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

TL;DR

This work addresses the numerical simulation of anisotropic surface diffusion flows on closed surfaces by developing structure-preserving, high-order PFEM schemes. It extends the minimal deformation (MD) formulation to anisotropic SDF and couples it with backward differentiation (BDFk) time stepping, alongside invariant-preserving techniques such as scalar auxiliary variables (SAV) and Lagrange multipliers (LM). Key contributions include a new MD formulation with a stabilizing matrix $Z_k(n)$, the MD-BDFk family, and SAV-MD-BDFk, LM-MD-BDFk, and LM-SAV-MD-BDFk schemes, with proven or demonstrated energy stability and (approximate) volume conservation, validated by extensive numerical experiments across multiple anisotropic energies. The results show improved mesh quality and stable long-time evolution, enabling accurate simulations of anisotropic SDF relevant to materials science applications, while offering flexible options to balance stability, accuracy, and computational cost.

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.

Figures (14)

• Figure 1: An illustration of surface diffusion on a closed surface $\mathcal{S}(t)$ with anisotropic surface energy density $\gamma(\vec{n})$ in three dimensions, where $\vec{n}$ is the unit outward normal vector, $\vec{\tau}_1$ and $\vec{\tau}_2$ are a set of orthonormal basis for the local tangential plane.
• Figure 2: Convergence rates of MD-BDF1 method, MD-BDF2 method and MD-BDF3 method at times $T = 2$ with different surface energy densities: $\gamma(\vec{n})=\sqrt{{n_1^2+n_2^2+2n_3^2}}$, $\gamma(\vec{n})=1+0.125 \left(n_1^3+n_2^3+n_3^3\right)$ and $\gamma(\vec{n})=1+0.05 \left(n_1^4+n_2^4+n_3^4\right)$.
• Figure 3: The relative volume loss of MD-BDFk methods under different surface energy densities: $\gamma(\vec{n})=1+0.05 \left(n_1^4+n_2^4+n_3^4\right)$ and $\gamma(\vec{n})=1+0.5 \left(n_1^4+n_2^4+n_3^4\right)$. Other parameters are chosen as $h = 1.5356\times 10^{-1}$ and $\Delta t = 10^{-3}$.
• Figure 4: The relative volume loss of SAV-MD-BDFk methods under different surface energy densities: $\gamma(\vec{n})=1+0.05 \left(n_1^4+n_2^4+n_3^4\right)$ (the first row) and $\gamma(\vec{n})=1+0.5 \left(n_1^4+n_2^4+n_3^4\right)$ (the second row). The first column: $r=5$. Other parameters are chosen as $h = 1.5356\times 10^{-1}$ and $\Delta t = 10^{-3}$.
• Figure 5: The relative volume loss of VC-LM-MD-BDFk methods under different surface energy densities: $\gamma(\vec{n})=1+0.05 \left(n_1^4+n_2^4+n_3^4\right)$ and $\gamma(\vec{n})=1+0.5 \left(n_1^4+n_2^4+n_3^4\right)$. Other parameters are chosen as $h = 1.5356\times 10^{-1}$ and $\Delta t = 10^{-3}$.

Theorems & Definitions (19)

• Remark 1
• Remark 2
• Remark 3
• Theorem 3.1
• proof
• Remark 4
• Theorem 3.2
• proof
• Remark 5
• Remark 6