Mathematical Sciences: Uniform Numerical Methods for Singularly Perturbed Equations

FARREL 9622744研究员与爱尔兰和俄罗斯的同事一起，开发了奇异摄动方程的一致收敛的有限差分格式。与经典格式不同，这种格式的优点是误差不依赖于小参数epsilon的逆幂，因此当epsilon趋于零时不会变得无界。这个项目的目的是发展一些非线性常微分方程组，以及线性和非线性椭圆型和抛物型偏微分方程组的方法，使用在边界层中浓缩(细化)的网格。这些网格通常至少包含一个自由参数。他们研究了这些参数对线性和非线性求解器的近似精度和收敛速度的影响。调查者在并行(共享内存)和分布式计算机上实现了这些方法。这涉及到关于区域分解的Schwarz型方法及其并行变体的收敛的理论工作，以及各种并行和顺序方法的效率的比较。他们还研究了由这些问题产生的线性系统的有效求解器，这些问题对于小epsilon来说可能是高度不对称的。奇异摄动的微分方程在科学和工程问题的数学应用中无处不在。其中包括高雷诺数流体流动的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，以及液晶材料和化学反应的数学模型。因此，对于这类方程发展有效的可推广的方法，其误差保证界与小扰动参数值无关，在许多重要领域具有相当重要的意义。

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.