We consider stable semidiscrete approximations of parameterized 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 C0 attachment
and C1 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.