Adaptive Finite Element Methods with Error Control for Nonlinear PDE

点击翻译按钮获取中文摘要

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.