Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves

Deckelnick and Dziuk (2009) proved a stability bound for a continuous-in-time semidiscrete parametric finite element approximation of Willmore flow/elastic flow of closed curves in R, d ≥ 2. We extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in Barrett, Garcke, and Nürnberg (2007b, 2008b, 2010b), in particular an equidistribution mesh property. For this new approximation, we obtain also a stability bound for a continuous-in-time semidiscrete scheme. Apart from the isotropic situation, we also consider the case of an anisotropic elastic energy. In addition to the evolution of closed curves, we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions.