The study of surface evolution under geometric flows is a fascinating area of research with applications in computer graphics, material science, and biological modeling. Geometric flows describe how surfaces evolve over time according to certain rules, often driven by curvature. One of the most well-known geometric flows is the mean curvature flow, where each point on the surface moves in the direction of the surface normal with a speed proportional to the mean curvature at that point: \begin{equation} \partial_t X = -H n \end{equation} where $X$ is the position vector of the surface, $H$ is the mean curvature, and $n$ is the unit normal vector.
This project aims to simulate the evolution of surfaces under geometric flows using parametric finite element methods.
An artificial tangential velocity is introduced into the evolving finite element methods for mean curvature flow and Willmore flow proposed by Kovács et al. (Numer Math 143(4), 797-853, 2019, Numer Math 149, 595-643, 2021) in order to improve the mesh quality in the computation. The artificial tangential velocity is constructed by considering a limiting situation in the method proposed by Barrett et al. (J Comput Phys 222(1), 441-467, 2007, J Comput Phys 227(9), 4281-4307, 2008, SIAM J Sci Comput 31(1), 225-253, 2008) . The stability of the artificial tangential velocity is proved. The optimal-order convergence of the evolving finite element methods with artificial tangential velocity are proved for both mean curvature flow and Willmore flow. Extensive numerical experiments are presented to illustrate the convergence of the method and the performance of the artificial tangential velocity in improving the mesh quality.
@article{HL2022NM,title={Evolving finite element methods with an artificial tangential velocity for mean curvature flow and {{Willmore}} flow},author={Hu, Jiashun and Li, Buyang},date={2022},volume={152},number={1},pages={127--181},doi={10.1007/s00211-022-01309-9},langid={english},journal={Numerische Mathematik},year={2022},dimensions={true},}
