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.

Alcohol income and health: a complicated but desirable mix

The aim of this thesis is to examine if a difference exists in income for different categories of drinkers in Ireland using the 2007 Slán data set. The possible impact of alcohol consumption on health status and health care utilisation is also examined. Potential endogeneity and selection bias is accounted for throughout. Endogeneity is where an independent variable included in the model is determined within the context of the model (Chenhall and Moers, 2007). An endogenous relationship between income and alcohol and between health and alcohol is accounted for by the use of separate income equations and separate health status equations for each category of drinker similar to what was done in previous studies into the effects of alcohol on earnings (Hamilton and Hamilton, 1997; Barrett, 2002). Sample selection bias arises when a sector selection is non-random due to individuals choosing a particular sector because of their personal characteristics (Heckman, 1979; Zhang, 2004). In relation to alcohol consumption, selection bias may arise as people may select into a particular drinker group due to the fact that they know that by doing so it will not have a negative effect on their income or health (Hamilton and Hamilton, 1997; Di Pietro and Pedace, 2008; Barrett, 2002). Selection bias of alcohol consumption is accounted for by using the Multinomial Logit OLS Two Step Estimate as proposed by Lee (1982), which is an extension of the Heckman Probit OLS Two Step Estimate. Alcohol status as an ordered variable is examined and possible methods of estimation accounting for this ordinality while also accounting for selection bias are looked at. Limited Information Methods and Full Information Methods of estimation of simultaneous equations are assessed and compared. Findings show that in Ireland moderate drinkers have a higher income compared with abstainers or heavy drinkers. Some studies such as Barrett (2002) argue that this is as a consequence of alcohol improving ones health, which in turn can influence ones productivity which may ultimately be reflected in earnings, due to the fact that previous studies have found that moderate levels of alcohol consumption are beneficial towards ones health status. This study goes on to examine the relationship between health status and alcohol consumption and whether the correlation between income and the consumption of alcohol is similar in terms of sign and magnitude to the correlation between health status and the consumption of alcohol. Results indicate that moderate drinkers have a higher income than non or heavy drinkers, with the weekly household income of moderate drinkers being €660.10, non drinkers being €546.75 and heavy drinkers being €449.99. Moderate Drinkers also report having a better health status than non drinkers and a slightly better health status than heavy drinkers. More non-drinkers report poor health than either moderate or heavy drinkers. As part of the analysis into the effect of alcohol consumption on income and on health status, the relationship between other socio economic variables such as gender, age, education among others, with income, health and alcohol status is examined.