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.

Uniformly convergent finite element methods for singularly perturbed parabolic partial differential equations

This thesis is concerned with uniformly convergent finite element methods for numerically solving singularly perturbed parabolic partial differential equations in one space variable. First, we use Petrov-Galerkin finite element methods to generate three schemes for such problems, each of these schemes uses exponentially fitted elements in space. Two of them are lumped and the other is non-lumped. On meshes which are either arbitrary or slightly restricted, we derive global energy norm and L2 norm error bounds, uniformly in the diffusion parameter. Under some reasonable global assumptions together with realistic local assumptions on the solution and its derivatives, we prove that these exponentially fitted schemes are locally uniformly convergent, with order one, in a discrete L∞norm both outside and inside the boundary layer. We next analyse a streamline diffusion scheme on a Shishkin mesh for a model singularly perturbed parabolic partial differential equation. The method with piecewise linear space-time elements is shown, under reasonable assumptions on the solution, to be convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 outside layers and almost order 3/4 inside the boundary layer. Numerical results for the above schemes are presented. Finally, we examine a cell vertex finite volume method which is applied to a model time-dependent convection-diffusion problem. Local errors away from all layers are obtained in the l2 seminorm by using techniques from finite element analysis.

On classical and other methods of discriminant analysis and estimation of log-odds

For two multinormal populations with equal covariance matrices the likelihood ratio discriminant function, an alternative allocation rule to the sample linear discriminant function when n1 ≠ n2 ,is studied analytically. With the assumption of a known covariance matrix its distribution is derived and the expectation of its actual and apparent error rates evaluated and compared with those of the sample linear discriminant function. This comparison indicates that the likelihood ratio allocation rule is robust to unequal sample sizes. The quadratic discriminant function is studied, its distribution reviewed and evaluation of its probabilities of misclassification discussed. For known covariance matrices the distribution of the sample quadratic discriminant function is derived. When the known covariance matrices are proportional exact expressions for the expectation of its actual and apparent error rates are obtained and evaluated. The effectiveness of the sample linear discriminant function for this case is also considered. Estimation of true log-odds for two multinormal populations with equal or unequal covariance matrices is studied. The estimative, Bayesian predictive and a kernel method are compared by evaluating their biases and mean square errors. Some algebraic expressions for these quantities are derived. With equal covariance matrices the predictive method is preferable. Where it derives this superiority is investigated by considering its performance for various levels of fixed true log-odds. It is also shown that the predictive method is sensitive to n1 ≠ n2. For unequal but proportional covariance matrices the unbiased estimative method is preferred. Product Normal kernel density estimates are used to give a kernel estimator of true log-odds. The effect of correlation in the variables with product kernels is considered. With equal covariance matrices the kernel and parametric estimators are compared by simulation. For moderately correlated variables and large dimension sizes the product kernel method is a good estimator of true log-odds.

Symptom burden in individuals with inflammatory bowel disease: a mixed methods study

To investigate the symptom burden experiences of individuals with inflammatory bowel disease (IBD). An explanatory sequential mixed methods study was conducted. A cross-sectional, correlational survey was first undertaken. Symptom burden was measured using a modified disease specific version of the Memorial Symptom Assessment Scale, which was administered to a consecutive sample of individuals with IBD (n = 247) at an IBD Outpatients department in one urban teaching hospital in Ireland. Disease activity was determined using clinical disease activity indices, which were completed by the consulting physician. A sequential qualitative, descriptive study was then conducted aimed at explaining noteworthy quantitative findings. A criterion-related purposeful sample of seven participants from the quantitative study was recruited. Semi-structured face to face interviews were conducted using an interview guide and data were analysed using content analysis. Findings revealed that participants experienced a median of 10 symptoms during the last week, however as many as 16 symptoms were experienced during active disease. The most burdensome symptoms were lack of energy, bowel urgency, diarrhoea, feeling bloated, flatulence and worry. Total symptom burden was found to be low with a mean score of 0.56 identified out of a possible range from 0 to 4. Participants with active disease (M = 0.81, SD = 0.48; n = 68) had almost double mean total symptom burden scores than participants with inactive disease (M = 0.46, SD = 0.43; n = 166) (p < 0.001). Mean total psychological symptom burden was found to be significantly greater than mean total physical symptom burden (rho = 0.73, n = 247, p < 0.001). Self-reported disease control, gender, number of flare ups in the last two years, and smoking status was found to be significant predictors of total symptom burden, with self-reported disease control identified as the strongest predictor. Qualitative data revealed tiredness, pain, bowel symptoms, worry and fear as being burdensome. Furthermore, symptom burden experiences were described in terms of its impact on restricting aspects of daily activities, which accumulated into restrictions on general life events. Psychological symptom burden was revealed as more problematic than physical symptom burden due to its constant nature, with physical and psychological symptoms described to occur in a cyclical manner. Participants revealed that disease control was evaluated not only in terms of symptoms, but also in terms of their abilities to control the impact of symptoms on their lives. This study highlights the considerable number of symptoms and the most burdensome symptoms experienced by individuals with IBD, both during active and inactive disease. This study has important implications on symptom assessment in terms of the need to encompass both physical and psychological symptoms. In addition, greater attention needs to be placed on psychological aspects of IBD care.

