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Convergence analysis of three semi-discrete numerical schemes for nonlocal geometric flows including perimeter terms

Jiang Wei, Su Chunmei, Zhang Ganghui

TL;DR

The authors study three semi-discrete schemes for nonlocal plane-curve flows with normal velocity $\mathcal V=(\kappa-f(L))\mathcal N$, proving quadratic $H^1$-convergence for the finite-difference scheme and linear $H^1$-convergence for two finite-element variants (one with tangential motion). A unifying insight shows the nonlocal term error can be controlled by the local term error, and convergence under manifold distance is linked to $L^{\infty}$-norm convergence. Comprehensive numerical experiments validate the theoretical rates across area-preserving and area-changing nonlocal flows, with the tangential-motion scheme significantly improving mesh quality and equidistribution. The work also establishes a foundational connection between standard norms and shape-based metrics, with implications for robust geometric-flow simulations.

Abstract

We present and analyze three distinct semi-discrete schemes for solving nonlocal geometric flows incorporating perimeter terms. These schemes are based on the finite difference method, the finite element method, and the finite element method with a specific tangential motion. We offer rigorous proofs of quadratic convergence under $H^1$-norm for the first scheme and linear convergence under $H^1$-norm for the latter two schemes. All error estimates rely on the observation that the error of the nonlocal term can be controlled by the error of the local term. Furthermore, we explore the relationship between the convergence under $L^\infty$-norm and manifold distance. Extensive numerical experiments are conducted to verify the convergence analysis, and demonstrate the accuracy of our schemes under various norms for different types of nonlocal flows.

Convergence analysis of three semi-discrete numerical schemes for nonlocal geometric flows including perimeter terms

TL;DR

The authors study three semi-discrete schemes for nonlocal plane-curve flows with normal velocity , proving quadratic -convergence for the finite-difference scheme and linear -convergence for two finite-element variants (one with tangential motion). A unifying insight shows the nonlocal term error can be controlled by the local term error, and convergence under manifold distance is linked to -norm convergence. Comprehensive numerical experiments validate the theoretical rates across area-preserving and area-changing nonlocal flows, with the tangential-motion scheme significantly improving mesh quality and equidistribution. The work also establishes a foundational connection between standard norms and shape-based metrics, with implications for robust geometric-flow simulations.

Abstract

We present and analyze three distinct semi-discrete schemes for solving nonlocal geometric flows incorporating perimeter terms. These schemes are based on the finite difference method, the finite element method, and the finite element method with a specific tangential motion. We offer rigorous proofs of quadratic convergence under -norm for the first scheme and linear convergence under -norm for the latter two schemes. All error estimates rely on the observation that the error of the nonlocal term can be controlled by the error of the local term. Furthermore, we explore the relationship between the convergence under -norm and manifold distance. Extensive numerical experiments are conducted to verify the convergence analysis, and demonstrate the accuracy of our schemes under various norms for different types of nonlocal flows.
Paper Structure (11 sections, 8 theorems, 87 equations, 9 figures)

This paper contains 11 sections, 8 theorems, 87 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Finite difference method
  4. Finite element methods
  5. FEM with only the normal motion
  6. FEM with tangential motions
  7. Convergence under manifold distance
  8. Numerical results
  9. Accuracy test
  10. Dynamics and evolution of geometric quantities
  11. Conclusions

Key Result

Lemma 2.1

Deckelnick2022 Under Assumption 2.1, there exists $h_0>0$ such that for $0<h\le h_0$, we have where $\tau_{j+1/2}:=\frac{\tau_j+\tau_{j+1}}{|\tau_j+\tau_{j+1}|}$ represents the averaged vertex tangent.

Figures (9)

  • Figure 1: Illustration of (a) the definition of $\Gamma_{X},\Gamma_{Y},\Gamma_{\delta_w}^{\mathrm{int}}$ and $\Gamma_{\delta_w}^\mathrm{{ext}}$, (b) the comparison of the projection distance $\delta_w$ and the function $L^\infty$-norm $\|X-Y\|_{L^\infty}$.
  • Figure 2: Numerical errors under different norms for the FDM \ref{['Discrete,FDM']} at $T=1/4$: (a) Case 1; (b) Case 2; (c) Case 3.
  • Figure 3: Numerical errors under different norms of the FEM \ref{['Discrete,FEM']} at $T=1/4$: (a) Case 1; (b) Case 2; (c) Case 3.
  • Figure 4: Numerical errors under different norms of the FEM-TM \ref{['Discrete,FEMt']} at $T=1/4$: (a) Case 1 with $\alpha=1$; (b) Case 1 with $\alpha=0.5$; (c) Case 2 with $\alpha=1$; (d) Case 3 with $\alpha=1$.
  • Figure 5: Snapshots of the curve evolution using the FDM (first row), FEM (second row) and FEM-TM (third row) with $\alpha=1$ for Case 1. The parameters are chosen as $N=80$ and $\tau=1/160$.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Lemma 2.1
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Definition 4.1
  • Remark 4.2
  • Theorem 4.3
  • Lemma 4.4
  • Definition 4.5
  • Theorem 4.6
  • ...and 2 more