Computational Error Estimation and Adaptive Error Control for Multiscaled Differential Equations

多尺度物理系统涉及在不同尺度上作用的过程，在各种重要应用中出现。分析这样的系统对数学分析提出了巨大的挑战，因此数值模拟已经成为一个重要的工具。然而，多尺度问题也使当前计算能力和资源的极限紧张，因为需要精确地近似小尺度的行为。主要研究者将通过计算误差估计和自适应误差控制来解决这个问题。他将开发基于残差的后验估计的用户选择的功能的解决方案，根据计算的数值解决方案。该方法通过变分分析和对原问题的对偶问题来解释稳定性的全局效应。计算估计将用于指导离散化分辨率，以便有效地计算所需精度的数值解。本提案中的研究将包括计算误差估计的发展，估计的可靠性和准确性的分析，自适应有限元代码的实施，以及使用这些代码来研究物理系统。推动这项研究的潜在应用将是研究在低雷诺数流动中悬浮的少量颗粒的行为，以及在剪切流动问题中产生的反应扩散系统，广义相对论，生物系统中模式形成的建模，以及种群动力学。一些物理系统涉及的过程在很大程度上是不同的。一个例子是悬浮在缓慢流动的流体中的小颗粒的运动，例如发生在河床中的运动。在这种情况下，粒子的相互作用可以在非常快的时间尺度上发生，因为它们彼此接近，而不是流体的运动。其他多尺度系统包括生物系统的模式形成，如斑马的条纹，种群中疾病的动态，以及涉及广义相对论的问题。这种系统的计算机模拟已经成为理解和预测其行为的主要工具。然而，由于需要精确地近似小尺度上的行为，多尺度方面使电流计算能力紧张。主要研究者将通过开发计算误差估计来解决这个问题，该估计用于指导空间和时间上每个点所需的分辨率，以获得所需精度的模拟。随着系统的发展，该决议将通过反馈机制进行调整。建议的研究包括估计误差和调整近似分辨率的技术的发展，将这些技术实现到将向公众提供的计算机程序中，以及使用这些计算机程序来调查真实的物理系统，包括上面提到的那些。

Multiscaled physical systems involving processes acting on vastly different scales arise in a variety of important applications. Analyzing such systems present overwhelming challenges to mathematical analysis and therefore numerical simulation has become a centrally important tool. However, multi- scaled problems also strain the limit of current computational abilities and resources because of the resolution required to approximate behavior on the small scales accurately. The principle investigator will attack this problem by means of computational error estimation and adaptive error control. He will develop residual-based a posteriori estimates of user-chosen functionals of the solution given in terms of the computed numerical solution. The proposed approach accounts for the global effects of stability through a variational analysis and the dual problem to the original problem. The computational estimates will be used to guide the discretization resolution in order to efficiently compute numerical solutions of a desired accuracy. The research in this proposal will encompass development of computational error estimates, analysis of the reliability and accuracy of the estimates, implementation of adaptive finite element codes, and the use of these codes to investigate physical systems. The underlying applications driving this research will be investigations into the behavior of a small number of particles suspended in a low-Reynolds number flow and reaction- diffusion systems arising in shear flow problems, general relativity, modeling of pattern formation in biological systems, and population dynamics.Some physical systems involve processes that act on vastly different scales. An example is given by the motion of small particles suspended in a slow flowing fluid, as for example occurs in riverbeds. In this situation, the interactions of the particles can occur on a very rapid time scale as they approach each other compared to motion of the fluid. Other multi-scaled systems include pattern formation in biological systems like stripes on Zebras, the dynamics of diseases in populations, and problems involving general relativity. Computer simulations of such systems have become a main tool for understanding and predicting their behavior. Yet the multi-scaled aspects strain current computational ability because of the resolution needed to approximate the behavior on small scales accurately. The principle investigator will attack this problem by developing computational error estimates to be used to guide the resolution needed at each point in space and time to obtain simulations of a desired accuracy. The resolution will be adjusted through a feedback mechanism as a system evolves. The proposed research encompasses development of techniques to estimate the error and adjust the resolution of the approximation, implementation of these techniques into computer programs that will be made available to the public, and the use of these computer programs to investigate real physical systems including those mentioned above.