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.

Fundamental studies of sourdoughs fermented with Weissella cibaria and Lactobacillus plantarum: influence on baking characteristics, sensory profiles and in vitro starch digestibility of gluten-free breads

The application of sourdough can improve texture, structure, nutritional value, staling rate and shelf life of wheat and gluten-free breads. These quality improvements are associated with the formation of organic acids, exopolysaccharides (EPS), aroma or antifungal compounds. Initially, the suitability of two lactic acid bacteria strains to serve as sourdough starters for buckwheat, oat, quinoa, sorghum and flours was investigated. Wheat flour was chosen as a reference. The obligate heterofermentative lactic acid bacterium (LAB) Weissella cibaria MG1 (Wc) formed the EPS dextran (a α-1,6-glucan) from sucrose in situ with a molecular size of 106 to 107 kDa. EPS formation in all breads was analysed using size exclusion chromatography and highest amounts were formed in buckwheat (4 g/ kg) and quinoa sourdough (3 g/ kg). The facultative heterofermentative Lactobacillus plantarum FST1.7 (Lp) was identified as strong acidifier and was chosen due to its ubiquitous presence in gluten-free as well as wheat sourdoughs (Vogelmann et al. 2009). Both Wc and Lp, showed highest total titratable acids in buckwheat (16.8 ml; 26.0 ml), teff (16.2 ml; 24.5 ml) and quinoa sourdoughs (26.4 ml; 35.3 ml) correlating with higher amounts of fermentable sugars and higher buffering capacities. Sourdough incorporation reduced the crumb hardness after five days of storage in buckwheat (Wc -111%), teff (Wc -39%) and wheat (Wc -206%; Lp -118%) sourdough breads. The rate of staling (N/ day) was reduced in buckwheat (Ctrl 8 N; Wc 3 N; Lp 6 N), teff (Ctrl 13 N; Wc 9 N; Lp 10 N) and wheat (Ctrl 5 N; Wc 1 N; Lp 2 N) sourdough breads. Bread dough softening upon Wc and Lp sourdough incorporation accounted for increased crumb porosity in buckwheat (+10.4%; +4.7), teff (+8.1%; +8.3%) and wheat sourdough breads (+8.7%; +6.4%). Weissella cibaria MG1 sourdough improved the aroma quality of wheat bread but had no impact on aroma of gluten-free breads. Microbial shelf life however, was not prolonged in any of the breads regardless of the starter culture used. Due to the high prevalence of insulin-dependent diabetes mellitus particular amongst coeliac patients, glycaemic control is of great (Berti et al. 2004). The in vitro starch digestibility of gluten-free breads with and without sourdough addition was analysed to predict the GI (pGI). Sourdough can decrease starch hydrolysis in vitro, due to formation of resistant starch and organic acids. Predicted GI of gluten-free control breads were significantly lower than for the reference white wheat bread (GI=100). Starch granule size was investigated with scanning electron microscopy and was significantly smaller in quinoa flour (<2 μm). This resulted in higher enzymatic susceptibility and hence higher pGI for quinoa bread (95). Lowest hydrolysis indexes for sorghum and teff control breads (72 and 74, respectively) correlate with higher gelatinisation peak temperatures (69°C and 71°C, respectively). Levels of resistant starch were not increased by addition of Weissella cibaria MG1 (weak acidifier) or Lactobacillus plantarum FST1.7 (strong acidifier). The pGI was significantly decreased for both wheat sourdough breads (Wc 85; Lp 76). Lactic acid can promote starch interactions with gluten hence decreasing starch susceptibility (Östman et al. 2002). For most gluten-free breads, the pGI was increased upon sourdough addition. Only sorghum and teff Lp sourdough breads (69 and 68, respectively) had significantly decreased pGI. Results suggest that the increase of starch hydrolysis in gluten-free breads was related to mechanism other than presence of organic acids and formation of resistant starch.

A qualitative study of trends in patient preferences for the management of partial dentition

Objective: To identify factors influencing attitudes of partially dentate adults towards dental treatment in Ireland. Background: People are retaining more teeth later in life than ever before. Management of partially dentate older adults will be a major requirement for the future and it is important to determine factors which may influence patients’ attitudes to care. Methods: Subjects: A purposive sample of 22 partially dentate patients was recruited; 12 women and 12 men, ranging in age from 45 to 75 years. Data Collection: Semi-structured individual interviews. Results: Dental patients have increasing expectations in relation to (i) a more sophisticated approach to the management of missing teeth and (ii) their right to actively participate in decision making regarding the management of their tooth loss. There is some evidence of a cohort effect with younger patients (45–64 years) having higher expectations. Conclusions: The evidence of a cohort effect within this study in relation to higher patient expectations indicates that both contemporary and future patients are likely to seek a service based on conservation and restoration of missing teeth by fixed prostheses.