Algorithms for continuum processes on complex, moving surfaces

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***Partial differential equation (PDE) models for continuum processes arise throughout the applied and natural sciences. There has been a great effort made to develop numerical methods for many important classes of PDEs in one, two or three spatial dimensions. However, a remarkable variety of problems involve differential equations on curved surfaces. Numerical methods for PDEs on curved surfaces are much more complicated than in the standard Cartesian coordinate spaces R2 or R3. A popular class of methods for solving such PDEs are the embedding methods. In recent research, we introduced the closest point method, which is an embedding method that decouples surface geometry from the underlying differential operators. This research is part of the work that led to the 2011 Germund Dahquist Prize. *******The closest point method solves rather general PDEs on smooth, stationary surfaces in an accurate manner, using standard numerical PDE methods in the embedding space. However, many practical problems involve nonsmooth, moving surfaces. In this grant, a number of geometric enhancements to the closest point method will be introduced: (1) Research will be conducted on algorithms for general PDEs on surfaces with folds, junctions and other nonsmooth features, (2) robust methods for approximating PDEs on complex, evolving surfaces will be developed, (3) research on accurate methods for general PDEs on point cloud surfaces will also be carried out. Alongside these geometric improvements, efficiency enhancements will be developed using domain decomposition and optimized time-stepping methods. Analytical research will also be conducted to provide new insight into the convergence of the method, especially in situations where its performance is unexpectedly accurate. Throughout, highly qualified personnel will participate in the development of mathematical software, the analysis of numerical algorithms, and the design of numerical methods to solve PDEs on surfaces. This provides valuable experience for a broad range of careers in academic institutions and technology-related industries.*******The proposed research will seek algorithms and software that are accurate and efficient, yet are simple in the sense that they compute solutions to different continuum models as uniformly as possible while making strong use of existing techniques and software in 3D. It is anticipated that the methods obtained under this grant will enable researchers and end-users to numerically investigate realistic models of general continuum processes on complex moving surfaces with great accuracy.**** *****