Mathematical Sciences: Uniform Numerical Methods for Singularly Perturbed Equations

点击翻译按钮获取中文摘要

Farrell 9622744 The investigator, together with colleagues in Ireland and Russia, develops epsilon-uniformly convergent finite difference schemes for singularly perturbed equations. Unlike classical schemes, such schemes have the advantage that the errors do not depend on an inverse power of the small parameter epsilon, and hence do not become unbounded as epsilon approaches zero. The aim of this project is to develop methods for a number of nonlinear ordinary differential equations, and linear and nonlinear partial differential equations of elliptic and parabolic type, using meshes that are condensed (refined) in the boundary layers. These meshes usually involve at least one free parameter. They investigate the influence of these parameters on the accuracy of the approximation and the speed of convergence of linear and nonlinear solvers. The investigator implements these methods on parallel (shared memory) and distributed computers. This involves theoretical work on convergence of Schwarz type methods of domain decomposition and their parallel variants, as well as comparisons of the efficiency of various parallel and sequential methods. They also study efficient solvers for the linear systems arising from these problems, which can be highly nonsymmetric for small epsilon. Singularly perturbed differential equations are pervasive in applications of mathematics to problems in the sciences and engineering. Among these are the Navier-Stokes equations of fluid flow at high Reynolds number, the drift-diffusion equations of semiconductor device physics, the Michaelis-Menten theory for enzyme reactions, and mathematical models of liquid crystal materials and of chemical reactions. The development of efficient generalizable methods for such equations, which yield guaranteed bounds on the error independent of the value of the small perturbation parameter, is thus of considerable significance in a number of important areas.