Collaborative Research: Numerical Methods for Nonlinear Diffusion Problems

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This proposal is concerned with the approximate solution of systemsof nonlinear elliptic and parabolic partial differential equations posedon complicated domains. Such problems are used to model a wide varietyof physical phenomena in various fields of engineering, mathematics, andscience. Particular applications considered in this project arise inchemical electrostatics, in the topological classification of generalsurfaces in geometric analysis, and in semiconductor physics. Commoncharacteristics of such nonlinear problems presenting hurdles tocomputational 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 ofapplicability.The results of this project will have a broad impact in mathematicsas well as in science and engineering. In this project, the investigator is focused on applications in geometric analysis that are used to address fundamentalquestions in geometry, and on applications to highly complex mathematicalmodels arising in modern biology, chemistry, and physics. However, thecomputational difficulties associated with complicated, multidimensionaldomains and multi-scaled, multi-physics differential equations that requirenew computational approaches arise in many contexts. The methods developedhere will contribute to the advancement of numerical methods for complexthree dimensional nonlinear dynamical simulations. The simulationtechnology we produce will provide exciting and powerful tools for theexploration of models in physics, chemistry, and biology, as well as insome areas of pure mathematics such as geometric analysis.