A structure-preserving parametric finite element method for geometric flows with anisotropic surface energy

Abstract

We propose and analyze a structure-preserving parametric finite element method (SP-PFEM) for the evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy density $\gamma (\varvec{n})$, where $\varvec{n}\in \mathbb {S}^1$ represents the outward unit normal vector. We begin with the anisotropic surface diffusion which possesses two well-known geometric structures—area conservation and energy dissipation—during the evolution of the closed curve. By introducing a novel surface energy matrix $\varvec{G}_k(\varvec{n})$ depending on $\gamma (\varvec{n})$ and the Cahn-Hoffman $\varvec{\xi }$-vector as well as a nonnegative stabilizing function $k(\varvec{n})$, we obtain a new conservative geometric partial differential equation and its corresponding variational formulation for the anisotropic surface diffusion. Based on the new weak formulation, we propose a full discretization by adopting the parametric finite element method for spatial discretization and a semi-implicit temporal discretization with a proper and clever approximation for the outward normal vector. Under a mild and natural condition on $\gamma (\varvec{n})$, we can prove that the proposed full discretization is structure-preserving, i.e. it preserves the area conservation and energy dissipation at the discretized level, and thus it is unconditionally energy stable. The proposed SP-PFEM is then extended to simulate the evolution of a close curve under other anisotropic geometric flows including anisotropic curvature flow and area-conserved anisotropic curvature flow. Extensive numerical results are reported to demonstrate the efficiency and unconditional energy stability as well as good mesh quality (and thus no need to re-mesh during the evolution) of the proposed SP-PFEM for simulating anisotropic geometric flows.

1 Introduction

Anisotropic surface energy along surface/interface is ubiquitous in solids and materials science due to the lattice orientational difference [16, 29]. It thus generates anisotropic evolution process of interface/surface (or geometric flows with anisotropic surface energy) in materials science [31, 41, 44], imaging science [19, 22, 37], and computational geometry [16, 20, 45]. In fact, anisotropic geometric flows have significant and broader applications in materials science, solid-state physics and computational geometry, such as grain boundary growth [13], foam bubble/film [40], surface phase formation [49], epitaxial growth [27, 29], heterogeneous catalysis [39], solid-state dewetting [5, 7, 44], and computational graphics [19, 22, 37].

Fig. 1

An illustration of a closed curve $\Gamma$ in 2D with an anisotropic surface energy density $\gamma (\varvec{n})$ depending on the outward unit normal vector $\varvec{n}$

Assume $\Gamma$ be a closed curve in two dimensions (2D) associated with a given anisotropic surface energy density $\gamma (\varvec{n})$, where $\varvec{n}=(n_1,n_2)^T\in \mathbb {S}^1$ represents the outward unit normal vector, see Fig. 1. Define its corresponding free energy functional $W(\Gamma )$ as [4, 6, 20, 33, 42]:

$$\begin{aligned} W(\Gamma ):=\int _{\Gamma }\gamma (\varvec{n})ds, \end{aligned}$$

(1.1)

where s denotes the arc-length parameter of $\Gamma$. By applying the thermodynamic variation, one can obtain the chemical potential $\mu :=\mu (s)$ (or weighted curvature denoted as $\kappa _\gamma :=\kappa _\gamma (s)$) generated from the energy functional $W(\Gamma )$ as [33, 42]

$$\begin{aligned} \mu =\kappa _\gamma :=\frac{\delta W(\Gamma )}{\delta \Gamma }= \lim _{\varepsilon \rightarrow 0}\frac{W(\Gamma ^\varepsilon )-W(\Gamma )}{\varepsilon }, \end{aligned}$$

(1.2)

where $\Gamma ^\varepsilon$ is a small perturbation of $\Gamma$ [33]. Different geometric flows associate with the anisotropic surface energy density $\gamma (\varvec{n})$ can be easily defined by providing the normal velocity $V_n$ for the evolution of $\Gamma$, which is usually generated from the chemical potential $\mu$ [15, 43]. Typical anisotropic geometric flows are widely used in different applications including anisotropic curvature flow, area-conserved anisotropic curvature flow and anisotropic surface diffusion with the corresponding normal velocity $V_n$ given as [1, 14, 15, 18, 43]

$$\begin{aligned} V_n=\left\{ \begin{array}{ll} -\mu , &{}\qquad \hbox {anisotropic curvature flow},\\ -\mu +\lambda , &{}\qquad \hbox {area-conserved anisotropic curvature flow},\\ \partial _{ss}\mu , &{}\qquad \hbox {anisotropic surface diffusion},\\ \end{array}\right. \end{aligned}$$

(1.3)

where the Lagrange multiplier $\lambda$ is chosen such that the area of the region enclosed by $\Gamma$ is conserved, which is given as

$$\begin{aligned} \lambda =\frac{\int _{\Gamma } \mu \,ds}{\int _{\Gamma }1\,ds} \quad \Leftrightarrow \quad \int _{\Gamma }V_n\, ds=\int _{\Gamma }(-\mu +\lambda ) ds=0. \end{aligned}$$

(1.4)

We remark here that if $\gamma (\varvec{n})\equiv 1$ for $\varvec{n}\in \mathbb {S}^1$, i.e. in the case of the isotropic surface energy density, the chemical potential $\mu$ defined in (1.2) collapses to the curvature $\kappa$ of $\Gamma$. Then the geometric flows defined in (1.3) collapse to curvature flow (or curve-shortening flow), area-conserved curvature flow and surface diffusion, respectively [14].

