Adaptive Finite Element Methods for Nonlinear Multiscale Problems

对多尺度偏微分方程所描述的非线性现象进行适当的数值处理是一个巨大的计算挑战。 现代算法应该能够在不过度解析其他物理量的情况下解析某些物理量的精细尺度，从而优化计算工作并使真实的3d模拟可行。 材料科学中的外延生长和晶体生长以及粘性不可压缩流体是本提案中所涉及的典型而又非常独特的例子。 本项目的目标是设计、测试和分析可靠和有效的自适应有限元方法来解决这类问题，该方法具有时空误差控制，并基于细化/粗化网格修改。本项目融合了相当精细的分析和计算问题，并将其应用于自由边界问题、约束问题、几何PDE和不可压缩流体的Navier-Stokes方程。 科学计算已经将理论和实验结合起来，形成了科学探究的三个核心方面。计算数学目前的优势在于，计算建模被广泛接受，可以作为许多领域物理测试的补充，甚至是替代。 在这种情况下，研究人员开发可靠和高效的计算工具，可能是有用的几个领域的战略重要性，如纳米技术，材料科学和高性能计算。 这个项目是一个合作的奋进，涉及一些科学家在美国和国外，以及一些学生和博士后。 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 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.