Structure-preserving parametric finite element methods for anisotropic surface diffusion flow with minimal deformation formulation

Abstract: High mesh quality plays a crucial role in maintaining the stability of solutions in geometric flow problems. Duan and Li [Duan & Li, SIAM J. Sci. Comput. 46 (1) (2024) A587-A608] applied the minimal deformation (MD) formulation to propose an artificial tangential velocity determined by harmonic mapping to improve mesh quality. In this work, we extend the method to anisotropic surface diffusion flows, which, similar to isotropic curvature flow, also preserves excellent mesh quality. Furthermore, developing a numerical algorithm for the flow with MD formulation that guarantees volume conservation and energy stability remains a challenging task. We, in this paper, successfully construct several structure-preserving algorithms, including first-order and high-order temporal discretization methods. Extensive numerical experiments show that our methods effectively preserve mesh quality for anisotropic SDFs, ensuring high-order temporal accuracy, volume conservation or/and energy stability.