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Error analysis for temporal second-order finite element approximations of axisymmetric mean curvature flow of genus-1 surfaces

Meng Li, Lining Wang, Yiming Wang

TL;DR

This work develops and analyzes two second-order parametric finite element schemes, Crank-Nicolson and BDF2, for genus-1 axisymmetric mean curvature flow reduced to a 1D generating curve with DeTurck-type tangential motion to preserve mesh quality. Under suitable regularity and time-space constraints, the authors prove optimal $L^2$ and $H^1$ error bounds and a $H^1$-convergence superiority for the fully discrete solutions, along with a rigorous initialization strategy. Numerical experiments on torus-like and spiral genus-1 surfaces validate the theoretical rates and reveal significant mesh-quality advantages over first-order schemes and BGN-based second-order methods. The results provide a solid theoretical and practical foundation for stable, accurate simulations of axisymmetric geometric evolutions and open avenues for general boundary conditions and higher-order temporal methods. Overall, the paper advances the reliability and efficiency of curvature-flow simulations in axisymmetric, genus-1 settings with rigorous error control.

Abstract

Existing studies on the convergence of numerical methods for curvature flows primarily focus on first-order temporal schemes. In this paper, we establish a novel error analysis for parametric finite element approximations of genus-1 axisymmetric mean curvature flow, formulated using two classical second-order time-stepping methods: the Crank-Nicolson method and the BDF2 method. Our results establish optimal error bounds in both the L^2-norm and H^1-norm, along with a superconvergence result in the H^1-norm for each fully discrete approximation. Finally, we perform convergence experiments to validate the theoretical findings and present numerical simulations for various genus-1 surfaces. Through a series of comparative experiments, we also demonstrate that the methods proposed in this paper exhibit significant mesh advantages.

Error analysis for temporal second-order finite element approximations of axisymmetric mean curvature flow of genus-1 surfaces

TL;DR

This work develops and analyzes two second-order parametric finite element schemes, Crank-Nicolson and BDF2, for genus-1 axisymmetric mean curvature flow reduced to a 1D generating curve with DeTurck-type tangential motion to preserve mesh quality. Under suitable regularity and time-space constraints, the authors prove optimal and error bounds and a -convergence superiority for the fully discrete solutions, along with a rigorous initialization strategy. Numerical experiments on torus-like and spiral genus-1 surfaces validate the theoretical rates and reveal significant mesh-quality advantages over first-order schemes and BGN-based second-order methods. The results provide a solid theoretical and practical foundation for stable, accurate simulations of axisymmetric geometric evolutions and open avenues for general boundary conditions and higher-order temporal methods. Overall, the paper advances the reliability and efficiency of curvature-flow simulations in axisymmetric, genus-1 settings with rigorous error control.

Abstract

Existing studies on the convergence of numerical methods for curvature flows primarily focus on first-order temporal schemes. In this paper, we establish a novel error analysis for parametric finite element approximations of genus-1 axisymmetric mean curvature flow, formulated using two classical second-order time-stepping methods: the Crank-Nicolson method and the BDF2 method. Our results establish optimal error bounds in both the L^2-norm and H^1-norm, along with a superconvergence result in the H^1-norm for each fully discrete approximation. Finally, we perform convergence experiments to validate the theoretical findings and present numerical simulations for various genus-1 surfaces. Through a series of comparative experiments, we also demonstrate that the methods proposed in this paper exhibit significant mesh advantages.

Paper Structure

This paper contains 6 sections, 1 theorem, 136 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Temporal second-order schemes and main results
  3. Error estimates for the CN method
  4. Error estimate for the BDF2 method
  5. Numerical results
  6. Conclusions

Key Result

Theorem 2.1

Suppose that eqn:model has a solution $\boldsymbol{x}(\rho, t): \mathbb{I}\times [0, T]\rightarrow \mathbb R^2$, satisfying that as well as Then there exist $\Delta t_0$, $h_0$, $\gamma_1$ and $\gamma_2$, such that when $0<h\leq h_0$, $0<\Delta t\leq \Delta t_0$, $\Delta t\leq \gamma_1\sqrt[4]{h}$ and $h\leq \gamma_2\sqrt{\Delta t}$, the CN method eqn:cn and the BDF2 method eqn:bdf have a unique

Figures (7)

  • Figure 1: Sketch of $\Gamma$ and $\mathcal{S}$, as well as the unit vectors $\boldsymbol{e}_1$, $\boldsymbol{e}_2$ and $\boldsymbol{e}_3$.
  • Figure 7: Evolution for a genus-1 surface generated by a spiral, with the use of the CN method. Plots are at times t = 0, 0.05, 0.1, 0.18. Below we visualize the axisymmetric surfaces generated by the curves.
  • Figure 8: Evolution for a genus-1 surface generated by a spiral, with the use of the BDF2 method. Plots are at times t = 0, 0.2, 0.3, 0.54. Below we visualize the axisymmetric surfaces generated by the curves.
  • Figure 9: Evolution for a genus-1 surface generated by an ellipse, with the use of the CN-BGN method and the CN method. Plots are at times t = 0, 0.8, 1.03.
  • Figure 10: Evolution for a genus-1 surface generated by an ellipse, with the use of the BDF2-BGN method and the BDF2 method. Plots are at times t = 0, 0.8, 1.05.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Remark 1
  • Theorem 2.1
  • proof
  • Remark 2
  • Example 1
  • Example 2
  • Example 3
  • Example 4