Evolving finite element methods with an artificial tangential velocity for mean curvature flow and Willmore flow
ABSTRACT
An artificial tangential velocity is introduced into the evolving surface finite element methods for mean curvature flow and Willmore flow in order to improve the mesh quality in approximating the surface evolution under these geometric flows. The artificial tangential velocity is constructed by considering a limiting situation in the methods proposed by Barrett, Garcke & Nürnberg in 2007 and 2008. The stability of the newly proposed artificial tangential motion 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.