For the geometric flows (1.3) with the isotropic surface energy density, i.e. $\gamma (\varvec{n})\equiv 1$ for $\varvec{n}\in \mathbb {S}^1$, based on different parametrizations of $\Gamma$ and different mathematical formulations for the flows, many different numerical methods have been proposed in the literature. These numerical methods include the level set method and the phase-field method [24], the marker particle method [23, 47], the finite element method based on graph formulation [20, 21], the discontinuous Galerkin method [48], the crystalline method [17, 28], the evolving surface finite element method (ESFEM) [34], and the parametric finite element method (PFEM) [9, 11, 20, 25, 26]. Among these numerical methods, most of them need to frequently carry out remeshing during the evolution in order to avoid the collision of mesh points, given that only the normal velocity is provided in the geometric flows (1.3). Therefore, different artificial tangential velocities are introduced in different numerical methods. One notable exception is the energy-stable PFEM (ES-PFEM) proposed by Barrett, Garcke, and Nürnberg (also called as BGN scheme in the literature). It demonstrates several good properties including efficiency, accuracy and unconditional energy stability as well as the surprising asymptotic equal-mesh distribution property for isotropic geometric flows [9, 11]. Recently, by introducing a clever approximation of the normal vector via an average of the two normal vectors at two adjacent time steps, Bao and Zhao proposed a structure-preserving PFEM (SP-PFEM) [2, 3, 8] for the isotropic surface diffusion. The SP-PFEM preserves the two geometric structures—mass (or the area of the region enclosed by the curve) conservation and energy (or the length of the curve) dissipation—in the fully discretized level.

Different techniques have been presented to extend the ES-PFEM and/or SP-PFEM from isotropic geometric flows to anisotropic geometric flows in the literature [10, 12, 32, 33, 35, 45]. When $\gamma (\varvec{n})$ is taken as the Riemannian-like metric anisotropic surface energy, by adapting a modified variational formulation with a proper and clever surface energy matrix, Barrett, Garcke and Nürnberg [10] extended successfully the ES-PFEM to anisotropic surface diffusion with this very special surface energy density. Recently, by introducing conservative geometric partial differential equations for the anisotropic surface diffusion with a general form of $\gamma (\varvec{n})= \gamma (-\sin \theta ,\cos \theta ):=\hat{\gamma }(\theta )$ with $\theta$ being the angle between $\varvec{n}$ and the vertical axis (cf. Fig 1), we extended the ES-PFEM to the anisotropic surface diffusion under a very strong condition on $\hat{\gamma }(\theta )$ [35]. Very recently, by introducing a symmetric positive definite surface matrix $\varvec{Z}_k(\varvec{n})$ depending on $\gamma (\varvec{n})$ and the Cahn-Hoffman $\varvec{\xi }$-vector as well as a stabilizing function $k(\varvec{n}):\ \mathbb {S}^1\rightarrow \mathbb {R}^+$ [4, 6], we successfully and systematically extended the ES-PFEM and SP-PFEM methods from isotropic surface diffusion to anisotropic surface diffusion in two ($d=2$) and three dimensions ($d=3$) under a symmetric condition on $\gamma (\varvec{n})$ as:

$$\begin{aligned} \gamma (-\varvec{n})=\gamma (\varvec{n}), \qquad \varvec{n}\in \mathbb {S}^{d-1}, \qquad \hbox {with} \quad d=2,3. \end{aligned}$$

(1.5)

To our best knowledge, it is still an open question to extend systematically either ES-PFEM or SP-PFEM for solving the anisotropic surface diffusion with arbitrary form of $\gamma (\varvec{n})$, especially when $\gamma (\varvec{n})$ is NOT symmetric, such as for the 3-fold anisotropic surface energy density arising in solid-state dewetting [5, 45].

The main objective of this paper is to extend the SP-PFEM (and ES-PFEM) for solving the anisotropic geometric flow (1.3) with the surface energy density $\gamma (\varvec{n})$ satisfying a relatively mild and simple condition as

$$\begin{aligned} \gamma (-\varvec{n})<3\gamma (\varvec{n}), \quad \varvec{n}\in \mathbb {S}^{1}, \qquad \gamma (\varvec{p})\in C^2(\mathbb {R}^2\setminus \{\varvec{0}\}), \end{aligned}$$

(1.6)

where $\gamma (\varvec{p})$ is a one-homogeneous extension of $\gamma (\varvec{n})$ from $\mathbb {S}^{1}$ to $\mathbb {R}^2$ defined as [4, 6]

$$\begin{aligned} \gamma (\varvec{p}):= {\left\{ \begin{array}{ll} |\varvec{p}|\,\gamma \left( \frac{\varvec{p}}{|\varvec{p}|}\right) ,\qquad &{}\forall \varvec{p}=(p_1,p_2)^T\in \mathbb {R}^2_*:=\mathbb {R}^2\setminus \{\varvec{0}\},\\ 0, &{}\varvec{p}=\varvec{0}, \end{array}\right. } \end{aligned}$$

(1.7)

where $|\varvec{p}|=\sqrt{p_1^2+p_2^2}$. In fact, if $\gamma (\varvec{n})$ is symmetric, i.e. it satisfies (1.5), then it satisfies (1.6) automatically since $0<\gamma (-\varvec{n})=\gamma (\varvec{n}) <3\gamma (\varvec{n})$ for $\varvec{n}\in \mathbb {S}^{1}$, and thus the SP-PFEM proposed in this paper works for both symmetric and non-symmetric surface energy density $\gamma (\varvec{n})$. The main ingredients in the proposed method are based on: (i) the introduction of a novel surface energy matrix $\varvec{G}_k(\varvec{n})$ depending on $\gamma (\varvec{n})$ and the Cahn-Hoffman $\varvec{\xi }$-vector as well as a stabilizing function $k(\varvec{n}):\ \mathbb {S}^1\rightarrow \mathbb {R}^+$, which can be explicitly decoupled into a symmetric positive definite matrix and an anti-symmetric matrix; (ii) a new conservative geometric partial differential equation (PDE) for the anisotropic geometric flows (1.3); (iii) a new variational formulation; and (iv) a proper approximation of the normal vector. Under the mild and simple condition (1.6) on the anisotropic surface energy density $\gamma (\varvec{n})$, we can prove rigorously that the proposed SP-PFEM is structure-preserving—area conservation and energy dissipation—for the anisotropic surface diffusion in the discretized level. Then the proposed SP-PFEM is extended to simulate the evolution of a closed curve under other anisotropic geometric flows including anisotropic curvature flow and area-conserved anisotropic curvature flow.

