Stable fully discrete finite element methods with BGN tangential motion for Willmore flow of planar curves

Abstract

We propose and analyze stable finite element approximations for Willmore flow of planar curves. The presented schemes are based on a novel weak formulation which combines an evolution equation for curvature with the curvature formulation originally proposed by Barrett, Garcke and Nürnberg (BGN) in \cite{BGN07}. Under discretization in space with piecewise linear elements this leads to a stable continuous-in-time semidiscrete scheme, which retains the equidistribution property from the BGN methods. Furthermore, two fully discrete schemes can be shown to satisfy unconditional energy stability estimates. Numerical examples are presented to showcase the good properties of the introduced schemes, including an asymptotic equidistribution of vertices.

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• A Stable Parametric Finite Element Discretization of Two-Phase Navier--Stokes Flow
• Computational p-Willmore Flow with Conformal Penalty
• Arbitrary Lagrangian-Eulerian finite element approximations for axisymmetric two-phase flow
• Finite element schemes with tangential motion for fourth order geometric curve evolutions in arbitrary codimension