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Predictor-corrector, BGN-based parametric finite element methods for surface diffusion

Wei Jiang, Chunmei Su, Ganghui Zhang, Lian Zhang

TL;DR

The paper addresses accurate numerical simulation of surface diffusion flows by introducing a predictor-corrector time-stepping strategy for the BGN parametric finite element method, achieving second-order temporal accuracy for isotropic planar diffusion while preserving mesh equidistribution and avoiding mesh regularization. The approach is extended to curve shortening flow, area-preserving variants, anisotropic surface diffusion, and diffusion on surfaces in ${\mathbb R}^3$, with a rigorous treatment of well-posedness and mesh properties. Extensive numerical tests show that the predictor-corrector BGN schemes deliver robust second-order convergence and superior accuracy compared to existing second-order methods, across 2D and 3D geometric evolutions. The results provide a practical, efficient toolkit for simulating geometric flows with high fidelity in materials science and computational geometry, with future work aimed at higher-order, structure-preserving schemes and broader flow classes.

Abstract

We present a novel parametric finite element approach for simulating the surface diffusion of curves and surfaces. Our core strategy incorporates a predictor-corrector time-stepping method, which enhances the classical first-order temporal accuracy to achieve second-order accuracy. Notably, our new method eliminates the necessity for mesh regularization techniques, setting it apart from previously proposed second-order schemes by the authors (J. Comput. Phys. 514 (2024) 113220). Moreover, it maintains the long-term mesh equidistribution property of the first-order scheme. The proposed techniques are readily adaptable to other geometric flows, such as (area-preserving) curve shortening flow and surface diffusion with anisotropic surface energy. Comprehensive numerical experiments have been conducted to validate the accuracy and efficiency of our proposed methods, demonstrating their superiority over previous schemes.

Predictor-corrector, BGN-based parametric finite element methods for surface diffusion

TL;DR

The paper addresses accurate numerical simulation of surface diffusion flows by introducing a predictor-corrector time-stepping strategy for the BGN parametric finite element method, achieving second-order temporal accuracy for isotropic planar diffusion while preserving mesh equidistribution and avoiding mesh regularization. The approach is extended to curve shortening flow, area-preserving variants, anisotropic surface diffusion, and diffusion on surfaces in , with a rigorous treatment of well-posedness and mesh properties. Extensive numerical tests show that the predictor-corrector BGN schemes deliver robust second-order convergence and superior accuracy compared to existing second-order methods, across 2D and 3D geometric evolutions. The results provide a practical, efficient toolkit for simulating geometric flows with high fidelity in materials science and computational geometry, with future work aimed at higher-order, structure-preserving schemes and broader flow classes.

Abstract

We present a novel parametric finite element approach for simulating the surface diffusion of curves and surfaces. Our core strategy incorporates a predictor-corrector time-stepping method, which enhances the classical first-order temporal accuracy to achieve second-order accuracy. Notably, our new method eliminates the necessity for mesh regularization techniques, setting it apart from previously proposed second-order schemes by the authors (J. Comput. Phys. 514 (2024) 113220). Moreover, it maintains the long-term mesh equidistribution property of the first-order scheme. The proposed techniques are readily adaptable to other geometric flows, such as (area-preserving) curve shortening flow and surface diffusion with anisotropic surface energy. Comprehensive numerical experiments have been conducted to validate the accuracy and efficiency of our proposed methods, demonstrating their superiority over previous schemes.

Paper Structure

This paper contains 12 sections, 3 theorems, 45 equations, 16 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. A predictor-corrector second-order algorithm for planar isotropic surface diffusion flow
  3. Extensions to other geometric flows
  4. For curve shortening flow
  5. For anisotropic surface diffusion flow of planar curves
  6. For surface diffusion flow of surfaces in ${\mathbb R}^3$
  7. Numerical results
  8. For isotropic surface diffusion flow of planar curves
  9. For curve shortening flow
  10. For anisotropic surface diffusion flow of planar curves
  11. For surface diffusion flow of surfaces in ${\mathbb R}^3$
  12. Conclusions

Key Result

Theorem 2.1

For $m\ge 1$, we assume that the discrete polygon $\widetilde{\Gamma}^{m+\frac{1}{2}}:=\widetilde{\mathbf{X}}^{m+\frac{1}{2}}$ satisfies the following conditions within the BGN/PC algorithm: Then, the BGN/PC scheme SDF:BGN/PC is well-posed, i.e., there exists a unique solution $(\mathbf{X}^{m+1},\kappa^{m+1})\in [V^h]^2\times V^h$ to SDF:BGN/PC.

Figures (16)

  • Figure 1: Log-log plot of the numerical errors of the classical BGN scheme \ref{['SDF:BGN1']}, the BGN/CNLF scheme \ref{['SDF:BGN2']}, the BGN/BDF2 scheme \ref{['SDF:BDF2']} and the BGN/PC scheme \ref{['SDF:BGN/PC']} for solving the isotropic planar SDF associated with an initially elliptic curve at three time levels: (a) $T=0.05$, (b) $T=0.5$, (c) $T=5$.
  • Figure 2: Evolution of several geometric quantities using the classical BGN scheme \ref{['SDF:BGN1']}, the BGN/CNLF scheme \ref{['SDF:BGN2']}, the BGN/BDF2 scheme \ref{['SDF:BDF2']} and the BGN/PC scheme \ref{['SDF:BGN/PC']}: (a) normalized perimeter; (b) relative area loss; (c) mesh ratio, where the initial curve is an ellipse, with discretization parameters set as $N=80$ and $\tau=1/160$.
  • Figure 3: Evolution of the curves and the corresponding geometric quantities by using the BGN/PC scheme \ref{['SDF:BGN/PC']} for three different initial curves: a rectangular curve (first row); a 'flower' curve (second row); a non-convex curve (third row), where the discretization parameters are set as $N=320$, $\tau=1/80,1/160,1/320$.
  • Figure 4: Evolution of an elliptic curve driven by the CSF using the BGN/PC scheme: (a) snapshots of the evolution; (b) evolution of the normalized perimeter and relative area loss; (c) temporal accuracy of the scheme at different time instances $T=0.05,0.25,0.75$, with the spatial mesh size fixed as $N=10^4$.
  • Figure 5: Evolution of an elliptic curve driven by the AP-CSF via the BGN/PC scheme: (a) snapshots of the evolution; (b) evolution of the normalized perimeter and relative area loss; (c) evolution of the mesh ratio function; (d) temporal convergence rates at $T=0.05,0.5,2$, with the spatial mesh size fixed as $N=10^4$.
  • ...and 11 more figures

Theorems & Definitions (18)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1: Well-posedness
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1: Well-posedness
  • proof
  • Example 4.1: Convergence order test
  • Example 4.2: Comparison of computational cost
  • ...and 8 more