Adaptive Finite Element Methods for Nonlinear Multiscale PDE

对具有不同时空尺度的偏微分方程（PDE）所控制的非线性现象进行适当的数值处理是一个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，使现实的三维模拟可行。 本项目主要研究自适应有限元法（AFEM）的设计、测试和分析的基本数学问题，以及AFEM在多尺度问题中的应用，AFEM是其中最强大的计算技术之一。在生物物理学（如生物膜）、流体和固体力学（如Navier-Stokes方程）、图像处理和金融等领域，这些领域都有着广泛的应用。 他们是典型的，但相当独特的，多尺度现象的例子，表现出奇异性，快速瞬变，和topological changes.This建议的基础上，事实上扩展和增强，以前的NSF Grant DMS-0204670。 它由许多看似独立的小型项目组成，但这些项目通过非线性、误差估计、数值分析和计算（提案的统一主题）的相互作用而相互关联。 这是一个合作的奋进，涉及一些科学家在美国和国外，以及一些研究生和博士后。 请拨资源以部分支助这些机构。 在教育和人力资源开发方面投入了大量努力。

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.