Development and implementation of numerical algorithm for variational methods and generalized gradient flows for geometric evolution problems of higher order for surface processing in computer graphics
Final Report Abstract
In this project we developed a novel approach to solve geometric partial differential equations combining ideas from numerical homogenization relying on very recent results in applied mathematics and geometric flows. We identified this research project which we believed had the most potential. The idea is to construct localized approximation bases functions by solving an elliptic PDE in a neighborhood of the support of the classical nodal basis functions on a coarse mesh. The fine grid is generated via a subdivision scheme. This allows us to bypass the stringent time step size restrictions of nonlinear PDE integration schemes, thus rendering the integration of nonlinear surface flows more numerically robust and efficient. The spacial grid size of the coarse mesh can be chosen arbitrary. Typically, it equals the square root of the spatial grid size of the fine mesh. Then, if the time step size is chosen of the order of the spatial grid size squared of the coarse mesh, the actual time step is of the order of the spatial grid size of the fine mesh. We apply these techniques to second order geometric partial differential equations. Computational results underline the robustness of the new scheme, in particular with respect to large time steps. Extending this idea to fourth order problems in the future we will consider variational methods and geometric evolution problems for surface processing applications including surface fairing and surface restoration for surfaces with corners and edges in the future.
