Numerical solution for the anisotropic Willmore flow of graphs

TL;DR

This paper develops a numerical method for solving the anisotropic Willmore flow of graphs, deriving a weak solution, proving energy equality, and demonstrating stability and convergence through experiments.

Contribution

It introduces a finite volume and finite difference based numerical scheme for anisotropic Willmore flow of graphs, including stability analysis and convergence results.

Findings

Numerical scheme is stable and convergent.

Experimental results confirm the accuracy for various anisotropies.

Energy equality is preserved in the discrete scheme.

Abstract

The 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 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.