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

Estep提出分析和实现奇异摄动反应扩散方程和耗散常微分方程系统的自适应有限元方法。这类问题的解决方案通常在多个尺度上演变，为了有用，数值结果需要在参数的物理意义范围内一致准确。他试图通过建立一个基于后验和先验误差分析的自适应误差控制理论来克服这些困难。后验结果从逼近的规律性和解的稳定性方面衡量了误差，并为适应离散化提供了依据。先验结果从解的正则性和格式的稳定性方面限制了误差，保证了收敛性。Estep还研究了数值格式在产生具有正确动力学行为的近似的背景下的动力学性质。最后，他正在研究并行计算机有限元代码的自适应误差控制的实现。该项目的最终目标是公开发布一个并行代码，该代码可以用最小的用户界面在二维和三维中求解反应扩散方程系统。应用科学中的许多模型产生非线性反应扩散微分方程，其中包含源项与扩散能量项的平衡。这种平衡通常是微妙的，难以用数学方法处理，因此数值近似是一个重要的工具。然而，这类问题的解决方案通常在几个尺度上演变，即一些有趣的行为发生在时空的非常局部的区域，而另一些行为则在很长时间和更大的空间区域演变。在实际应用中使用统一的数值离散化会导致大量的计算，即使是最大的计算机也会感到吃力。Estep的目标是产生能够适应目标解的局部行为的数值方案，以便使计算尽可能准确和高效。在数学上，他试图理解如何使用近似提供的信息来适应离散化，也就是说，使计算自治。他还致力于用并行代码实现这一理论，以最少的用户输入解决非常普遍的问题。该项目的成功将导致代码的公开发布，从而使工程和科学界受益匪浅。

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.