A second-order in time, BGN-based parametric finite element method for geometric flows of curves

TL;DR

A fully discrete, temporal second-order parametric finite element method, which integrates with two different mesh regularization techniques, for solving geometric flows of curves and demonstrates that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics.

Abstract

Over the last two decades, the field of geometric curve evolutions has attracted significant attention from scientific computing. One of the most popular numerical methods for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and Nürnberg (J. Comput. Phys., 222 (2007), pp.~441--467), due to its favorable properties (e.g., its computational efficiency and the good mesh property). However, the BGN scheme is limited to first-order accuracy in time, and how to develop a higher-order numerical scheme is challenging. In this paper, we propose a fully discrete, temporal second-order parametric finite element method, which integrates with two different mesh regularization techniques, for solving geometric flows of curves. The scheme is constructed based on the BGN formulation and a semi-implicit Crank-Nicolson leap-frog time stepping discretization as well as a linear finite element approximation in space. More importantly, we point out that the shape metrics, such as manifold distance and Hausdorff distance, instead of function norms, should be employed to measure numerical errors. Extensive numerical experiments demonstrate that the proposed BGN-based scheme is second-order accurate in time in terms of shape metrics. Moreover, by employing the classical BGN scheme as mesh regularization techniques, our proposed second-order schemes exhibit good properties with respect to the mesh distribution. In addition, an unconditional interlaced energy stability property is obtained for one of the mesh regularization techniques.

Figures (8)

• Figure 1: Log-log plot of the numerical errors at time $T=0.25$ measured by the manifold distance for BGN1, equi-BGN1, BGN2 and equi-BGN2 schemes for solving the CSF with two different initial curves: (a) Shape 1 and (b) Shape 2, respectively, where the number of nodes is fixed as $N=10000$.
• Figure 2: Log-log plot of the numerical errors at time $T=0.25$, measured by the manifold distance, for solving two different flows with Shape 2 as the initial curve: (a) AP-CSF and (b) SDF, respectively.
• Figure 3: (a) Several snapshots of the curve evolution controlled by the SDF, starting with Shape 2 as its initial shape. (b) The relative area loss as a function of time. (c) The normalized perimeter as a function of time. (d) The mesh ratio function $\Psi(t)$ (in blue line) and the number of mesh regularizations (in red line) for the BGN2 scheme. (e) The mesh ratio function $\Psi(t)$ (in blue line) and the number of iteration numbers (in red line) at each time step for the equi-BGN2 scheme
• Figure 4: (a) Several snapshots of the curve evolution controlled by the SDF, starting with Shape 3 as its initial shape. (b) The relative area loss as a function of time. (c) The normalized perimeter as a function of time. (d) The mesh distribution function $\Psi(t)$ (in blue line) and the number of mesh regularizations (in red line) for the BGN2 scheme. (e) The mesh ratio function $\Psi(t)$ (in blue line) and the number of iteration numbers (in red line) at each time step for the equi-BGN2 scheme
• Figure 5: Evolution of the three geometrical quantities when the initial data is prepared as in Algorithm \ref{['CSF:BGN initial data 1']}: (a) the relative area loss, (b) the normalized perimeter, (c) the mesh distribution function $\Psi(t)$, for the BGN2 scheme.

Theorems & Definitions (9)

• Remark 2.1
• Remark 2.2
• Theorem 2.1: Well-posedness
• proof
• Theorem 2.2: Interlaced energy stability
• proof
• Remark 2.3
• Remark 3.1
• Remark 3.2