Temporal high-order structure-preserving parametric finite element methods for curvature flows

The quality of the mesh is crucial for simulating curvature flows, as standard approaches may fail due to mesh distortion. We first present a series of high-order parametric finite element methods based on the Barrett-Garcke--Nurnberg formulation for solving various types of flows involving curves and surfaces. Extensive numerical experiments demonstrate the anticipated high-order accuracy while maintaining favorable mesh quality throughout the evolution process. Secondly, for flows involving multiple geometric structures, such as surface diffusion—which reduces area while preserving volume—we propose a type of structure-preserving method that incorporates two scalar Lagrange multipliers along with two evolution equations related to area and volume, respectively. These schemes effectively preserve the geometric structure at a fully discrete level. Comprehensive numerical experiments illustrate that our methods achieve the desired temporal accuracy, while simultaneously preserving the geometric structure of the surface diffusion.