Adaptive Finite Element Methods for Nonlinear Multiscale Problems

对多尺度偏微分方程控制的非线性现象进行适当的数值处理是一项艰巨的计算挑战。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，从而优化计算工作并使现实的3d模拟可行。材料科学中的外延和晶体生长，以及粘性不可压缩流体是本提案中讨论的典型但非常独特的例子。本项目的目标是设计、测试和分析可靠、高效的自适应有限元方法来解决这类问题，并采用时空误差控制和基于细化/粗化网格修正的方法。这个项目混合了相当微妙的分析和计算问题，并将它们应用于自由边界问题，约束问题，几何偏微分方程和不可压缩流体的Navier-Stokes方程。科学计算把理论和实验结合在一起，形成了科学探究的三个中心方面。计算数学目前的优势在于，计算建模在广泛的领域被广泛接受，作为物理测试的补充，甚至是替代。在这方面，研究者开发了可靠和高效的计算工具，这些工具可能在几个具有战略重要性的领域有用，如纳米技术、材料科学和高性能计算。本项目是一项合作项目，涉及美国和国外的一些科学家，以及一些学生和博士后。在教育和人力资源开发方面作出了大量努力。

The adequate numerical treatment of nonlinear phenomenagoverned by partial differential equations with several disparatescales is a formidable computational challenge. Modern algorithmsshould be able to resolve fine scales for certain physicalquantities without overresolving others, thereby optimizing thecomputational effort and making realistic 3d simulationsfeasible. Epitaxial and crystal growth in materials science, andviscous incompressible fluids are typical yet quite distinctexamples addressed in this proposal. The goal of this project isto design, test, and analyze reliable and efficient adaptivefinite element methods for such problems, with space-time errorcontrol and based on refinement/coarsening mesh modification.This project blends quite delicate analytical and computationalissues, and applies them to free boundary problems, constrainedproblems, geometric PDE and the Navier-Stokes equations ofincompressible fluids. Scientific computing has joined theory and experiment toform together the three central aspects of scientific inquiry.The current strengths in computational mathematics draw on thewidespread acceptance of computational modeling as a complementto, and even a replacement for, physical tests in a broad numberof fields. In this vein, the investigator develops reliable andefficient computational tools that may be useful in several areasof strategic importance such as nanotechnology, materialsscience, and high-performance computing. This project is acollaborative endeavor, involving a number of scientists in theUS and abroad, as well as several students and postdocs. Asubstantial effort is devoted to education and human resourcedevelopment.