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.

An investigation of implicit measurement techniques amongst low risk and forensic samples

There are difficulties with utilising self- report and physiological measures of assessment amongst forensic populations. This study investigates implicit based measures amongst sexual offenders, nonsexual offenders and low risk samples. Implicit measurement is a term applied to measurement methods that makes it difficult to influence responses through conscious control. The test battery includes the Implicit Association Test (IAT), Rapid Serial Visual Presentation (RSVP), Viewing Time (VT) and the Structured Clinical interview for disorders. The IAT proposes that people will perform better on a task when they depend on well-practiced cognitive associations. The RSVP task requires participants to identify a single target image that is presented amongst a series of rapidly presented visual images. RSVP operates on the premise that if two target images are presented within 500milliseconds of each other, the possibility that the participant will recognize the second target is significantly reduced when the first target is of salience to the individual. This is the attentional blink phenomenon. VT is based on the principle that people will look longer at images that are of salience. Results showed that on the VT task, child sexual offenders took longer to view images of children than low risk groups. Nude over clothed images induced a greater attentional blink amongst low risk and offending samples on the RSVP task. Sexual offenders took longer than low risk groups on word pairing tasks where sexual words were paired with adult words on the IAT. The SCID highlighted differences between the offending and non offending groups on the sub scales for personality disorders. More erotic stimulus items on the VT and RSVP measures is recommended to better differentiate sexual preference between offending and non offending samples. A pictorial IAT is recommended. Findings provide the basis for further development of implicit measures within the assessment of sexual offenders.