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Structure-preserving parametric finite element method for curve diffusion based on Lagrange multiplier approaches

Harald Garcke, Wei Jiang, Chunmei Su, Ganghui Zhang

TL;DR

This work develops high-order, structure-preserving parametric finite element methods for curve diffusion by embedding two scalar Lagrange multipliers to enforce perimeter reduction and area preservation. By combining a BGN-compatible continuous formulation with spatial linear finite elements and various high-order time stepping schemes (Crank–Nicolson, BDF), the authors obtain fully discrete schemes that maintain geometric invariants and mesh quality, solvable via Newton iterations. They further address long-time evolution by switching to single-multiplier (AP-type) formulations, ensuring area preservation and improved mesh equidistribution. Extensive numerical tests confirm the expected convergence orders, structure preservation, and robust long-time behavior, highlighting practical benefits for sharp-interface problems and suggesting extensions to 3D and anisotropic flows.

Abstract

We propose a novel formulation for parametric finite element methods to simulate surface diffusion of closed curves, which is also called as the curve diffusion. Several high-order temporal discretizations are proposed based on this new formulation. To ensure that the numerical methods preserve geometric structures of curve diffusion (i.e., the perimeter-decreasing and area-preserving properties), our formulation incorporates two scalar Lagrange multipliers and two evolution equations involving the perimeter and area, respectively. By discretizing the spatial variable using piecewise linear finite elements and the temporal variable using either the Crank-Nicolson method or the backward differentiation formulae method, we develop high-order temporal schemes that effectively preserve the structure at a fully discrete level. These new schemes are implicit and can be efficiently solved using Newton's method. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy, as measured by the manifold distance, while simultaneously preserving the geometric structure of the curve diffusion.

Structure-preserving parametric finite element method for curve diffusion based on Lagrange multiplier approaches

TL;DR

This work develops high-order, structure-preserving parametric finite element methods for curve diffusion by embedding two scalar Lagrange multipliers to enforce perimeter reduction and area preservation. By combining a BGN-compatible continuous formulation with spatial linear finite elements and various high-order time stepping schemes (Crank–Nicolson, BDF), the authors obtain fully discrete schemes that maintain geometric invariants and mesh quality, solvable via Newton iterations. They further address long-time evolution by switching to single-multiplier (AP-type) formulations, ensuring area preservation and improved mesh equidistribution. Extensive numerical tests confirm the expected convergence orders, structure preservation, and robust long-time behavior, highlighting practical benefits for sharp-interface problems and suggesting extensions to 3D and anisotropic flows.

Abstract

We propose a novel formulation for parametric finite element methods to simulate surface diffusion of closed curves, which is also called as the curve diffusion. Several high-order temporal discretizations are proposed based on this new formulation. To ensure that the numerical methods preserve geometric structures of curve diffusion (i.e., the perimeter-decreasing and area-preserving properties), our formulation incorporates two scalar Lagrange multipliers and two evolution equations involving the perimeter and area, respectively. By discretizing the spatial variable using piecewise linear finite elements and the temporal variable using either the Crank-Nicolson method or the backward differentiation formulae method, we develop high-order temporal schemes that effectively preserve the structure at a fully discrete level. These new schemes are implicit and can be efficiently solved using Newton's method. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy, as measured by the manifold distance, while simultaneously preserving the geometric structure of the curve diffusion.
Paper Structure (18 sections, 8 theorems, 73 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 18 sections, 8 theorems, 73 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Connection to the BGN formulation and spatial semi-discretization
  3. Connection to the BGN formulation
  4. Spatial semi-discrete scheme
  5. Time stepping discretization
  6. Backward Euler time discretization
  7. Crank-Nicolson time discretization
  8. Backward differentiation formulae for time discretization
  9. Newton's iteration method and modification for long-time evolution
  10. Perimeter-decreasing or area-preserving scheme with a single Lagrange multiplier
  11. Perimeter-decreasing schemes for long-time evolution
  12. Area-preserving schemes for long-time evolution
  13. Modification for SP-type schemes by AP-type schemes
  14. Numerical results
  15. Cauchy test for the convergence order
  16. ...and 3 more sections

Key Result

Proposition 2.1

\newlabelProp:cont Assuming that $(\mathbf{X},\kappa,\lambda,\eta)$ constitutes a solution to Lag and $\Gamma(t)=\mathbf{X}({\mathbb I},t)$ represents a closed $C^2$-curve, it follows that $\lambda=\eta=0$ if $\kappa(\cdot, t)$ is not constant with respect to the spatial variable.

Figures (8)

  • Figure 5.1: Log-log plot of the numerical errors with an ellipse as its initial shape at time $T=0.25$. For the SP-Euler scheme \ref{['Lageuler']}, the Cauchy path is chosen as $\tau=h^2$. For the SP-CN scheme \ref{['LagCN']}, the SP-BDF2 scheme \ref{['LagBDFk']} and the SP-BDF2-variant scheme with replacement \ref{['Lag3:BDFk']} replaced by \ref{['Lag3:BDFk:change']}, the Cauchy paths are chosen as $\tau=0.05h$.
  • Figure 5.2: Snapshots of curve evolution with numerical curvature using the SP-BDF2 scheme \ref{['LagBDFk']}, starting with an ellipse initial curve: (a) $t=0$, (b) $t=0.2$, (c) $t=0.4$, (d) $t=0.8$. The discretization parameters are chosen as $N=160$, $\tau=1/640$.
  • Figure 5.3: Evolution of geometric quantities and Lagrange multipliers as functions of time for three schemes, with the initial curve being an ellipse. (a) The normalized perimeter. (b) The relative area loss. (c) The iteration number of Newton's method. (d) The Lagrange multiplier $\lambda(t)$. (e) The Lagrange multiplier $\eta(t)$. (f) The mesh ratio function $\Psi(t)$. The discretization parameters are chosen as $N=160$, $\tau=1/640$.
  • Figure 5.4: Snapshots of the curve evolution with its numerical curvature by the SP-BDF2 scheme \ref{['Lag1:BDFk']}-\ref{['Lag4:BDFk']}, starting with Mikula-Ševčovič's curve. (a) $t=0$, (b) $t=0.005$, (c) $t=0.03$, (d) $t=0.15$. The parameters are chosen as $N=160$, $\tau=1/6400$.
  • Figure 5.5: Evolution of geometric quantities and Lagrange multipliers as functions of time for three schemes, starting with Mikula-Ševčovič's curve. (a) The normalized perimeter. (b) The relative area loss. (c) The iteration number of Newton's method. (d) The Lagrange multiplier $\lambda(t)$. (e) The Lagrange multiplier $\eta(t)$. (f) The mesh ratio function $\Psi(t)$. The parameters are chosen as $N=160$, $\tau=1/6400$.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.1
  • Theorem 3.1
  • Remark 3.1
  • Theorem 3.2
  • ...and 8 more