Anisotropic mean curvature flow for two-dimensional surfaces in higher codimension: a numerical scheme

In this paper we consider evolution of parametric two-dimensional surfaces by anisotropic mean curvature in arbitrary codimension. After deriving a classical and a weak formulation of the flow, we discretize the problem by the finite element method, provide a fully discrete stable scheme and present numerical tests and examples. Anisotropic mean curvature flow has been widely studied, both analytically and numerically. Much of the research concentrated so far on the case of codimension one, i.e. on the flow for hypersurfaces. Given the amount of work done in this field, we do not attempt to give a list of references but limit ourselves to point to the survey article [9]. There the interested reader can get a flavour of recent progress on some important topics concerning the theoretical and numerical analysis of the mean curvature flow, and the list of references helps finding many of the most important contributions in the field; applications are also briefly discussed. For the case of higher codimension there are some works in the isotropic setting (see for example [22] and [2]), but the anisotropic case seems to have been less studied. The evolution by anisotropic mean curvature of parametric curves in R is investigated by the author in [15]; in [18] a level set approach to motion of manifolds of arbitrary codimension is studied. The survey article [14], on the existence and regularity theory for Cartan functionals (i.e. for general parameter invariant double integrals defined on parametric surfaces with arbitrary codimension), can also be considered as related to our studies. The aim of this work is to extend the algorithm given in [7, §4.2] for the anisotropic mean curvature flow of parametric surfaces in R3 to the the case of arbitrary codimension. The generalization is not quite straightforward, since the formulation of the three-dimensional flow already heavily relies on the fact that the codimension is equal to one. An extension of the algorithm to arbitrary codimension is made possible by embedding the problem in the field of Minkowski (and its related Finsler) geometry. In this context the Euclidean space R is endowed with an additional norm which defines the anisotropic geometry of the space. Concepts of area and content are well defined and come with a range of powerful tools (see for example the beautiful book by Thompson [20], and [1]).