The remainder of this paper is organized as follows: In Sect. 2, by introducing the surface energy matrix $\varvec{G}_k(\varvec{n})$, we derive a new conservative PDE and its corresponding variational formulation for the anisotropic surface diffusion, and propose the semi-discretization in space and the full-discretization by SP-PFEM. In Sect. 3, we establish the unconditional energy dissipation of the proposed SP-PFEM for the anisotropic surface diffusion under the condition (1.6). In Sect. 4, we extend the proposed SP-PFEM to the anisotropic curvature flow and area-conserved anisotropic curvature flow. Extensive numerical results are reported in Sect. 5 to illustrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM. Finally, some concluding remarks are drawn in Sect. 6.

2 The structure-preserving PFEM for anisotropic surface diffusion

In this section, by taking the anisotropic surface diffusion in (1.3), we introduce a surface energy matrix $\varvec{G}_k(\varvec{n})$, obtain a conservative geometric PDE and derive its corresponding variational formulation. An SP-PFEM for the variational problem is presented and its structure-preserving property is stated under the simple and mild condition (1.6) on the anisotropic surface energy density $\gamma (\varvec{n})$.

2.1 The geometric PDE

Let $\Gamma :=\Gamma (t)$ be parameterized by $\varvec{X}:=\varvec{X}(s, t)=(x(s,t),y(s,t))^T\in {\mathbb R}^2$ with s denoting the time-dependent arc-length parametrization (or ‘Lagrangian coordinate’) of $\Gamma$ and t representing the time. Then the anisotropic surface diffusion of $\Gamma$ can be described by

$$\begin{aligned} \partial _t \varvec{X}= (\partial _{ss} \mu ) \, \varvec{n}, \end{aligned}$$

(2.1)

where $\varvec{n}=(n_1,n_2)^T\in {\mathbb S}^1$ represents the outward unit normal vector of $\Gamma$, and $\mu$ is the chemical potential given in (1.2). In practice, another popular way (or ‘Eulerian coordinate’) is to adopt $\rho \in \mathbb {T}=\mathbb {R}/\mathbb {Z}=[0, 1]$ being the periodic interval and then parameterize $\Gamma (t)$ on $\mathbb {T}$ by $\varvec{X}(\rho , t)$, which is given as

$$\begin{aligned} \Gamma (t):=\varvec{X}({\mathbb {T}}, t)\quad \varvec{X}(\cdot , t): \mathbb {T}\rightarrow \mathbb {R}^2, (\rho , t)\mapsto (x(\rho ,t), y(\rho ,t))^T. \end{aligned}$$

(2.2)

Then the arc-length parameter s is given as $s(\rho , t)=\int _0^\rho |\partial _\rho \varvec{X}(\rho , t)|\, d\rho$ satisfying $\partial _\rho s=|\partial _\rho \varvec{X}|$. We assume that the parametrization by $\rho$ is regular during the evolution, i.e., there is a constant $C>1$ such that $\frac{1}{C}\le |\partial _\rho s(\rho , t)|\le C$. The normal velocity $V_n$ can be given by this parametrization as

$$\begin{aligned} V_n=\varvec{n}\cdot \partial _t \varvec{X}. \end{aligned}$$

(2.3)

Based on the one-homogeneous extension $\gamma (\varvec{p})$ in (1.7), we can talk about the regularity of $\gamma (\varvec{n})$ by referring to the regularity of $\gamma (\varvec{p})$, i.e., $\gamma (\varvec{n})\in C^2(\mathbb {S}^1)\iff \gamma (\varvec{p})\in C^2(\mathbb {R}^2{\setminus } \{\varvec{0}\})$. The Cahn-Hoffman $\varvec{\xi }$-vector is widely adopted in the literature as [30, 46]

$$\begin{aligned} \varvec{\xi }:=\varvec{\xi }(\varvec{n}): \mathbb {S}^1\rightarrow \mathbb {R}^2, \quad \varvec{n}\mapsto \varvec{\xi }{(\varvec{n})}=(\xi _1, \xi _2)^T:=\nabla \gamma (\varvec{p})|_{\varvec{p}=\varvec{n}}=\gamma (\varvec{n})\varvec{n}+(\varvec{\xi }\cdot \varvec{\tau })\varvec{\tau }, \end{aligned}$$

(2.4)

here $\varvec{\tau }=\partial _s\varvec{X}=\varvec{n}^\perp$ is the unit tangential vector of $\Gamma$, and $\perp$ denotes the clockwise rotation by $\frac{\pi }{2}$. Then an explicit formulation of the chemical potential $\mu$ in (1.2) can be expressed in term of the $\varvec{\xi }$-vector as [33]

$$\begin{aligned} \mu :=\mu (\varvec{n})=-\varvec{n}\cdot \partial _s \varvec{\xi }^\perp . \end{aligned}$$

(2.5)

