Geometric evolutions can arise as simple models or fundamental building blocks in various applications with moving boundaries and time-dependent domains, such as grain boundaries in materials or deforming cell boundaries. Mesh-based methods require adaptation and smoothing, particularly in the case of strong deformations. We consider finite element schemes based on classical approaches for geometric evolution equations but augmented with the gradient of the Dirichlet energy or a variant of it, which is known to produce a tangential mesh movement beneficial for the mesh quality. We focus on the one-dimensional case, where convergence of semi-discrete schemes can be proved, and discuss two cases. For networks forming triple junctions, it is desirable to keep the impact of any additional, mesh smoothing terms on the geometric evolution as small as possible, which can be achieved with a perturbation approach. Regarding the elastic flow of curves, the Dirichlet energy can serve as a replacement of the usual penalty in terms of the length functional in that, modulo rescaling, it yields the same minimisers in the long run.
