vis-à-vis Article 41, marriage equality has been realised in Ireland.

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.