A perimeter-decreasing and area-conserving algorithm for surface diffusion flow of curves

Abstract

A fully discrete finite element method, based on a new weak formulation and a new time-stepping scheme, is proposed for the surface diffusion flow of closed curves in the two-dimensional plane. It is proved that the proposed method can preserve two geometric structures simultaneously at the discrete level, i.e., the perimeter of the curve decreases in time while the area enclosed by the curve is conserved. Numerical examples are provided to demonstrate the convergence of the proposed method and the effectiveness of the method in preserving the two geometric structures.

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Cited by in corpus (10)

• A structure-preserving parametric finite element method for surface diffusion
• Volume-preserving parametric finite element methods for axisymmetric geometric evolution equations
• Structure-preserving discretizations of two-phase Navier-Stokes flow using fitted and unfitted approaches
• A structure-preserving finite element approximation of surface diffusion for curve networks and surface clusters
• A structure preserving front tracking finite element method for the Mullins--Sekerka problem
• 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
• Unfitted finite element methods for axisymmetric two-phase flow
• Isoparametric finite element methods for mean curvature flow and surface diffusion
• A structure-preserving finite element method for the multi-phase Mullins-Sekerka problem with triple junctions