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Convergence analysis for the Barrett--Garcke--Nurnberg method of transport type for evolving curves

Genming Bai, Harald Garcke, Shravan Veerapaneni

TL;DR

This work establishes the first convergence result for a fully discrete Barrett--Garcke--Nürnberg scheme solving curve evolution under general, transport-dominated flows. By leveraging a projection-error framework and Gauss–Lobatto mass lumping, the authors derive discrete transport-energy estimates that compensate for the lack of $H^1$ parabolicity. They prove a sub-optimal $L^2$ convergence rate $\| \hat{e}_h^{m} \|_{L^2} \le C(\tau + h^k)$ for degree $k\ge 3$, under a mild time-step constraint, and show stability through high-order tangential estimates and an intrinsic $H^2$ stability of the discrete Laplacian. The results advance the theory of parametric finite element methods for transport-type interface evolution and provide a robust discretization framework for moving interfaces driven by general flows, with potential extensions to higher dimensions and related interface problems.

Abstract

In this paper, we propose a Barrett-Garcke-Nurnberg (BGN) method for evolving geometries under general flows and present the corresponding convergence analysis. Specifically, we examine the scenario where a closed curve evolves according to a prescribed background velocity field. Unlike mean curvature flow and surface diffusion, where the evolution velocities inherently exhibit parabolicity, this case is dominated by transport which poses a significant difficulty in establishing convergence proofs. To address the challenges imposed by this transport-dominant nature, we derive several discrete energy estimates of the transport type on discretized polynomial surfaces within the framework of the projection error. The use of the projection error is indispensable as it provides crucial additional stability through its orthogonality structure. We prove that the proposed method converges sub-optimally in the L2 norm, and this is the first convergence proof for a fully discrete numerical method solving the evolution of curves driven by general flows.

Convergence analysis for the Barrett--Garcke--Nurnberg method of transport type for evolving curves

TL;DR

This work establishes the first convergence result for a fully discrete Barrett--Garcke--Nürnberg scheme solving curve evolution under general, transport-dominated flows. By leveraging a projection-error framework and Gauss–Lobatto mass lumping, the authors derive discrete transport-energy estimates that compensate for the lack of parabolicity. They prove a sub-optimal convergence rate for degree , under a mild time-step constraint, and show stability through high-order tangential estimates and an intrinsic stability of the discrete Laplacian. The results advance the theory of parametric finite element methods for transport-type interface evolution and provide a robust discretization framework for moving interfaces driven by general flows, with potential extensions to higher dimensions and related interface problems.

Abstract

In this paper, we propose a Barrett-Garcke-Nurnberg (BGN) method for evolving geometries under general flows and present the corresponding convergence analysis. Specifically, we examine the scenario where a closed curve evolves according to a prescribed background velocity field. Unlike mean curvature flow and surface diffusion, where the evolution velocities inherently exhibit parabolicity, this case is dominated by transport which poses a significant difficulty in establishing convergence proofs. To address the challenges imposed by this transport-dominant nature, we derive several discrete energy estimates of the transport type on discretized polynomial surfaces within the framework of the projection error. The use of the projection error is indispensable as it provides crucial additional stability through its orthogonality structure. We prove that the proposed method converges sub-optimally in the L2 norm, and this is the first convergence proof for a fully discrete numerical method solving the evolution of curves driven by general flows.

Paper Structure

This paper contains 24 sections, 16 theorems, 162 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Numerical scheme and the main theorem
  3. Preliminaries
  4. Notations
  5. Lifts and interpolations
  6. Shape regularity constants and basic approximation properties
  7. Induction hypothesis
  8. Geometric relations
  9. Stability estimates
  10. Consistency error
  11. Naive estimates for linear forms
  12. High-order tangential stability estimates
  13. Estimates for $\delta {\hat{e}_h^m}$
  14. Refined estimates for linear forms
  15. Proof of Theorem \ref{['thm:main']}
  16. ...and 9 more sections

Key Result

theorem 2.1

We assume that the flow map $X: \Gamma^0\times[0,T]\rightarrow {\mathbb R}^2$ generated by eq:evol of a closed curve and its inverse map $X(\cdot,t)^{-1}:\Gamma(t)\rightarrow\Gamma^0$ are both sufficiently smooth, and the initial polynomial $\Gamma_h^0$ satisfies the approximation properties P0--P1. where the constant $C$ is independent of $\tau$ and $h$ (but may depend on $c$ and $T$).

Figures (3)

  • Figure 1: Rate of convergence
  • Figure 2: Evolution of a curve $\Gamma(t)$ under velocity field $v$ in \ref{['eq:exp_v']}
  • Figure 3: Evolution of a curve $\Gamma(t)$ under velocity field $v$ in \ref{['eq:exp_v1']}

Theorems & Definitions (32)

  • remark thmcounterremark
  • theorem 2.1: Convergence of the transport BGN method
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • remark thmcounterremark
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • remark thmcounterremark
  • ...and 22 more