Mathematical Sciences: Uniform Numerical Methods for Singularly Perturbed Equations

研究者和他在爱尔兰和俄罗斯的同事一起，发展了奇异摄动方程的一致收敛有限差分格式。与经典方案不同，这种方案的优点是误差不依赖于小参数epsilon的逆幂，因此当epsilon接近零时不会变得无界。该项目的目的是开发一些非线性常微分方程的方法，以及椭圆型和抛物型的线性和非线性偏微分方程，使用在边界层中浓缩（精炼）的网格。这些网格通常涉及至少一个自由参数。他们研究了这些参数对线性和非线性求解器的逼近精度和收敛速度的影响。研究者在并行（共享内存）和分布式计算机上实现了这些方法。这涉及域分解的Schwarz型方法及其并行变体的收敛性的理论工作，以及各种并行和顺序方法的效率比较。他们还研究了由这些问题引起的线性系统的有效解，这些问题对于小的是非对称的。奇摄动微分方程在科学和工程问题的数学应用中非常普遍。其中包括高雷诺数下流体流动的Navier-Stokes方程，半导体器件物理的漂移-扩散方程，酶反应的Michaelis-Menten理论，液晶材料和化学反应的数学模型。这类方程的有效推广方法的发展在许多重要的领域具有相当的意义，这些方法可以产生与小扰动参数值无关的误差保证界。

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.