Thus another equivalent geometric PDE for the anisotropic surface diffusion can be written as

$$\begin{aligned}&\varvec{n}\cdot \partial _t \varvec{X}=\partial _{ss} \mu , \end{aligned}$$

(2.6a)

$$\begin{aligned}&\mu =-\varvec{n}\cdot \partial _s \varvec{\xi }^\perp ,\qquad \varvec{\xi }(\varvec{n})=\nabla \gamma (\varvec{p})|_{\varvec{p}=\varvec{n}}. \end{aligned}$$

(2.6b)

2.2 The surface energy matrix

Introduce the surface energy matrix $\varvec{G}_k(\varvec{n})$ as

$$\begin{aligned} \varvec{G}_k(\varvec{n}):=\gamma (\varvec{n})I_2 -\varvec{n}\varvec{\xi }^T+\varvec{\xi }\varvec{n}^T +k(\varvec{n})\varvec{n}\varvec{n}^T:= \varvec{G}_k^{(s)}(\varvec{n})+\varvec{G}^{(a)}(\varvec{n}), \quad \forall \varvec{n}\in \mathbb {S}^1, \end{aligned}$$

(2.7)

where $I_2$ is the $2\times 2$ identity matrix, $k(\varvec{n}): \mathbb {S}^1\rightarrow \mathbb {R}^+$ is a nonnegative stabilizing function to be determined later, and $\varvec{G}_k^{(s)}(\varvec{n})$ and $\varvec{G}^{(a)}(\varvec{n})$ are a symmetric positive matrix and an anti-symmetric matrix, respectively, which are given as

$$\begin{aligned} \varvec{G}_k^{(s)}(\varvec{n}):=\gamma (\varvec{n})I_2 +k(\varvec{n})\varvec{n}\varvec{n}^T, \quad \varvec{G}^{(a)}(\varvec{n}):= -\varvec{n}\varvec{\xi }^T+\varvec{\xi }\varvec{n}^T, \quad \forall \varvec{n}\in \mathbb {S}^1. \end{aligned}$$

(2.8)

Then we get the following relationship between the weighted curvature $\mu$ and the newly constructed $\varvec{G}_k(\varvec{n})$.

Lemma 2.1

The weighted curvature $\mu$ defined in (2.5) has the following alternative explicit expression

$$\begin{aligned} \mu \varvec{n}=-\partial _s (\varvec{G}_k(\varvec{n})\partial _s \varvec{X}). \end{aligned}$$

(2.9)

Proof

From [4, Lemma 2.1], we know that $\varvec{\xi }$ and $\mu$ given in (2.4) and (2.5), respectively, satisfy

$$\begin{aligned} \mu \varvec{n}=-\partial _s \varvec{\xi }^\perp ,\quad \varvec{\xi }^\perp =\gamma (\varvec{n})\varvec{\tau } -(\varvec{\xi }\cdot \varvec{\tau })\varvec{n}. \end{aligned}$$

(2.10)

Thus it suffices to show $\varvec{G}_k(\varvec{n})\partial _s \varvec{X}=\varvec{\xi }^\perp$. Since $\partial _s\varvec{X}=\varvec{\tau }$, by using the definition of $\varvec{G}_k(\varvec{n})$ in (2.7), we can simplify $\varvec{G}_k(\varvec{n})\partial _s \varvec{X}$ as

$$\begin{aligned} \varvec{G}_k(\varvec{n})\partial _s \varvec{X}&=\gamma (\varvec{n})\varvec{\tau }-(\varvec{\xi }\cdot \varvec{\tau })\varvec{n}+(\varvec{n}\cdot \varvec{\tau })\varvec{\xi }+k(\varvec{n})(\varvec{n}\cdot \varvec{\tau })\varvec{n}\nonumber \\&=\gamma (\varvec{n})\varvec{\tau }-(\varvec{\xi }\cdot \varvec{\tau })\varvec{n}=\varvec{\xi }^\perp . \end{aligned}$$

(2.11)

Combining this with (2.10), we get the desired equality (2.9).$\square$

By applying (2.9), the geometric PDE for the anisotropic surface diffusion (2.6) can be rewritten as the following conservative form

$$\begin{aligned}&\varvec{n}\cdot \partial _t \varvec{X}=\partial _{ss} \mu , \end{aligned}$$

(2.12a)

$$\begin{aligned}&\mu \varvec{n}=-\partial _s(\varvec{G}_k(\varvec{n})\partial _s \varvec{X}). \end{aligned}$$

(2.12b)

Here the ‘conservative form’ refers to the right-hand side of (2.12b) is in conservative form with respect to s.

Remark 2.1

When $\gamma (\varvec{n})\equiv 1$ for $\varvec{n}\in \mathbb {S}^1$, i.e., isotropic case, it is well known that $\varvec{\xi } = \varvec{n}$. Taking $k(\varvec{n})\equiv 0$ for $\varvec{n}\in \mathbb {S}^1$ in (2.7), we get $\varvec{G}_k(\varvec{n})=I_2$. Then the conservative form (2.9) (and (2.12)) collapses to the standard forms used in the literature for the isotropic case [9]. Thus our new formulation (2.9) (and (2.12)) is a natural extension from the isotropic case to anisotropic case.

On the other hand, when $\gamma (\varvec{n})\ne 1$, i.e., anisotropic case, it is well known that $\varvec{0}\ne \varvec{\xi } \ne \varvec{n}$, and then $-\varvec{n}\varvec{\xi }^T+\varvec{\xi }\varvec{n}^T\ne -\varvec{n}\varvec{\xi }^T-\varvec{\xi }\varvec{n}^T$ since $\varvec{\xi }\varvec{n}^T\ne \textbf{0}$. Thus our new surface energy matrix $\varvec{G}_k(\varvec{n})$ introduced in (2.7) is quite different with the surface energy matrix $\varvec{Z}_k(\varvec{n})$ introduced in [4, 6] mainly for anisotropic case with symmetric surface energy density $\gamma (-\varvec{n})=\gamma (\varvec{n})$.

