Adaptive Finite Element Methods with Error Control for Nonlinear PDE

对具有多个不同尺度的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，使现实的3d模拟可行。材料科学中的相变、外延生长、粘性不可压缩流体和化学趋化是典型但又非常独特的例子。该项目的目标是设计、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，并进行时空误差控制和基于细化/粗化网格修正。这个项目建立在我们对进化问题的后验误差估计的新理论的基础上；实际上，它扩展并增强了先前的NSF GrantDMS-9623394。它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，约束问题和平流扩散方程。该项目与材料科学、微电子学和高性能计算等战略领域有着密切的联系，因为它解决了所有这些领域的核心问题，如开发可靠和高效的算法。这是一个合作项目，涉及许多美国和国外的科学家，以及一些学生。大力发展教育和人力资源。

The adequate numerical treatment of nonlinear phenomena governedby partial differentialequations with severaldisparate scales is a formidable computational challenge. Modernalgorithms should be able to resolve fine scales for certain physicalquantities without overresolving others, thereby optimizing thecomputational effort and making realistic 3d simulations feasible.Phase transitions in materials science, epitaxial growth,viscous incompressible fluids, and chemotaxisare typical yet quite distinct examples. The goal of this project is to design, test, andanalyze reliable and efficient adaptive finite element methods for such problems, with space-time error control and based onrefinement/coarsening mesh modification.This project buildsupon our novel theory for a posteriorierror estimation of evolution problems; in fact it extends and enhances the prior NSF GrantDMS-9623394. It blends quite delicate analytical and computational issues, and applies them to free boundary problems, constrainedproblems, and advection-diffusion equations.This project has strong connections with areas of strategic interestsuch as Materials Science, Micro Electronics and High-PerformanceComputing, in that it addresses central issues to all of themsuch as the development of reliable and efficient algorithms. This is a collaborative project involving a number of scientistsin the US and abroad, as well as several students. A substantial effort is devoted to educationand human resource development.