Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves
Provides theoretical guarantees for a numerical method to compute elastic knots, relevant for researchers in geometric mechanics and knot theory.
The paper proves stability of a semi-implicit numerical scheme for minimizing bending energy of curves within isotopy classes, demonstrating energy decay and maintenance of arclength parametrization. Numerical experiments explore global minimizers (elastic knots).
We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of the bending energy and the tangent-point functional. Based on estimates for the second derivative of the latter and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization. Finally we present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in knot classes, so-called elastic knots.