NEW ARTIFICIAL TANGENTIAL MOTIONS FOR PARAMETRIC FINITE ELEMENT APPROXIMATION OF SURFACE EVOLUTION

A new class of parametric finite element methods, with a new type of artificial tangential velocity constructed at the continuous level, is proposed for solving surface evolution under geometric flows. The method is constructed by coupling the normal velocity of the geometric flow with an artificial tangential velocity determined by a harmonic map from a fixed reference surface M to the unknown surface (t), formulated at the continuous level as a system of geometric partial differential equations in terms of a Lagrange multiplier. Since the harmonic map is almost angle-preserving, the new method could preserve the mesh quality, i.e., the shapes of the triangles, as long as the mesh quality of the reference surface is good. Extensive numerical experiments and benchmark examples are presented to demonstrate the convergence of the proposed method and the advantages of the method in preserving the mesh quality of the surfaces for mean curvature flow and surface diffusion, in comparison with other available methods such as the parametric finite element methods proposed by Barrett, Garcke, and N urnberg in 2008 and the DeTurck flow techniques proposed by Elliott and Fritz in 2017.