Discrete duality finite volume method with tangential redistribution of points for surfaces evolving by mean curvature

Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 811 07 Bratislava, Slovak Republic

^* Corresponding author: tomek@math.sk

Received: 11 October 2018
Accepted: 6 June 2019

Abstract

We propose a new discrete duality finite volume method for solving mean curvature flow of surfaces in ℝ³. In the cotangent scheme, which is widely used discretization of Laplace–Beltrami operator, a two-dimensional surface is usually approximated by a triangular mesh. In the cotangent scheme the unknowns are the vertices of the triangulation. A finite volume around each vertex is constructed as a surface patch bounded by a piecewise linear curve with nodes in the midpoints of the neighbouring edges and a representative point of each adjacent triangle. The basic idea of our new approach is to include the representative points into the numerical scheme as supplementary unknowns and generalize discrete duality finite volume method from ℝ² to 2D surfaces embedded in ℝ³. To improve the quality of the mesh we use an area-oriented tangential redistribution of the grid points. We derive the numerical scheme for both closed surfaces and surfaces with boundary, and present numerical experiments. Surface evolution models are applied to construction of minimal surfaces with given set of boundary curves.