2025/01/28 by Yifei Li, Wenjun Ying, Li, Yifei +3
Computer Science · Engineering · #35K55 #53C44 #65M12 #65M60 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2501.16660
openalex publication_date 2025/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose and analyze a structure-preserving parametric finite element method (SP-PFEM) for the evolution of closed curves under anisotropic surface diffusion with surface energy density γ(θ). Our primary theoretical contribution establishes that the condition 3γ(θ)-γ(θ-π)≥ 0 is both necessary and sufficient for unconditional energy stability within the framework of local energy estimates. The proposed method introduces a symmetric surface energy matrix \boldsymbolZk(θ) with a stabilizing function k(θ), leading to a conservative weak formulation. Its fully discretization via SP-PFEM rigorously preserves the two geometric structures: enclosed area conservation and energy dissipation unconditionally under our energy stability condition. Numerical results are reported to demonstrate the efficiency and accuracy of the proposed method, along with its area conservation and energy dissipation properties.