Energy-Stable Full Discretization of the Modified Elastic Flow of Closed Curves

Abstract

An energy-stable full discretization for the modified elastic flow of closed curves is proposed. This is a gradient flow of a modified elastic energy combining bending and Dirichlet energies. The minimization of Dirichlet energy can lead to improved mesh quality. Gradient flows for both isotropic and anisotropic cases are considered. We derive new evolution equations for the parameterization and curvature vector of curves in arbitrary codimension. The proposed formulation is discretized by a parametric finite element method in space and a first-order implicit scheme in time. We establish the unconditional energy stability for the fully discretized scheme. Additionally, the second-order accuracy of the BDF2 scheme is demonstrated. Numerical examples in two and three dimensions illustrate the efficiency, energy stability, and asymptotic mesh distribution of the method for simulating the modified elastic flow.