Remark 2.2

As it is illustrated in Fig. 1, we have

$$\begin{aligned} \begin{aligned}&\varvec{n}=(n_1,n_2)^T=(-\sin \theta , \cos \theta )^T:=\varvec{n}(\theta ), \\ {}&\varvec{\tau }=(\tau _1,\tau _2)^T=(\cos \theta , \sin \theta )^T:=\varvec{\tau }(\theta ), \qquad \theta \in [-\pi ,\pi ],\\&\gamma (\varvec{n})=\gamma (-\sin \theta , \cos \theta ):=\hat{\gamma }(\theta ). \end{aligned} \end{aligned}$$

(2.13)

Plugging (1.7) into (2.4) and taking $\varvec{p}=|\varvec{p}|(-\sin \theta ,\cos \theta )^T$, we get

$$\begin{aligned} \begin{aligned} \varvec{\xi }&=\left. \varvec{\xi }(\varvec{n}) =\nabla \gamma (\varvec{p})\right| _{\varvec{p}=\varvec{n}} =\left. \left[ \gamma \left( \frac{\varvec{p}}{|\varvec{p}|}\right) \nabla |\varvec{p}|+|\varvec{p}|\nabla \gamma \left( \frac{\varvec{p}}{|\varvec{p}|}\right) \right] \right| _{\varvec{p}=\varvec{n}}\\&=\gamma (-\sin \theta ,\cos \theta )\varvec{n}(\theta )+\nabla \gamma (-\sin \theta ,\cos \theta )\\&=\hat{\gamma }(\theta ) \varvec{n}(\theta ) -\hat{\gamma }^\prime (\theta ) \varvec{\tau }(\theta ): =\hat{\varvec{\xi }}(\theta ). \end{aligned} \end{aligned}$$

(2.14)

Substituting (2.14) into (2.7), we have

$$\begin{aligned} \varvec{G}_k(\varvec{n}) = \begin{pmatrix} \hat{\gamma }(\theta ) &{}-\hat{\gamma }'(\theta )\\ \hat{\gamma }'(\theta ) &{}\hat{\gamma }(\theta ) \end{pmatrix} +\hat{k}(\theta ) \begin{pmatrix} \sin ^2\theta &{}-\sin \theta \cos \theta \\ -\sin \theta \cos \theta &{}\cos ^2\theta \end{pmatrix}:=\hat{\varvec{G}}_k(\theta ), \end{aligned}$$

(2.15)

where $\hat{k}(\theta ):=k(\varvec{n})=k(-\sin \theta ,\cos \theta )$ is a stabilizing function. If we take $\hat{k}(\theta )\equiv 0$ in (2.15), then $\hat{\varvec{G}}_k(\theta )$ collapses to the surface energy matrix $G(\theta )$ proposed in [35]. Thus our formulation is also a natural extension of the method in [35] which adopts the surface energy density in $\hat{\gamma }(\theta )$ formulation and thus cannot be extended to three dimensions (3D). On the contrary, since we adopt the surface energy density in the $\gamma (\varvec{n})$ formulation, our method can be extended to 3D straightforwardly via the Cahn-Hoffman $\varvec{\xi }$-vector.

Moreover, due to the stabilizing function $k(\varvec{n})=\hat{k}(\theta )$, the energy stable condition (1.6) becomes

$$\begin{aligned} \hat{\gamma }(\theta +\pi )<3\hat{\gamma }(\theta ), \quad \forall \theta \in [-\pi , \pi ], \qquad \hat{\gamma }(\theta )\in C^2([-\pi , \pi ]). \end{aligned}$$

(2.16)

This energy stable condition significantly improve the conditions required in [35] without using the stabilizing function, i.e., $\hat{k}(\theta )\equiv 0$. For example, for the m-fold anisotropic surface energy density $\hat{\gamma }(\theta )=1+\beta \cos m\theta$ for $m=2,3,4,6$: the energy stable condition in [35] (without using the stabilizing function $\hat{k}(\theta )$) requires $|\beta |\le \frac{1}{m^2+1}$ for all m; while it only needs $|\beta |<1$ and $|\beta |<\frac{1}{2}$ for m being even and odd, respectively. Thus by introducing the stabilizing function $k(\varvec{n})$ (or $\hat{k}(\theta )$), we can improve the energy stable condition significantly.

2.3 A variational formulation and its properties

We then derive the variational formulation for the conservative form (2.12). Suppose the usual $L^2$ space over $\mathbb {T}$ is

$$\begin{aligned} L^2(\mathbb {T}):=\left\{ u: \mathbb {T}\rightarrow \mathbb {R}\;|\; \int _{\mathbb {T}} |u(\rho )|^2 d\rho <\infty \right\} . \end{aligned}$$

(2.17)

The functional space with respect to $\Gamma :=\Gamma (t)$ can be given as

$$\begin{aligned} L^2(\Gamma (t)):=\left\{ u: \mathbb {T}\rightarrow \mathbb {R}\;|\; \int _{\mathbb {T}} |u(\rho )|^2\partial _\rho s(\rho , t) d\rho <\infty \right\} , \end{aligned}$$

(2.18)

with the following weighted $L^2$-inner product $(\cdot , \cdot )_{\Gamma (t)}$

$$\begin{aligned} \Bigl (u, v\Bigr )_{\Gamma (t)}:=\int _{\Gamma (t)} u(s) v(s)ds=\int _{\mathbb {T}} u(\rho ) v(\rho ) \partial _\rho s(\rho ,t)\, d\rho , \quad \forall u,v\in L^2(\mathbb {T}). \end{aligned}$$

