| Abstract: |
| The Willmore energy has widespread applications in differential geometry, cell membranes, optical lenses, materials science, among others. The Willmore flow, as the $L^2$ gradient flow dissipating the Willmore energy, serves as a fundamental tool for its analysis. Despite its importance, the development of energy-stable parametric methods for the Willmore flow remains open. In this talk, we present a novel energy-stable numerical approximation for the Willmore flow. We begin by introducing our method for planar curves, then demonstrating the underlying ideas -- the new transport equation and the time derivative of the mean curvature, that ensure energy stability. Finally, we discuss the extension of our approach to surfaces in 3D. |
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