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.

Negotiating the boundaries between home and work practices: The case of home-workers

When people work from home, the domains of home and work are co-located, often under one roof. Home-workers have to cope with the meeting of two practices that have traditionally been physically separated. In light of this, we need to understand: how do people who work from home negotiate the boundaries between their home and work practices? What kinds of boundaries do people construct? How do boundaries affect the relationship between home and work as domains? What kinds of boundaries are available to home-workers? Are home-workers in charge of their boundaries or do they co-create them with others? How does this position home-workers in their domains? In order to address these questions, I analysed a variety of data, including newspaper columns, online forum discussions, interviews, and personal diary entries, using a discourse analytic approach that lends itself to issues of positioning. Current literature clashes over whether home-workers are in control of their boundaries, and over the relationship between home and work that arises out of boundary negotiations, i.e. whether home and work are dichotomous or layered. I seek to contribute to boundary theory by adopting a practice theory stance (Wenger, 1998) to guide my analysis. By viewing home and work as practices, I show that boundary negotiations depend on how home-workers are positioned, e.g. if they are positioned as peripheral in a domain, they lack influence over boundaries. I demonstrate that home and work constitute a number of different practices, rather than a rigid dichotomy, and that the way home and work are related are not the same for all home-workers. The application of practice concepts further shows how relationships between practices are created. The contribution of this work is a reconceptualisation of current boundary theory away from individual and cognitive notions (Nippert-Eng, 1996) into the realm of positioning.