Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusion equation on an interval. The discretization is based on the equation's gradient flow structure with respect to the Wasserstein distance. The scheme inherits various properties of the continuous flow, like entropy monotonicity, mass preservation, metric contraction and minimum/maximum principles. As the main result, we give a proof of convergence in the limit of vanishing mesh size under a CFL-type condition. We also present results from numerical experiments.Comment: 28 pages, 6 figure

FULLY-DISCRETE VARIATIONAL LAGRANGIAN SCHEMES FOR NONLINEAR DRIFT-DIFFUSION EQUATIONS

Non UBCUnreviewedAuthor affiliation: TU MunchenGraduat

Analysis of a nonlinear cross-diffusion population model using the finite element method

Abweichender Titel laut Übersetzung der Verfasserin/des VerfassersZsfassung in engl. SpracheZiel dieser Diplomarbeit ist die analytische Ausformulierung der Arbeit "Finite element approximation of a nonlinear cross-diffusion population model" der Autoren John W. Barrett und James F. Blowey.In dieser Arbeit wird ein Populationsmodell behandelt, welches aus einem System zweier parabolischer, nichtlinearer, partieller Differentialgleichungen besteht, die im Diffusionsteil gekoppelt sind.Hierfür diskretisieren die Autoren das Problem mittels Finiter Elemente Methode und zeigen Konvergenz der dadurch approximierten Lösung gegen eine schwache Lösung des ursprünglichen Problems.This diploma thesis writes out the paper "Finite element approximation of a nonlinear cross-diffusion population model" from John W. Barrett and James F. Blowey in full.The paper describes a system of two nonlinear parabolic partial differential equations with cross-diffusion, which is motivated by a population model. The authors made a discretization using the Finite Element Method and show convergence towards a solution of the original problem in a weak sense.13

Hearing-impaired children in the UK, I: Cochlear implantation, auditory receptive capabilities, communication skills, educational achievements and quality of life.

OBJECTIVES: The objectives of this study were to identify variables that are associated with differences in outcome among hearing-impaired children and to control those variables while assessing the impact of cochlear implantation. STUDY DESIGN: In a cross-sectional study, the parents and teachers of a representative sample of hearing-impaired children were invited to complete questionnaires about children's auditory performance, spoken communication skills, educational achievements, and quality of life. Multiple regression was used to measure the strength of association between these outcomes and variables related to the child (average hearing level, age at onset of hearing impairment, age, gender, number of additional disabilities), the family (parental occupational skill level, ethnicity, and parental hearing status), and cochlear implantation. RESULTS: Questionnaires were returned by the parents of 2858 children, 468 of whom had received a cochlear implant, and by the teachers of 2241 children, 383 of whom had received an implant. Across all domains, reported outcomes were better for children with fewer disabilities in addition to impaired hearing. Across most domains, reported outcomes were better for children who were older, female, with a more favorable average hearing level, with a higher parental occupational skill level, and with an onset of hearing-impairment after 3 years. When these variables were controlled, cochlear implantation was consistently associated with advantages in auditory performance and spoken communication skills, but less consistently associated with advantages in educational achievements and quality of life. Significant associations were found most commonly for children who were younger than 5 years when implanted, and had used implants for more than 4 years. These children, whose mean (preoperative, unaided) average hearing level was 118 dB, were reported to perform at the same level as nonimplanted children with average hearing levels in the range from 80 dB to 104 dB, depending on the outcome measure. CONCLUSIONS: When rigorous statistical control is exercised in comparing implanted and nonimplanted children, pediatric cochlear implantation is associated with reported improvements both in spoken communication skills and in some aspects of educational achievements and quality of life, provided that children receive implants before 5 years of age