A FEM for Mean Curvature and Related Flows

Figure 4. Singularity formation when boundary conditions are such that the catenoid solution does not exist. 7 Figure 3. Limiting catenoid surface obtained from an initially cylindrical surface. is the minimal surface with the same xed boundary. Figure 3 shows the required catenoid. This surface was obtained as the limiting surface of a FEM evolution starting from a cylindrical initial surface and evolving until no change could be seen in the surface. If the distance between the rings is increased, then the catenoid solution does not exist. The mean-curvature evolution then ows to the surface with a singularity as in Figure 4. It has been shown by Dziuk Dziuk, 1991a] that this singularity occurs at a single point and the behavior of the surface close to the singularity point is at least cubic in nature ((Huisken, 1991a]). It should be noted that numerical experiments of this form have been very helpful in the formulation of a number of theoretic results pertaining to the formation of singularities in the mean-curvature ow (see for instance Huisken, 1991c]). This nal experiment demonstrates a shortcoming of the method and a direction for further research. Triangles near the point of singularity become very long and thin and the corresponding conditioning of the matrix equation grows very large. We are at present considering various tangential ow equations associated with energy considerations (see Hutchinson, 1991]) which will be added to the overall evolution equation to ensure that the triangles remain non-degenerate. 4. CONCLUSION In this paper we have demonstrated a simple nite element method which has been used to study mean-curvature ow, in both the case of singularity formation and in the case of convergence to minimal surfaces. The code has been used to help in the intuitive understanding of mean-curvature ow and singularity formation and has helped in the formulation of a number of theoretic results (see Huisken, 1991c], Huisken, 1991b]). The method has also been used to solve Laplace-Beltrami equations and associated parabolic evolution equations on xed surfaces and we are now in the process of implementing a non-linear parabolic evolution equation on the sphere which is associated with the formation of space-like surfaces in general relativity. In conclusion this method allows us to study a number of naturally arising elliptic and parabolic problems associated with geometric evolution equations. 6 S. Roberts (a) (b) (c) (d) Figure 2. Mean-curvature evolution of an octahedron. (a) Initial …