Computational Error Estimation and Adaptive Error Control for Multiscaled Differential Equations

多尺度物理系统涉及的过程作用在非常不同的尺度上出现在各种重要的应用。分析这样的系统提出了压倒性的挑战，数学分析，因此数值模拟已成为一个非常重要的工具。然而，多尺度问题也紧张的限制，目前的计算能力和资源，因为分辨率需要近似的行为在小尺度上准确。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泛函的解决方案中给出的计算数值解。该方法通过变分分析和原问题的对偶问题来考虑稳定性的全局影响。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现，以及使用这些代码来调查物理系统。推动这项研究的潜在应用将是研究少量悬浮在低雷诺数流动中的颗粒的行为，以及剪切流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。在这种情况下，与流体的运动相比，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。其他多尺度系统包括生物系统中的模式形成，如斑马上的条纹，种群中疾病的动力学，以及涉及广义相对论的问题。对这类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，多尺度方面应变电流的计算能力，因为分辨率需要近似的行为在小尺度上准确。主要研究人员将通过开发计算误差估计来解决这个问题，以用于指导空间和时间中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，分辨率将通过反馈机制进行调整。拟议的研究包括技术的发展，以估计误差和调整的分辨率的近似，这些技术的实施到计算机程序，将提供给公众，并使用这些计算机程序来调查真实的物理系统，包括上述。

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.