Inclusive education policy, the general allocation model and dilemmas of practice in primary schools

Background: Inclusive education is central to contemporary discourse internationally reflecting societies’ wider commitment to social inclusion. Education has witnessed transforming approaches that have created differing distributions of power, resource allocation and accountability. Multiple actors are being forced to consider changes to how key services and supports are organised. This research constitutes a case study situated within this broader social service dilemma of how to distribute finite resources equitably to meet individual need, while advancing inclusion. It focuses on the national directive with regard to inclusive educational practice for primary schools, Department of Education and Science Special Education Circular 02/05, which introduced the General Allocation Model (GAM) within the legislative context of the Education of Persons with Special Educational Needs (EPSEN) Act (Government of Ireland, 2004). This research could help to inform policy with ‘facts about what is happening on the ground’ (Quinn, 2013). Research Aims: The research set out to unearth the assumptions and definitions embedded within the policy document, to analyse how those who are at the coalface of policy, and who interface with multiple interests in primary schools, understand the GAM and respond to it, and to investigate its effects on students and their education. It examines student outcomes in the primary schools where the GAM was investigated. Methods and Sample The post-structural study acknowledges the importance of policy analysis which explicitly links the ‘bigger worlds’ of global and national policy contexts to the ‘smaller worlds’ of policies and practices within schools and classrooms. This study insists upon taking the detail seriously (Ozga, 1990). A mixed methods approach to data collection and analysis is applied. In order to secure the perspectives of key stakeholders, semi-structured interviews were conducted with primary school principals, class teachers and learning support/resource teachers (n=14) in three distinct mainstream, non-DEIS schools. Data from the schools and their environs provided a profile of students. The researcher then used the Pobal Maps Facility (available at www.pobal.ie) to identify the Small Area (SA) in which each student resides, and to assign values to each address based on the Pobal HP Deprivation Index (Haase and Pratschke, 2012). Analysis of the datasets, guided by the conceptual framework of the policy cycle (Ball, 1994), revealed a number of significant themes. Results: Data illustrate that the main model to support student need is withdrawal from the classroom under policy that espouses inclusion. Quantitative data, in particular, highlighted an association between segregated practice and lower socioeconomic status (LSES) backgrounds of students. Up to 83% of the students in special education programmes are from lower socio-economic status (LSES) backgrounds. In some schools 94% of students from LSES backgrounds are withdrawn from classrooms daily for special education. While the internal processes of schooling are not solely to blame for class inequalities, this study reveals the power of professionals to order children in school, which has implications for segregated special education practice. Such agency on the part of key actors in the context of practice relates to ‘local constructions of dis/ability’, which is influenced by teacher habitus (Bourdieu, 1984). The researcher contends that inclusive education has not resulted in positive outcomes for students from LSES backgrounds because it is built on faulty assumptions that focus on a psycho-medical perspective of dis/ability, that is, placement decisions do not consider the intersectionality of dis/ability with class or culture. This study argues that the student need for support is better understood as ‘home/school discontinuity’ not ‘disability’. Moreover, the study unearths the power of some parents to use social and cultural capital to ensure eligibility to enhanced resources. Therefore, a hierarchical system has developed in mainstream schools as a result of funding models to support need in inclusive settings. Furthermore, all schools in the study are ‘ordinary’ schools yet participants acknowledged that some schools are more ‘advantaged’, which may suggest that ‘ordinary’ schools serve to ‘bury class’ (Reay, 2010) as a key marker in allocating resources. The research suggests that general allocation models of funding to meet the needs of students demands a systematic approach grounded in reallocating funds from where they have less benefit to where they have more. The calculation of the composite Haase Value in respect of the student cohort in receipt of special education support adopted for this study could be usefully applied at a national level to ensure that the greatest level of support is targeted at greatest need. Conclusion: In summary, the study reveals that existing structures constrain and enable agents, whose interactions produce intended and unintended consequences. The study suggests that policy should be viewed as a continuous and evolving cycle (Ball, 1994) where actors in each of the social contexts have a shared responsibility in the evolution of education that is equitable, excellent and inclusive.