We present a finite element approximation of motion by minus the Laplacian
of curvature and related flows. The proposed scheme covers both the closed curve
case, and the case of curves that are connected via triple junctions. On introducing
a parametric finite element approximation, we prove stability bounds and compare
our scheme with existing approaches. It turns out that the new scheme has very
good properties with respect to area conservation and the equidistribution of mesh
points. We state also an extension of our scheme to Willmore flow of curves and
discuss possible further generalizations.