We propose and analyze a structure-preserving parametric finite element method (SP-PFEM) for the evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy density \(\gamma (\varvec{n})\), where \(\varvec{n}\in \mathbb {S}^1\) represents the outward unit normal vector. We begin with the anisotropic surface diffusion which possesses two well-known geometric structures—area conservation and energy dissipation—during the evolution of the closed curve. By introducing a novel surface energy matrix \(\varvec{G}_k(\varvec{n})\) depending on \(\gamma (\varvec{n})\) and the Cahn-Hoffman \(\varvec{\xi }\)-vector as well as a nonnegative stabilizing function \(k(\varvec{n})\), we obtain a new conservative geometric partial differential equation and its corresponding variational formulation for the anisotropic surface diffusion. Based on the new weak formulation, we propose a full discretization by adopting the parametric finite element method for spatial discretization and a semi-implicit temporal discretization with a proper and clever approximation for the outward normal vector. Under a mild and natural condition on \(\gamma (\varvec{n})\), we can prove that the proposed full discretization is structure-preserving, i.e. it preserves the area conservation and energy dissipation at the discretized level, and thus it is unconditionally energy stable. The proposed SP-PFEM is then extended to simulate the evolution of a close curve under other anisotropic geometric flows including anisotropic curvature flow and area-conserved anisotropic curvature flow. Extensive numerical results are reported to demonstrate the efficiency and unconditional energy stability as well as good mesh quality (and thus no need to re-mesh during the evolution) of the proposed SP-PFEM for simulating anisotropic geometric flows.

我们提出并分析了一种结构保持参数有限元方法（SP-PFEM），用于在具有任意各向异性表面能量密度\(\gamma (\varvec{n})\) 的不同几何流动下闭合曲线的演化，其中\ (\varvec{n}\in \mathbb {S}^1\)表示向外的单位法线向量。我们从各向异性表面扩散开始，它在闭合曲线的演化过程中具有两个众所周知的几何结构——面积守恒和能量耗散。通过引入一个新颖的表面能量矩阵\(\varvec{G}_k(\varvec{n})\)取决于\(\gamma (\varvec{n})\)和 Cahn-Hoffman \(\varvec{\ xi }\) -向量以及非负稳定函数\(k(\varvec{n})\)，我们得到了一个新的保守几何偏微分方程及其相应的各向异性表面扩散的变分公式。基于新的弱公式，我们提出了一种完全离散化，采用参数有限元方法进行空间离散化，并采用对向外法向量进行适当且巧妙的近似的半隐式时间离散化。在\(\gamma (\varvec{n})\)上温和自然的条件下，我们可以证明所提出的完全离散化是结构保持的，即它在离散水平上保持了面积守恒和能量耗散，从而它是无条件能量稳定的。然后将所提出的 SP-PFEM 扩展到模拟其他各向异性几何流（包括各向异性曲率流和面积守恒各向异性曲率流）下闭合曲线的演化。大量的数值结果证明了所提出的用于模拟各向异性几何流的 SP-PFEM 的效率和无条件能量稳定性以及良好的网格质量（因此在演化过程中无需重新网格划分）。