This research is an attempt to conduct an extensive and comprehensive numerical study of geometrical flows, such as the mean curvature flow, the inverse mean curvature flow, the Gauss curvature flow, the surface diffusion flow, the Willmore flow, and the Ricci flow, from differential geometry, fluid mechanics, materials science, and cosmology, and to address theoretically and numerically challenging issues arising from geometrical flow computations. The goal of this project is to develop accurate, robust, and efficient adaptive finite element discretization methods, parallel iterative solution algorithms and computer codes for computing geometrical flows based on both level set and phase field formulations. The investigator aims to carry out a balanced numerical study for the geometrical flows by emphasizing both qualitative analysis and quantitative computation, and thus to provide reliable computational tools for discovering and analyzing fine properties such as dynamics of the singularities of the geometrical flows, which often are difficult and even may not be possible to predicate and characterize by analytical means. The methods and algorithms resulting from this research will have the following attractive features: high accuracy, strong stability, low cost, and high efficiency. In addition, the proposed methods are also capable of accurately and efficiently approximating the geometrical flows not only before but also beyond the onset of singularities.
As critical applications from fluid mechanics, cosmology, and materials science are directly tied to the solutions of geometrical flows, it is expected that successful completion of the proposed research has the potential to significantly impact these applied sciences not only by presenting new methods for solving underlying mathematical problems but also providing insights for the understanding of each of these applications. Furthermore, the methods to be developed will find applications in other fields such as cell biology, geophysics, image processing, and computer vision. The educational component of the project consists of graduate graduate course development, training and mentoring both graduate and undergraduate students through the project.
