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.

Challengers to provisional republicanism: the official republican movement, people’s democracy and the Irish republican socialist party, 1968–98

The Provisional IRA and its political wing Sinn Féin have attracted by far the greatest scholarly interest of all the players in the Northern Irish conflict. This emphasis is perfectly legitimate, given the centrality of the Provos to so many turning-points in the conflict, from the collapse of Stormont in the early 1970s to the hunger strikes of the following decade and the ceasefires which were followed by the Belfast Agreement. My project, however, looks at political groups that at one time or another challenged the Provos for leadership of the militant, anti-state constituency in Northern Ireland (chiefly based in the Catholic working class). Although never as large or influential as the Provisional republicans, groups such as the Official IRA and the Irish Republican Socialist Party sometimes had a discernible impact on the course of events which is overlooked by most studies, and often pioneered ideas and tactics that were later adopted by the Provos themselves. The idea that republicans should embrace political action and work in broad campaigning alliances was promoted by the IRSP and socialist groups such as People’s Democracy before it was taken up by Gerry Adams and his allies, while the Official IRA supported the principle of a settlement based on democratization of the Northern Irish state, which was later accepted by Sinn Féin in the form of the Belfast Agreement. The goal of my research is to provide a novel perspective on the conflict in Northern Ireland, while engaging with theoretical debates about its character.