(2.19)

Since we assume that the parameterization is regular; the space $L^2(\Gamma (t))$ with respect to $\Gamma (t)$ is equivalent to the usual $L^2(\mathbb {T})$, which is independent of t. The Sobolev space $H^1(\mathbb {T})$ is defined as

$$\begin{aligned} H^1(\mathbb {T}):=\left\{ u: \mathbb {T}\rightarrow \mathbb {R}\;|\; u\in L^2(\mathbb {T}), \partial _s u\in L^2(\mathbb {T})\right\} . \end{aligned}$$

(2.20)

Here $\partial _s u$ is the weak derivative. Moreover, we can extend the above definitions to the functions in $[L^2(\mathbb {T})]^2$ and $[H^1(\mathbb {T})]^2$.

Multiplying a test function $\phi \in H^1(\mathbb {T})$ to (2.12a), integrating over $\Gamma (t)$ and taking integration by parts, we obtain

$$\begin{aligned} \Bigl (\varvec{n}\cdot \partial _t\varvec{X}, \phi \Bigr )_{\Gamma (t)}=-\Bigl (\partial _s\mu , \partial _s\phi \Bigr )_{\Gamma (t)}. \end{aligned}$$

(2.21)

Similarly, by multiplying a test function $\varvec{\omega }=(\omega _1, \omega _2)^T\in [H^1(\mathbb {T})]^2$ to (2.12b), we deduce that

$$\begin{aligned} \Bigl (\mu \varvec{n}, \varvec{\omega }\Bigr )_{\Gamma (t)}=\Bigl (\varvec{G}_k(\varvec{n})\partial _s \varvec{X}, \partial _s\varvec{\omega }\Bigr )_{\Gamma (t)}. \end{aligned}$$

(2.22)

Based on the two Eqs. (2.21) and (2.22), the new variational formulation for the conservative form (2.12) can be stated as follows: Suppose the initial curve $\Gamma _0:=\varvec{X}(\cdot , 0)=(x(\cdot , 0), y(\cdot , 0))^T\in [H^1(\mathbb {T})]^2$ and the initial weighted curvature $\mu (\cdot , 0):=\mu _0(\cdot )\in H^1(\mathbb {T})$, then for any $t>0$, find the solution $(\varvec{X}(\cdot , t), \mu (\cdot , t))\in [H^1(\mathbb {T})]^2\times H^1(\mathbb {T})$ satisfying

$$\begin{aligned}&\Bigl (\varvec{n}\cdot \partial _{t}\varvec{X}, \varphi \Bigr )_{\Gamma (t)}+\Bigl (\partial _{s}\mu , \partial _s\varphi \Bigr )_{\Gamma (t)}=0,\qquad \forall \varphi \in H^1(\mathbb {T}), \end{aligned}$$

(2.23a)

$$\begin{aligned}&\Bigl (\mu \varvec{n},\varvec{\omega }\Bigr )_{\Gamma (t)}-\Bigl ( \varvec{G}_k(\varvec{n})\partial _s \varvec{X},\partial _s\varvec{\omega }\Bigr )_{\Gamma (t)}= 0,\quad \forall \varvec{\omega }\in [H^1(\mathbb {T})]^2. \end{aligned}$$

(2.23b)

Denote the area of the region enclosed by $\Gamma (t)$ as A(t) and the free interfacial energy of $\Gamma (t)$ as W(t), which are given by

$$\begin{aligned} A(t):=\int _{\Gamma (t)}y(\rho , t) \partial _\rho x(\rho , t) d\rho ,\quad W(t):=\int _{\Gamma (t)}\gamma (\varvec{n})\,ds, \qquad t\ge 0. \end{aligned}$$

(2.24)

We can show that the two geometric properties, i.e. area conservation and energy dissipation, still hold for the weak formulation (2.23).

Proposition 2.1

(area conservation and energy dissipation) Suppose $\Gamma (t)$ is given by the solution $(\varvec{X}(\cdot , t), \mu (\cdot , t))$ of the variational formulation (2.23), we have

$$\begin{aligned} A(t)\equiv A(0),\qquad W(t)\le W(t_1)\le W(0), \qquad \forall t\ge t_1\ge 0. \end{aligned}$$

(2.25)

Proof

By taking $\varphi =1$ in (2.23a) and using (2.17) in [35], we know

$$\begin{aligned} \frac{dA(t)}{dt}=\Bigl (\partial _t\varvec{X}, \varvec{n}\Bigr )_{\Gamma (t)}=-\Bigl (\partial _s\mu , \partial _s 1\Bigr )_{\Gamma (t)}=0. \end{aligned}$$

Therefore we have $A(t)\equiv A(0), \forall t\ge 0$. For the energy dissipation, from the proof of [4, Proposition 2.2], it holds

$$\begin{aligned} \frac{dW(t)}{dt}=\Bigl (\gamma (\varvec{n})\partial _s \varvec{X}-(\varvec{\xi }\cdot \partial _s\varvec{X})\varvec{n}, \partial _{s}\partial _t\varvec{X}\Bigr )_{\Gamma (t)}. \end{aligned}$$

(2.26)

