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.

Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

Interaction of additive and quantization noises in digital PLLs

Phase-locked loops (PLLs) are a crucial component in modern communications systems. Comprising of a phase-detector, linear filter, and controllable oscillator, they are widely used in radio receivers to retrieve the information content from remote signals. As such, they are capable of signal demodulation, phase and carrier recovery, frequency synthesis, and clock synchronization. Continuous-time PLLs are a mature area of study, and have been covered in the literature since the early classical work by Viterbi [1] in the 1950s. With the rise of computing in recent decades, discrete-time digital PLLs (DPLLs) are a more recent discipline; most of the literature published dates from the 1990s onwards. Gardner [2] is a pioneer in this area. It is our aim in this work to address the difficulties encountered by Gardner [3] in his investigation of the DPLL output phase-jitter where additive noise to the input signal is combined with frequency quantization in the local oscillator. The model we use in our novel analysis of the system is also applicable to another of the cases looked at by Gardner, that is the DPLL with a delay element integrated in the loop. This gives us the opportunity to look at this system in more detail, our analysis providing some unique insights into the variance `dip' seen by Gardner in [3]. We initially provide background on the probability theory and stochastic processes. These branches of mathematics are the basis for the study of noisy analogue and digital PLLs. We give an overview of the classical analogue PLL theory as well as the background on both the digital PLL and circle map, referencing the model proposed by Teplinsky et al. [4, 5]. For our novel work, the case of the combined frequency quantization and noisy input from [3] is investigated first numerically, and then analytically as a Markov chain via its Chapman-Kolmogorov equation. The resulting delay equation for the steady-state jitter distribution is treated using two separate asymptotic analyses to obtain approximate solutions. It is shown how the variance obtained in each case matches well to the numerical results. Other properties of the output jitter, such as the mean, are also investigated. In this way, we arrive at a more complete understanding of the interaction between quantization and input noise in the first order DPLL than is possible using simulation alone. We also do an asymptotic analysis of a particular case of the noisy first-order DPLL with delay, previously investigated by Gardner [3]. We show a unique feature of the simulation results, namely the variance `dip' seen for certain levels of input noise, is explained by this analysis. Finally, we look at the second-order DPLL with additive noise, using numerical simulations to see the effects of low levels of noise on the limit cycles. We show how these effects are similar to those seen in the noise-free loop with non-zero initial conditions.

A study of the Hobson and Rogers volatility model

We firstly examine the model of Hobson and Rogers for the volatility of a financial asset such as a stock or share. The main feature of this model is the specification of volatility in terms of past price returns. The volatility process and the underlying price process share the same source of randomness and so the model is said to be complete. Complete models are advantageous as they allow a unique, preference independent price for options on the underlying price process. One of the main objectives of the model is to reproduce the `smiles' and `skews' seen in the market implied volatilities and this model produces the desired effect. In the first main piece of work we numerically calibrate the model of Hobson and Rogers for comparison with existing literature. We also develop parameter estimation methods based on the calibration of a GARCH model. We examine alternative specifications of the volatility and show an improvement of model fit to market data based on these specifications. We also show how to process market data in order to take account of inter-day movements in the volatility surface. In the second piece of work, we extend the Hobson and Rogers model in a way that better reflects market structure. We extend the model to take into account both first and second order effects. We derive and numerically solve the pde which describes the price of options under this extended model. We show that this extension allows for a better fit to the market data. Finally, we analyse the parameters of this extended model in order to understand intuitively the role of these parameters in the volatility surface.

First principles calculation of electron-phonon and alloy scattering in strained SiGe

First-principles electronic structure methods are used to predict the mobility of n-type carrier scattering in strained SiGe. We consider the effects of strain on the electron-phonon deformation potentials and the alloy scattering parameters. We calculate the electron-phonon matrix elements and fit them up to second order in strain. We find, as expected, that the main effect of strain on mobility comes from the breaking of the degeneracy of the six Δ and L valleys, and the choice of transport direction. The non-linear effects on the electron-phonon coupling of the Δ valley due to shear strain are found to reduce the mobility of Si-like SiGe by 50% per % strain. We find increases in mobility between 2 and 11 times that of unstrained SiGe for certain fixed Ge compositions, which should enhance the thermoelectric figure of merit in the same order, and could be important for piezoresistive applications.

Giant enhancement of n-type carrier mobility in highly strained germanium nanostructures

First-principles electronic structure methods are used to predict the rate of n-type carrier scattering due to phonons in highly-strained Ge. We show that strains achievable in nanoscale structures, where Ge becomes a direct bandgap semiconductor, cause the phonon-limited mobility to be enhanced by hundreds of times that of unstrained Ge, and over a thousand times that of Si. This makes highly tensile strained Ge a most promising material for the construction of channels in CMOS devices, as well as for Si-based photonic applications. Biaxial (001) strain achieves mobility enhancements of 100 to 1000 with strains over 2%. Low temperature mobility can be increased by even larger factors. Second order terms in the deformation potential of the Gamma valley are found to be important in this mobility enhancement. Although they are modified by shifts in the conduction band valleys, which are caused by carrier quantum confinement, these mobility enhancements persist in strained nanostructures down to sizes of 20 nm.

