  The main goal of this paper is to present results of comparison study for the
level set and direct Lagrangian methods for computing evolution of the Willmore
flow of embedded planar curves. To perform such a study we construct new
numerical approximation schemes for both Lagrangian as well as level set
methods based on semi-implicit in time and finite/complementary volume in space
discretizations. The Lagrangian scheme is stabilized in tangential direction by
the asymptotically uniform grid point redistribution. Both methods are
experimentally second order accurate. Moreover, we show precise coincidence of
both approaches in case of various elastic curve evolutions provided that
solving the linear systems in semi-implicit level set method is done in a
precise way, redistancing is performed occasionally and the influence of
boundary conditions on the level set function is eliminated.
