Numerical solution for the anisotropic Willmore flow of graphs

Abstract

Laplace-Beltrami operator Method of lines Complementary finite volume method Finite difference methodThe Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we prove an energy equality. We approximate the solution numerically by the complementary finite volume method. To show the stability, we re-formulate the resulting scheme in terms of the finite difference method. By using simple framework of the finite difference method (FDM) we show discrete version of the energy equality. The time discretization is done by the method of lines and the resulting system of ODEs is solved by the Runge-Kutta-Merson solver with adaptive integration step. We also show experimental order of convergence as well as results of the numerical experiments, both for several different anisotropies.

A p-Adaptive Local Discontinuous Galerkin Level Set Method for Willmore Flow

“…A semi-implicit numerical scheme for the Willmore flow of graphs with continuous finite element discretization and the convergence analysis has been provided by Deckelnick and Dziuk in [9]. Finite difference discretization for the Willmore flow of graphs has been discussed in [12]. Xu and Shu [19] developed a local discontinuous Galerkin method for Willmore flow of graphs, and time discretization was by the forward Euler method with a suitably small time step for stability.…”

Dynamics of dislocations described as evolving curves interacting with obstacles

“…In addition, the level-set method requires frequent reinitialization (see [32]) of the level-set function maintaining accuracy the the level-set detection.…”