Nonlinear Multiscale Phenomena: Analysis, Control, and Computation

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Capturing the essential behavior of nonlinear phenomena with the simplest possible models is of paramount 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 devices, and the analysis of both models and algorithms. These crucial aspects of modern research are blended together in this research project, which deals with modeling, formulation, and numerical analysis of physical and biological phenomena at a scale where surface tension competes with bulk effects and could in principle be manipulated (or controlled) to produce scientifically interesting and practically useful dynamical behavior. Applications of the work include nano and microtechnology (such as the design and control of micro electro-mechanical systems (MEMS)), biotechnology (such as the study of biomembranes), and high performance computing (such as the design of novel efficient numerical methods). Results of the work will enhance modeling and prediction capabilities and help educate students and postdocs in exciting, mathematically and computationally challenging, and practically relevant areas of research. This project investigates models, such as biomembranes, ferrofluids, liquid crystals, and bilayer actuators, that are governed by nonlinear geometric partial differential equations defined on deformable domains that are unknown beforehand. Numerical approximation is carried out via adaptive finite element methods, with a posteriori error estimation and multilevel solvers, which allow for the resolution of problems with very disparate space-time scales with relatively modest computational resources. The project will advance understanding of adaptive approximation methods and the role of geometry in key questions concerning:1. Convergence and complexity of adaptive finite element methods (FEM) for elliptic PDE; study of fractional diffusion, hybridizable discontinuous Galerkin methods, hp-FEM and isogeometric methods, and the Laplace-Beltrami operator on parametric surfaces. 2. Design of high order arbitrary Lagrangian-Eulerian methods for parabolic PDE on deformable domains and surfaces. 3. Control of problems involving surface tension and magnetic effects, with or without free boundaries, relevant for device design in technology and biomedicine. 4. Computational modeling and analysis of ferrofluids and liquid crystals; these are technologically useful and mathematically intriguing complex fluids which can be actuated by magnetic and electric fields, and thus manipulated and controlled for specific purposes. 5. Novel FEM for geometric PDE: handling of large deformations with isometry constraints, typical of bilayer actuators, and dealing with fully nonlinear PDE.