Adaptive Finite Element Methods for Nonlinear Multiscale PDE

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The adequate numerical treatment of nonlinear phenomena governed by partial differential equations (PDE) with several disparate space-time scales is a formidable mathematical and 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 three-dimensional simulations feasible. This proposal deals with fundamental mathematical questions for the design, testing, and analysis of adaptive finite element methods (AFEM) as well as their application to a variety of multiscale problems for which AFEM are among the most powerful computational techniques.This project considers mathematical models in materials science (such as epitaxial and crystal growth), in biophysics (such as biomembranes), in fluid and solid mechanics (such as the Navier-Stokes equations), in image processing and in finance. They are typical, yet quite distinct, examples of multiscale phenomena which exhibit singularities, fast transients, and topological changes.This proposal builds upon, and in fact extends and enhances, the prior NSF Grant DMS-0204670. It is organized in a number of small and seemingly independent projects, which are however related through the interplay of nonlinearity, error estimation, numerical analysis and computation, the unifying themes of the proposal. It is a collaborative endeavor involving a number of scientists in the US and abroad, as well as several graduate students and postdocs. Resources are requested to support them partially. A substantial effort is devoted to education and human resource development.