Adaptive Finite Element Methods for Nonlinear Multiscale PDE

对具有不同时空尺度的偏微分方程（PDE）控制的非线性现象进行适当的数值处理是一项艰巨的数学和计算挑战。现代算法应该能够解决某些物理量的精细尺度，而不会过度解决其他物理量，从而优化计算工作量并使现实的三维模拟可行。本建议涉及自适应有限元方法（AFEM）的设计、测试和分析的基本数学问题，以及它们在各种多尺度问题中的应用，其中AFEM是最强大的计算技术之一。该项目考虑了材料科学（如外延和晶体生长）、生物物理学（如生物膜）、流体和固体力学（如Navier-Stokes方程）、图像处理和金融中的数学模型。它们是多尺度现象的典型例子，表现出奇点、快速瞬态和拓扑变化。该提案建立在之前的NSF拨款DMS-0204670的基础上，实际上是扩展和增强的。它被组织在许多看似独立的小项目中，然而，这些项目通过非线性、误差估计、数值分析和计算的相互作用而相关，这是提案的统一主题。这是一项合作的努力，涉及美国和国外的一些科学家，以及一些研究生和博士后。要求提供资源来部分支持它们。在教育和人力资源发展方面作出了大量努力。

The adequate numerical treatment of nonlinear phenomena governed by partial differential equations (PDE) with several disparate space-time scales is a formidable mathematical and computational challenge. Modern algorithms should be able to resolve fine scales for certain physical quantities without overresolving others, thereby optimizing the computational effort and making realistic three-dimensional simulations feasible. This proposal deals with fundamental mathematical questions for the design, testing, and analysis of adaptive finite element methods (AFEM) as well as their application to a variety of multiscale problems for which AFEM are among the most powerful computational techniques.This project considers mathematical models in materials science (such as epitaxial and crystal growth), in biophysics (such as biomembranes), in fluid and solid mechanics (such as the Navier-Stokes equations), in image processing and in finance. They are typical, yet quite distinct, examples of multiscale phenomena which exhibit singularities, fast transients, and topological changes.This proposal builds upon, and in fact extends and enhances, the prior NSF Grant DMS-0204670. It is organized in a number of small and seemingly independent projects, which are however related through the interplay of nonlinearity, error estimation, numerical analysis and computation, the unifying themes of the proposal. It is a collaborative endeavor involving a number of scientists in the US and abroad, as well as several graduate students and postdocs. Resources are requested to support them partially. A substantial effort is devoted to education and human resource development.