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Stable BDF time discretization of BGN-based parametric finite element methods for geometric flows

Wei Jiang, Chunmei Su, Ganghui Zhang

TL;DR

This work develops high-order in time parametric finite element methods for geometric flows by embedding backward differentiation formula (BDF) time discretizations into the classical BGN framework. By carefully constructing a predictor polygon $\widetilde{\Gamma}^{m+1}$ from lower-order BGN/BDF steps, the BGN/BDF$k$ schemes achieve temporal orders $k=2,3,4$ while preserving mesh quality and enabling efficient linear solves at each step. The methods are demonstrated on curve and surface evolutions, including CSF, AP-CSF, G-MCF, WF, MCF, and SDF, with extensive numerical tests showing $k$-th order temporal convergence, stable mesh distribution, and improved geometric fidelity (e.g., reduced volume loss for SDF and accurate pinch-off times). The results indicate strong potential for robust, high-accuracy simulations of a broad class of geometric flows, with future work addressing perimeter/area preservation in certain flows and extensions to additional models.

Abstract

We propose a novel class of temporal high-order parametric finite element methods for solving a wide range of geometric flows of curves and surfaces. By incorporating the backward differentiation formulae (BDF) for time discretization into the BGN formulation, originally proposed by Barrett, Garcke, and Nürnberg (J. Comput. Phys., 222 (2007), pp.~441--467), we successfully develop high-order BGN/BDF$k$ schemes. The proposed BGN/BDF$k$ schemes not only retain almost all the advantages of the classical first-order BGN scheme such as computational efficiency and good mesh quality, but also exhibit the desired $k$th-order temporal accuracy in terms of shape metrics, ranging from second-order to fourth-order accuracy. Furthermore, we validate the performance of our proposed BGN/BDF$k$ schemes through extensive numerical examples, demonstrating their high-order temporal accuracy for various types of geometric flows while maintaining good mesh quality throughout the evolution.

Stable BDF time discretization of BGN-based parametric finite element methods for geometric flows

TL;DR

This work develops high-order in time parametric finite element methods for geometric flows by embedding backward differentiation formula (BDF) time discretizations into the classical BGN framework. By carefully constructing a predictor polygon from lower-order BGN/BDF steps, the BGN/BDF schemes achieve temporal orders while preserving mesh quality and enabling efficient linear solves at each step. The methods are demonstrated on curve and surface evolutions, including CSF, AP-CSF, G-MCF, WF, MCF, and SDF, with extensive numerical tests showing -th order temporal convergence, stable mesh distribution, and improved geometric fidelity (e.g., reduced volume loss for SDF and accurate pinch-off times). The results indicate strong potential for robust, high-accuracy simulations of a broad class of geometric flows, with future work addressing perimeter/area preservation in certain flows and extensions to additional models.

Abstract

We propose a novel class of temporal high-order parametric finite element methods for solving a wide range of geometric flows of curves and surfaces. By incorporating the backward differentiation formulae (BDF) for time discretization into the BGN formulation, originally proposed by Barrett, Garcke, and Nürnberg (J. Comput. Phys., 222 (2007), pp.~441--467), we successfully develop high-order BGN/BDF schemes. The proposed BGN/BDF schemes not only retain almost all the advantages of the classical first-order BGN scheme such as computational efficiency and good mesh quality, but also exhibit the desired th-order temporal accuracy in terms of shape metrics, ranging from second-order to fourth-order accuracy. Furthermore, we validate the performance of our proposed BGN/BDF schemes through extensive numerical examples, demonstrating their high-order temporal accuracy for various types of geometric flows while maintaining good mesh quality throughout the evolution.
Paper Structure (7 sections, 1 theorem, 48 equations, 15 figures, 3 algorithms)

This paper contains 7 sections, 1 theorem, 48 equations, 15 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Review of classical BGN scheme
  3. High-order in time, BGN-based algorithms
  4. Numerical results
  5. Curve evolution
  6. Surface evolution
  7. Conclusion

Key Result

theorem 1

For $k=2,3,4$, $m\ge k-1$, assume that the polygon $\widetilde{\Gamma}^{m+1}$ in the BGN/BDF$k$ schemes CSF:BDFk satisfies the following two conditions: (i) At least two vectors in $\{\widetilde{\mathbf{h}}^{m+1}_j\}_{j=1}^{N}$ are not parallel, i.e., (ii) No vertices degenerate on $\widetilde{\Gamma}^{m+1}$, i.e., Then the above BGN/BDF$k$ schemes CSF:BDFk are well-posed, i.e., there exists a u

Figures (15)

  • Figure 1: Log-log plot of the manifold distance errors via the classical BGN1 scheme and the BGN/BDF$k$ schemes ($2\le k\le 4$) at time $T=0.25$ for solving CSF with two various initial curves: (a) unit circle and (b) ellipse; and the evolution of the mesh distribution function $\Psi(t)$: (c) unit circle and (d) ellipse, where $N=640$ and $\tau=1/1280$.
  • Figure 2: Log-log plot of the manifold distance errors at time $T=0.25$ for solving CSF with a unit circle being initial curve: (a) uniform initial distribution; (b) nonuniform initial distribution; and the evolution of the mesh distribution function $\Psi(t)$: (c) uniform initial distribution and (d) nonuniform initial distribution, where $N=640$ and $\tau=1/1280$.
  • Figure 3: Log-log plot of the manifold distance errors for solving AP-CSF with ellipse being its initial shape at two different times: (a) $T=0.25$, (b) $T=1$. (c) The corresponding evolution of the mesh distributional function $\Psi(t)$, where $N=640$ and $\tau=1/1280$.
  • Figure 4: Evolution of AP-CSF for the 'flower' initial curve by using BGN/BDF3 algorithm with various choices of predictions $\widetilde{\mathbf{X}}^{m+1}$. Top row: Algorithm \ref{['BGN/BDF3 algorithm']}, i.e., approximating the prediction polygon by lower-order BGN/BDF2 scheme; Bottom row: approximating the prediction polygon by using extrapolation formulae \ref{['exbdf3']}. The parameters are chosen as $N = 80$ and $\tau=1/160$.
  • Figure 5: Evolution of geometric quantities of AP-CSF for 'flower' initial shape by using BGN/BDF3 algorithm: (a1)-(a2) the normalized perimeter; (b1)-(b2) the normalized area loss; (c1)-(c2) the mesh distribution function $\Psi(t)$. Top row: the prediction polygon $\widetilde{\mathbf{X}}^{m+1}$ is obtained by lower-order BGN/BDF2 scheme. Bottom row: $\widetilde{\mathbf{X}}^{m+1}$ is obtained by the extrapolation formulae \ref{['exbdf3']}. The parameters are chosen as $N = 80$ and $\tau=1/160$.
  • ...and 10 more figures

Theorems & Definitions (10)

  • theorem 1: Well-posedness
  • proof
  • Remark 3.1
  • Example 4.1: Convergence rate of BGN/BDFk scheme for CSF
  • Example 4.2: Extension to AP-CSF
  • Remark 4.1
  • Example 4.3: Extension to G-MCF
  • Example 4.4: Extension to WF
  • Example 4.5: Extension to MCF
  • Example 4.6: Extension to SDF