Discrete Gradient Flows for General Curvature Energies

Abstract

We consider the numerical approximation of the $L^2$--gradient flow of general curvature energies $\int G(|\vec\varkappa|)$ for a curve in $\mathbb{R^d}$, $d\geq 2$. Here the curve can be either closed, or it can be open and clamped at the end points. These general curvature energies, and the considered boundary conditions, appear in the modeling of the power loss within an optical fiber. We present two alternative finite element approximations, both of which admit a discrete gradient flow structure. Apart from being stable, in addition, one of the methods satisfies an equidistribution property. Numerical results demonstrate the robustness and the accuracy of the proposed methods.