Mathematical Sciences: Uniform Numerical Methods for Singularly Perturbed Equations

法雷尔9622744 研究人员，连同同事在爱尔兰和俄罗斯，开发ε一致收敛的奇摄动方程的有限差分格式。 与经典格式不同，这种格式的优点是误差不依赖于小参数ε的逆幂，因此当ε接近零时不会变得无界。 该项目的目的是使用在边界层中凝聚（细化）的网格，为一些非线性常微分方程以及椭圆和抛物型线性和非线性偏微分方程开发方法。 这些网格通常涉及至少一个自由参数。 他们调查这些参数的近似精度和线性和非线性求解器的收敛速度的影响。 研究者在并行（共享内存）和分布式计算机上实现这些方法。 这涉及到理论工作的收敛性施瓦茨型方法的区域分解及其并行变种，以及比较各种并行和顺序的方法的效率。 他们还研究了这些问题所产生的线性系统的有效求解器，这些问题对于小的非对称性来说可能是高度非对称的。 奇摄动微分方程是数学在科学和工程中的广泛应用。 其中包括高雷诺数下流体流动的Navier-Stokes方程、半导体器件物理学的漂移扩散方程、酶反应的Michaelis-Menten理论以及液晶材料和化学反应的数学模型。 发展有效的推广方法，这样的方程，产生保证的范围内的误差独立的小扰动参数的值，因此，在一些重要的领域具有相当大的意义。

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.