Capturing the essential behavior of nonlinear phenomena at the micro- and nanoscales with the simplest and crudest models is of fundamental importance in science and engineering. This allows for understanding of basic mechanisms, the design and implementation of efficient numerical methods for simulation and control of ensuing processes, and the analysis of both models and algorithms. Mathematical models in biophysics (such as biomembranes in both fluid and gel state), in materials science (such as crystal surface morphologies and bilayer actuators), and in shape optimization (including electro-wetting on dielectric) are typical yet quite distinct examples that the investigator studies in the project. The governing partial differential equations are geometric and exhibit disparate space-time scales: point and line singularities (interfaces), thin layers, and large domain deformations, perhaps leading to topology changes. The goal of the project is to model and control such multiscale phenomena, and to design, test, and analyze reliable and efficient adaptive finite element methods for them with space-time error control based on a posteriori error estimation.
Understanding the mechanisms of nonlinear phenomena at micro- and nanoscales is essential in many areas of science and engineering. The investigator develops mathematical models and reliable computational methods for studying a wide range of such problems. This project deals with applications of Federal strategic interest such as nano and microtechnology (such as the design and control of micro electro-mechanical system (MEMS)), biotechnology (such as the study of biomembranes), and high performance computing (such as the design of novel efficient numerical methods). It is a collaborative endeavor involving a number of scientists in the US and abroad, as well as several graduate students and postdocs. A substantial effort is devoted to education and human resource development.
Capturing the essential behavior of nonlinear phenomena at the micro and nano-scales with the simplest and crudest models is of fundamental importance in science and engineering. This allows for understanding of basic mechanisms, the design and implementation of efficient numerical methods for simulation and control of ensuing processes, and the analysis of both models and algorithms. This project contributed to all aspects of this endeavor: it was about modeling and control of such multiscale phenomena, as well as their numerical approximation via adaptive finite element methods (AFEM). It led to fundamental advances in several areas of modern research with impact in biophysics (modeling of biomembranes) and materials science (fractional diffusion, ferrofluids, surface tension effects, total variation flows), besides basic theory of AFEM and their application to geometric partial differential equations. Education, training and mentorship were also essential ingredients of this project. Three graduate students were partially supported, two received a Ph.D degree and one is expected to graduate in Spring 2015. Four postdocs were partially supported and contributed significantly to the research environment. Results were dissiminated in a number of formats. One book chapter and about twenty papers were published in top numerical analysis and applied analysis journals, with peer review, and three more papers were submitted for publication. Special lectures and summer schools were given around the world, as well as plenary talks in international meetings, conferences, and invited talks. Not only the PI but also the students are postdocs gave numerous presentations about their work. As a recognition of work developed over the years on the approximation of nonlinear phenomena, always supported by the NSF, the PI became a SIAM Fellow in 2011 and an AMS Fellow in 2012.