Elastic flow with junctions: Variational approximation and applications to nonlinear splines

We consider stable semidiscrete approximations of parametrized curve networks for gradient flows of elastic type functionals. Here meaningful and relevant conditions at junction points, such as double and triple junctions, need to be derived and suitably discretized. Examples for double junction types are C 0 attachment and C 1 continuity. We develop strong and weak formulations for the elastic flow for curve networks with such junction points. For junctions with three or more curves, the conditions at the junctions are derived here for the first time. Possible applications include a simplified one-dimensional model of geometric biomembranes, as well as nonlinear splines in two and higher dimensions. The numerical results presented in this paper demonstrate the practicality of the introduced finite element approximations. © 2012 World Scientific Publishing Company.