Mathematical Sciences: Adaptive Numerical Methods for Singularly-Perturbed Reaction-Diffusion Equations

数学科学：奇扰动反应扩散方程的自适应数值方法

• 批准号: 9208684
• 负责人: Donald Estep
• 金额: $ 3.71万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-15 至 1995-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9208684&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Adaptive; Numerical; Methods

Estep proposes to analyze and implement adaptive finite element methods for systems of singularly-perturbed reaction-diffusion equations and dissipative ordinary differential equations. Solutions of such problems typically evolve on multiple scales, and to be useful, numerical results need to be uniformly accurate for the physically meaningful range of the parameters. He is attempting to overcome these difficulties by constructing a theory of adaptive error control based on a posteriori and a priori error analyses. A posteriori results measure the error in terms of the regularity of the approximation and the stability of the solution and provide a basis for adapting the discretization. A priori results bound the error in terms of the regularity of the solution and the stability of the scheme and guarantee convergence. Estep is also examining the dynamical properties of numerical schemes in the context of producing approximations with the correct dynamical behavior. Finally, he is studying the implementation of adaptive error control in finite element codes for parallel computers. The ultimate goal of this project is the public release of a parallel code that can solve systems of reaction-diffusion equations in two and three dimensions with minimal user interface. Many models in applied science result in nonlinear reaction-diffusion differential equations that contain source terms balanced against terms that diffuse energy. This balance is usually delicate and difficult to handle mathematically, consequently numerical approximation is an important tool. Yet, solutions of such problems typically evolve on several scales, i.e. some interesting behavior occurs in very localized regions in space-time while other behavior evolves over long times and over larger regions in space. The use of a uniform numerical discretization for a real application results in huge computations that tax even the largest computers. Estep's goal is to produce numerical schemes that adapt themselves to the localized behavior of the target solution so as to make the computations both as accurate as desired and as efficient as possible. Mathematically, he is trying to understand how to use the information provided by an approximation to adapt the discretization, that is, make the computations self-governing. He is also working on the implementation of this theory into a parallel code that solves very general problems with minimum user input. Success of this project will lead to the public release of the code, to the great benefit of the engineering and scientific communities.

Estep提出分析和实现自适应有限元 奇摄动反应扩散方程组的方法 方程和耗散常微分方程。 这些问题的解决方案通常在多个尺度上发展， 为了有用，数值结果需要一致精确 参数的物理意义范围。他是 试图通过构建一个 基于后验和自适应误差控制理论 先验误差分析后验结果测量误差， 项的逼近的正则性和稳定性 为适应离散化提供了依据。 先验结果限制了误差的规律性， 方案的求解和稳定性及保证 收敛Estep也在研究 在产生近似的上下文中的数值方案 正确的动力学行为。最后，他正在研究 有限元码自适应差错控制实现 并行计算机。该项目的最终目标是 公开发布的并行代码，可以解决系统的 二维和三维反应扩散方程， 最小的用户界面。 应用科学中的许多模型导致非线性 含源反应扩散方程 与分散能量的术语相平衡的术语。这种平衡是 通常是精细的并且难以数学处理， 因此，数值近似是一种重要的工具。然而， 这些问题的解决方案通常在几个尺度上发展， 即，在非常局部化的区域中发生一些有趣的行为 在时空中，而其他行为在很长一段时间内进化， 在太空中更大的区域。使用统一的数值 真实的应用程序的离散化会导致 即使是最大的计算机也要负担的计算。Estep的目标是 产生数值方案， 目标溶液的局部化行为，以使 计算既像期望的那样精确， 可能在数学上，他试图理解如何使用 由近似提供的信息，以适应 离散化，也就是说，使计算自治。他 也在致力于将这一理论应用到 用最少的用户解决非常一般的问题的并行代码 输入.该项目的成功将导致公开发布 代码，以极大的工程和科学的利益， 社区.