This proposal is concerned with the approximate solution of systems of nonlinear elliptic and parabolic partial differential equations posed on complicated domains. Such problems are used to model a wide variety of physical phenomena in various fields of engineering, mathematics, and science. Particular applications considered in this project arise in chemical electrostatics, in the topological classification of general surfaces in geometric analysis, and in semiconductor physics. Common characteristics of such nonlinear problems presenting hurdles to computational science include: multi-domain and time-dependent domains, multi-scale and multi-physics, and potential loss of regularity and blowup. In this project, the investigator plans to develop some fundamental mathematical tools for dealing with these difficulties that will have a wide range of applicability.
The results of this project will have a broad impact in mathematics as well as in science and engineering. In this project, the investigator is focused on applications in geometric analysis that are used to address fundamental questions in geometry, and on applications to highly complex mathematical models arising in modern biology, chemistry, and physics. However, the computational difficulties associated with complicated, multidimensional domains and multi-scaled, multi-physics differential equations that require new computational approaches arise in many contexts. The methods developed here will contribute to the advancement of numerical methods for complex three dimensional nonlinear dynamical simulations. The simulation technology we produce will provide exciting and powerful tools for the exploration of models in physics, chemistry, and biology, as well as in some areas of pure mathematics such as geometric analysis.