By setting $\varphi =\mu$ in (2.23a) and $\varvec{\omega }=\partial _t\varvec{X}$ in (2.23b), and considering (2.11) and (2.26), we deduce that

$$\begin{aligned} \frac{dW(t)}{dt}=\Bigl (\varvec{G}_k(\varvec{n})\partial _s\varvec{X}, \partial _{s}\partial _t\varvec{X}\Bigr )_{\Gamma (t)}=\Bigl (\mu \varvec{n}, \partial _t\varvec{X}\Bigr )_{\Gamma (t)}=-\tau \Bigl (\partial _s \mu , \partial _{s}\mu \Bigr )_{\Gamma (t)}\le 0. \end{aligned}$$

Which indicates the energy dissipation $W(t)\le W(t_1)\le W(0), \, \forall t\ge t_1\ge 0$.$\square$

2.4 A semi-discretization in space

To obtain the spatial discretization, let $N>2$ be an integer and define the mesh size as $h=\frac{1}{N}$. Consider the uniform partition of $\mathbb {T}=[0, 1]=\cup _{j=1}^N I_j$, where $I_j=[\rho _{j-1}, \rho _j]$ and $\rho _j=jh$ for $j=0, 1, \ldots , N$. Here $\rho _N=\rho _0$ by periodicity. The closed curve $\Gamma (t)=\varvec{X}(\cdot , t)$ is approximated by the closed line segments $\Gamma ^h(t)=\varvec{X}^h(\cdot , t)=(x^h(\cdot , t), y^h(\cdot , t))^T$ satisfying $\varvec{X}^h(\rho _j, 0)=\varvec{X}(\rho _j, 0)$. And the discretization for the test function space $H^1(\mathbb {T})$ is given by the following space of piecewise linear finite element functions

$$\begin{aligned} \mathbb {K}^h=\mathbb {K}^h(\mathbb {T}):=\left\{ u\in C(\mathbb {T}) | \quad u|_{I_j}\in \mathcal {P}^1(I_j), \forall 1\le j\le N\right\} , \end{aligned}$$

(2.27)

where $\mathcal {P}^1(I_j)$ is the set of polynomials defined on $I_j$ of degree $\le 1$. The mass lumped inner product for two functions $u, v\in \mathbb {K}^h(\mathbb {T})$ with respect to $\Gamma ^h(t)$ is defined as

$$\begin{aligned} \left( u, v\right) ^h_{\Gamma ^h(t)}:=\frac{1}{2}\sum _{j=1}^N |\varvec{h}_j(t)|\,\left( (u\cdot v)(\rho _{j-1}^+)+(u\cdot v)(\rho _j^-)\right) . \end{aligned}$$

(2.28)

where $\varvec{h}_j(t)=\varvec{X}^h(\rho _j,t)-\varvec{X}^h(\rho _{j-1}, t)$, and $u(\rho _j^\pm )=\lim \limits _{\rho \rightarrow \rho _j^\pm } u(\rho )$. We note this mass lumped inner product is also applicable for $[\mathbb {K}^h]^2$ and the piecewise constant vector-valued functions with possible jump discontinuities at $\rho _j$ for $0\le j\le N$.

The discretized unit normal vector $\varvec{n}^h(t)$, unit tangential vector $\varvec{\tau }^h(t)$, and the $\varvec{\xi }$-vector $\varvec{\xi }^h(t)$ are such piecewise constant vectors on $\Gamma ^h(t)$, which are given as

$$\begin{aligned} \varvec{n}_j^h:=\varvec{n}^h|_{I_j}=-\frac{\varvec{h}_j^\perp }{|\varvec{h}_j|}, \quad \varvec{\tau }_j^h:=\varvec{\tau }^h|_{I_j}=(\varvec{n}_j^h)^\perp , \quad \varvec{\xi }_j^h:=\varvec{\xi }^h|_{I_j}=\nabla \gamma (\varvec{p})|_{\varvec{p}=\varvec{n}_j^h}. \end{aligned}$$

(2.29)

Here we use $\varvec{n}^h$ to represent $\varvec{n}^h(t)$ for short. To make sure $\varvec{n}^h, \varvec{\tau }^h, \varvec{\xi }^h$ are well defined, we need the following assumption on $\varvec{h}_j(t)$

$$\begin{aligned} \min _{1\le j\le N}|\varvec{h}_j(t)|>0,\quad \forall t\ge 0. \end{aligned}$$

(2.30)

After giving all the continuous objects their discretized versions, we can state the spatial semi-discretization as follows: Let $\Gamma ^h_0:= \varvec{X}^h(\mathbb {T}, 0)\in [\mathbb {K}^h]^2, \mu ^h(\cdot , 0)\in \mathbb {K}^h$ be the approximations of $\Gamma _0:=\varvec{X}_0(\mathbb {T}), \mu _0(\cdot )$, respectively, with $\varvec{X}^h(\rho _j, 0)=\varvec{X}(\rho _j, 0), \mu ^h(\rho _j, 0)=\mu _0(\rho _j)$ for $0\le j\le N$, find the solution $(\varvec{X}^h(\cdot , t), \mu ^h(\cdot , t))\in [\mathbb {K}^h]^2\times \mathbb {K}^h$, such that

$$\begin{aligned}&\Bigl (\varvec{n}^h\cdot \partial _{t}\varvec{X}^h, \varphi ^h\Bigr )_{\Gamma ^h(t)}^h+\Bigl (\partial _{s}\mu ^h, \partial _s\varphi ^h\Bigr )_{\Gamma ^h(t)}^h=0,\qquad \forall \varphi ^h\in \mathbb {K}^h, \end{aligned}$$

(2.31a)

$$\begin{aligned}&\Bigl (\mu ^h\varvec{n}^h,\varvec{\omega }^h\Bigr )_{\Gamma ^h(t)}^h-\Bigl ( \varvec{G}_k(\varvec{n}^h)\partial _s \varvec{X}^h,\partial _s\varvec{\omega }^h\Bigr )_{\Gamma ^h(t)}^h=0, \quad \forall \varvec{\omega }^h\in [\mathbb {K}^h]^2, \end{aligned}$$

