A Variational Discretization Method for Mean Curvature Flows by the Onsager Principle (CSIAM Transactions on Applied Mathematics, 2025)
The mean curvature flow describes the evolution of a surface (a curve) withnormal velocity proportional to the local mean curvature. It has many applications inmathematics, science and engineering. In this paper, we develop a numerical methodfor mean curvature flows by using the Onsager principle as an approximation tool.We first show that the mean curvature flow can be derived naturally from the Onsagervariational principle. Then we consider a piecewise linear approximation of the curveand derive a discrete geometric flow. The discrete flow is described by a system ofordinary differential equations for the nodes of the discrete curve. We prove that thediscrete system preserve the energy dissipation structure in the framework of the Onsager principle and this implies the energy decreasing property. The ODE system canbe solved by the improved Euler scheme and this leads to an efficient fully discretescheme. We first consider the method for a simple mean curvature flow and then extend it to the volume preserving mean curvature flow and also a wetting problem onsubstrates. Numerical examples show that the method has optimal convergence rateand works well for all the three problems.
Keywords: Mean curvature flow; the Onsager principle; moving finite element method.