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.

Environmental and biological characteristics of lagoons on the southwest coast of Ireland

Coastal lagoons are defined as shallow coastal water bodies partially separated from the adjacent sea by a restrictive barrier. Coastal lagoons are protected under Annex I of the European Habitats Directive (92/43/EEC). Lagoons are also considered to be “transitional water bodies” and are therefore included in the “register of protected areas” under the Water Framework Directive (2000/60/EC). Consequently, EU member states are required to establish monitoring plans and to regularly report on lagoon condition and conservation status. Irish lagoons are considered relatively rare and unusual because of their North Atlantic, macrotidal location on high energy coastlines and have received little attention. This work aimed to assess the physicochemical and ecological status of three lagoons, Cuskinny, Farranamanagh and Toormore, on the southwest coast of Ireland. Baseline salinity, nutrient and biological conditions were determined in order to provide reference conditions to detect perturbations, and to inform future maintenance of ecosystem health. Accumulation of organic matter is an increasing pressure in coastal lagoon habitats worldwide, often compounding existing eutrophication problems. This research also aimed to investigate the in situ decomposition process in a lagoon habitat together with exploring the associated invertebrate assemblages. Re-classification of the lagoons, under the guidelines of the Venice system for the classifications of marine waters according to salinity, was completed by taking spatial and temporal changes in salinity regimes into consideration. Based on the results of this study, Cuskinny, Farranamanagh and Toormore lagoons are now classified as mesohaline (5 ppt – 18 ppt), oligohaline (0.5 ppt – 5 ppt) and polyhaline (18 ppt – 30 ppt), respectively. Varying vertical, longitudinal and transverse salinity patterns were observed in the three lagoons. Strong correlations between salinity and cumulative rainfall highlighted the important role of precipitation in controlling the lagoon environment. Maximum effect of precipitation on the salinity of the lagoon was observed between four and fourteen days later depending on catchment area geology, indicating the uniqueness of each lagoon system. Seasonal nutrient patterns were evident in the lagoons. Nutrient concentrations were found to be reflective of the catchment area and the magnitude of the freshwater inflow. Assessment based on the Redfield molar ratio indicated a trend towards phosphorus, rather than nitrogen, limitation in Irish lagoons. Investigation of the decomposition process in Cuskinny Lagoon revealed that greatest biomass loss occurred in the winter season. Lowest biomass loss occurred in spring, possibly due to the high density of invertebrates feeding on the thick microbial layer rather than the decomposing litter. It has been reported that the decomposition of plant biomass is highest in the preferential distribution area of the plant species; however, no similar trend was observed in this study with the most active zones of decomposition varying spatially throughout the seasons. Macroinvertebrate analysis revealed low species diversity but high abundance, indicating the dominance of a small number of species. Invertebrate assemblages within the lagoon varied significantly from communities in the adjacent freshwater or marine environments. Although carried out in coastal lagoons on the southwest coast of Ireland, it is envisaged that the overall findings of this study have relevance throughout the entire island of Ireland and possibly to many North Atlantic coastal lagoon ecosystems elsewhere.

Project scoping for lessons learnt to apply to the Celtic Seas marine sub-region

This report involves a formal scoping exercise to identify lessons from a wide range of previous and current project and initiative experiences at the national, regional seas, European and global levels. An inventory of 77 projects and initiatives that are relevant with regard to the key activities proposed by the Celtic Seas Partnership has been compiled, as well as a short-list of 23 of the most pertinent projects, lessons learnt and contact names. This report has identified a number of synergies and collaborative opportunities from other project and initiatives in the Celtic Seas and beyond. It provides a strong starting point to clarify to stakeholders the uniqueness of the Celtic Seas Partnership project and delivers a sound baseline for the other preparatory actions (A2 Stakeholder Mapping and A3 Baseline Analysis for Marine Strategy Framework Directive Implementation) and the implementation actions. Furthermore, this report helps to avoid duplication of effort during the life-span of the project by highlighting what information pertaining to the Celtic Seas MSFD sub-region already exists.