Adaptive Finite Element Methods for Nonlinear Multiscale Problems

University Of Maryland, College Park, College Park MD

Investigators

Abstract

The adequate numerical treatment of nonlinear phenomena governed by partial differential equations with several disparate scales is a formidable 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 3d simulations feasible. Epitaxial and crystal growth in materials science, and viscous incompressible fluids are typical yet quite distinct examples addressed in this proposal. The goal of this project is to design, test, and analyze reliable and efficient adaptive finite element methods for such problems, with space-time error control and based on refinement/coarsening mesh modification. This project blends quite delicate analytical and computational issues, and applies them to free boundary problems, constrained problems, geometric PDE and the Navier-Stokes equations of incompressible fluids. Scientific computing has joined theory and experiment to form together the three central aspects of scientific inquiry. The current strengths in computational mathematics draw on the widespread acceptance of computational modeling as a complement to, and even a replacement for, physical tests in a broad number of fields. In this vein, the investigator develops reliable and efficient computational tools that may be useful in several areas of strategic importance such as nanotechnology, materials science, and high-performance computing. This project is a collaborative endeavor, involving a number of scientists in the US and abroad, as well as several students and postdocs. A substantial effort is devoted to education and human resource development.