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Numerical solution for the anisotropic Willmore flow of graphs. (English) Zbl 1309.65122

Summary: 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 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.

MSC:

65N08 Finite volume methods for boundary value problems involving PDEs
05C21 Flows in graphs
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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