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.

Diversity, difference and nation: indigenous peoples on screen in Mexico

This paper draws on constructivist theories of identity that regard the self as, paradoxically, coming into existence through interaction with the other, to investigate the discursive formation of indigenous people in the forging of Mexican national identity. The aim of the essay is to show how difference has been managed and deployed in the establishment of national Mexican identities from independence until the present. This is done with reference to visual culture and film and illustrated with examples from the ‘Golden Age’ as well as ‘the New Mexican Cinema’.

The geographies of encounter: investigating the spatiality and construction of 'meaningful' encounters with difference through the experiences of international students at University College Cork, Ireland

The purpose of this research is to investigate how international students negotiate encounters with Irish students and construct ‘meaning’ from those encounters in the spaces of the university and city. As cities are increasingly characterised by a multiplexity of diversity, the issue of living with difference is becoming more and more pertinent. In the wake of escalating socio-spatial polarisation, inter-cultural tension, racism, and xenophobia, the geographies of encounter seek to untangle the interactions that occur in the quotidian activities and spaces of everyday life to determine whether such encounters might reduce prejudice, antipathy and indifference and establish common social bonds (Amin 2002; Valentine 2008). Thus far, the literature has investigated a number of sites of encounter; public space, the home, neighbourhoods, schools, sports clubs, public transport, cafes and libraries (Wilson 2011; Schuermans 2013; Hemming 2011; Neal and Vincent 2011; Mayblin, Valentine and Anderrson 2015; Laurier and Philo 2006; Valentine and Sadgrove 2013; Harris, Valentine and Piekut 2014; Fincher and Iveson 2008). While these spaces produce a range of outcomes, the literature remains frustrated by a lack of clarity of what constitutes a ‘meaningful’ encounter and how such encounters might be planned for. Drawing on survey and interview data with full-time international students at University College Cork, Ireland, this study contributes to understanding how encounters are shaped by the construction and reproduction of particular identities in particular spaces, imbuing spaces with uneven power frameworks that produce diverse outcomes. Rather than identifying a singular ‘meaningful’ outcome of encounter as a potential panacea to the issues of exclusion and oppression, the contention here is to recognise a range of outcomes that are created by individuals in a range of ways. To define one outcome of encounter as ‘meaningful’ is to overlook the scale of intensity of diverse interactions and the multiplicity of ways in which people learn to live with difference.