Learning to 'understand backwards' in time: children's temporal cognition and the primary history curriculum

This study examines children’s temporal ways of knowing and it highlights the centrality of temporal cognition in the development of children’s historical understanding. It explores how young children conceptualise time and it examines the provision for temporal cognition at the levels of the intended, enacted and received history curriculum in the Irish primary school context. Positioning temporality as a prerequisite second-order concept, the study recognises the essential role of both first-order and additional second-order concepts in historical understanding. While the former can be defined as the basic, substantive content to be taught, the latter refers to a number of additional key concepts that are deemed fundamental to children's capacity to make meaningful sense of history. The study argues for due recognition to be given to temporality, in the belief that both sets of knowledge, the content and skills, are required to develop historical thinking (Lévesque, 2011). The study addresses a number of key research questions, using a mixed methods research design, comprising an analysis of history textbooks, a survey among final year student teachers about their teaching of history, and school-based interviews with primary school children: What opportunities are available for children to develop temporal ways of knowing? How do student teachers experience being apprenticed into the available culture for teaching history and understanding temporality at primary level? What insights do the cognitive-developmental and sociocultural perspectives on learning provide for understanding the dynamics of children’s temporal ways of knowing? The study argues that the skill of developing a deeper understanding of time is a key prerequisite in connecting with, and constructing, understandings and frameworks of the past. The study advances a view of temporality as complex, multi-faceted and developmental. The findings have a potential contribution to make in influencing policy and pedagogy in establishing an elaborated and well-defined curriculum framework for developing temporal cognition at both national and international levels.

A recessionary debate: who suffered more? A study of sectoral well-being and economic insecurity in Ireland

The aim of this thesis is to identify the relationship between subjective well-being and economic insecurity for public and private sector workers in Ireland using the European Social Survey 2010-2012. Life satisfaction and job satisfaction are the indicators used to measure subjective well-being. Economic insecurity is approximated by regional unemployment rates and self-perceived job insecurity. Potential sample selection bias and endogeneity bias are accounted for. It is traditionally believed that public sector workers are relatively more protected against insecurity due to very institution of public sector employment. The institution of public sector employment is made up of stricter dismissal practices (Luechinger et al., 2010a) and less volatile employment (Freeman, 1987) where workers are subsequently less likely to be affected by business cycle downturns (Clark and Postal-Vinay, 2009). It is found in the literature that economic insecurity depresses the well-being of public sector workers to a lesser degree than private sector workers (Luechinger et al., 2010a; Artz and Kaya, 2014). These studies provide the rationale for this thesis in testing for similar relationships in an Irish context. Sample selection bias arises when a selection into a particular category is not random (Heckman, 1979). An example of this is non-random selection into public sector employment based on personal characteristics (Heckman, 1979; Luechinger et al., 2010b). If selection into public sector employment is not corrected for this can lead to biased and inconsistent estimators (Gujarati, 2009). Selection bias of public sector employment is corrected for by using a standard Two-Step Heckman Probit OLS estimation method. Following Luechinger et al. (2010b), the propensity for individuals to select into public sector employment is estimated by a binomial probit model with the inclusion of the additional regressor Irish citizenship. Job satisfaction is then estimated by Ordinary Least Squares (OLS) with the inclusion of a sample correction term similar as is done in Clark (1997). Endogeneity is where an independent variable included in the model is determined within in the context of the model (Chenhall and Moers, 2007). The econometric definition states that an endogenous independent variable is one that is correlated with the error term (Wooldridge, 2010). Endogeneity is expected to be present due to a simultaneous relationship between job insecurity and job satisfaction whereby both variables are jointly determined (Theodossiou and Vasileiou, 2007). Simultaneity, as an instigator of endogeneity, is corrected for using Instrumental Variables (IV) techniques. Limited Information Methods and Full Information Methods of estimation of simultaneous equations models are assed and compared. The general results show that job insecurity depresses the subjective well-being of all workers in both the public and private sectors in Ireland. The magnitude of this effect differs among sectoral workers. The subjective well-being of private sector workers is more adversely affected by job insecurity than the subjective well-being of public sector workers. This is observed in basic ordered probit estimations of both a life satisfaction equation and a job satisfaction equation. The marginal effects from the ordered probit estimation of a basic job satisfaction equation show that as job insecurity increases the probability of reporting a 9 on a 10-point job satisfaction scale significantly decreases by 3.4% for the whole sample of workers, 2.8% for public sector workers and 4.0% for private sector workers. Artz and Kaya (2014) explain that as a result of many austerity policies implemented to reduce government expenditure during the economic recession, workers in the public sector may for the first time face worsening perceptions of job security which can have significant implications for their well-being (Artz and Kaya, 2014). This can be observed in the marginal effects where job insecurity negatively impacts the well-being of public sector workers in Ireland. However, in accordance with Luechinger et al. (2010a) the results show that private sector workers are more adversely impacted by economic insecurity than public sector workers. This suggests that in a time of high economic volatility, the institution of public sector employment held and was able to protect workers against some of the well-being consequences of rising insecurity. In estimating the relationship between subjective well-being and economic insecurity advanced econometric issues arise. The results show that when selection bias is corrected for, any statistically significant relationship between job insecurity and job satisfaction disappears for public sector workers. Additionally, in order to correct for endogeneity bias the simultaneous equations model for job satisfaction and job insecurity is estimated by Limited Information and Full Information Methods. The results from two different estimators classified as Limited Information Methods support the general findings of this research. Moreover, the magnitude of the endogeneity-corrected estimates are twice as large as those not corrected for endogeneity bias which is similarly found in Geishecker (2010, 2012). As part of the analysis into the effect of economic insecurity on subjective well-being, the effects of other socioeconomic variables and work-related variables are examined for public and private sector workers in Ireland.