(2.31b)

where

$$\begin{aligned} \varvec{G}_k(\varvec{n}^h)|_{I_j}=\gamma (\varvec{n}^h_j)I_2-\varvec{n}_j^h (\varvec{\xi }_j^h)^T+\varvec{\xi }_j^h(\varvec{n}^h_j)^T+k(\varvec{n}^h_j)\,\varvec{n}_j^h(\varvec{n}_j^h)^T, \end{aligned}$$

(2.32)

and the discretized derivative $\partial _s$ for a scalar and vector valued functions f and $\varvec{f}$, respectively, are given as

$$\begin{aligned} \partial _s f|_{I_j}:=\frac{f(\rho _j)-f(\rho _{j-1})}{|\varvec{h}_j|}, \quad \partial _s \varvec{f}|_{I_j}:=\frac{\varvec{f}(\rho _j)-\varvec{f}(\rho _{j-1})}{|\varvec{h}_j|}, \end{aligned}$$

(2.33)

and assumption (2.30) ensures $\partial _s f$ and $\partial _s \varvec{f}$ are piecewise constant functions with possible jump discontinuities at $\rho _j$ for $0\le j\le N$, thus the mass lumped inner product terms like $\Bigl (\partial _{s}\mu ^h, \partial _s\varphi ^h\Bigr )_{\Gamma ^h(t)}^h$ in (2.31) are well defined.

Denote the enclosed area and the total energy of the closed line segments $\Gamma ^h(t)$ as $A^h(t)$ and $W^h(t)$, respectively, which are given by

$$\begin{aligned} A^h(t)=\frac{1}{2}\sum _{j=1}^N(x_j^h(t)-x_{j-1}^h(t))(y_j^h(t)+y_{j-1}^h(t)), \quad W^h(t)=\sum _{j=1}^N|\varvec{h}_j(t)|\gamma (\varvec{n}^h_j), \end{aligned}$$

(2.34)

where $x_j^h(t):=x^h(\rho _j, t), y_j^h(t):=y^h(\rho _j, t), \forall 0\le j\le N$.

Given that the finite element space $\mathbb {K}^h$ is the subspace of $H^1(\mathbb {T})$, the spatial semi-discretization (2.31) is a conforming discretization. Therefore, the structure-preserving property (2.25) of the weak formulation still holds for (2.31).

Proposition 2.2

(Area conservation and energy dissipation) Suppose $\Gamma ^h(t)$ is given by the solution $(\varvec{X}^h(\cdot , t), \mu ^h(\cdot , t))$ of (2.31), then we have

$$\begin{aligned} A^h(t)\equiv A^h(0),\quad W^h(t)\le W^h(t_1)\le W^h(0), \quad \forall t\ge t_1\ge 0. \end{aligned}$$

(2.35)

The proof is similar to [35, Proposition 3.1], and is omitted for brevity.

2.5 A structure-preserving PFEM

We then consider the full discretization. Let $\tau$ be the uniform time step, and $\Gamma ^m=\varvec{X}^m(\mathbb {T})\in [\mathbb {K}^h]^2$ be the approximation of $\Gamma ^h(t_m)=\varvec{X}^h(\mathbb {T}, t_m), \forall m\ge 0$, where $t_m:=m\tau$. Suppose $\varvec{h}^m_j:=\varvec{X}^m(\rho _j)-\varvec{X}^m(\rho _{j-1})$, we can similarly give the definitions for the mass lumped inner product $\left( \cdot , \cdot \right) ^h_{\Gamma ^m}$ as well as the unit normal vector $\varvec{n}^m$, the unit tangential vector $\varvec{\tau }^m$, and the $\varvec{\xi }$-vector $\varvec{\xi }^m$ with respect to $\Gamma ^m$. By adopting the backward semi-implicit Euler method, the fully-implicit structure-preserving discretization of PFEM for anisotropic surface diffusion (2.6) can then be given as:

Suppose the initial approximation $\Gamma ^0=\varvec{X}^0(\mathbb {T})\in [\mathbb {K}^h]^2$ is given by $\varvec{X}^0(\rho _j)=\varvec{X}_0(\rho _j), \forall 0\le j\le N$. For any $m=0, 1, 2, \ldots$, find the solution

$(\varvec{X}^m(\cdot )=(x^m(\cdot ),y^m(\cdot ))^T, \mu ^m(\cdot ))\in [\mathbb {K}^h]^2\times \mathbb {K}^h$, such that

$$\begin{aligned}&\Bigl (\varvec{n}^{m+\frac{1}{2}}\cdot \frac{\varvec{X}^{m+1}-\varvec{X}^m}{\tau }, \varphi ^h\Bigr )_{\Gamma ^m}^h+\Bigl (\partial _{s}\mu ^{m+1}, \partial _s\varphi ^h\Bigr )_{\Gamma ^m}^h=0,\quad \forall \varphi ^h\in \mathbb {K}^h, \end{aligned}$$

(2.36a)

$$\begin{aligned}&\Bigl (\mu ^{m+1}\varvec{n}^{m+\frac{1}{2}},\varvec{\omega }^h\Bigr )_{\Gamma ^m}^h-\Bigl ( \varvec{G}_k(\varvec{n}^m)\partial _s \varvec{X}^{m+1},\partial _s\varvec{\omega }^h\Bigr )_{\Gamma ^m}^h=0, \quad \forall \varvec{\omega }^h\in [\mathbb {K}^h]^2, \end{aligned}$$

(2.36b)

where

$$\begin{aligned} \varvec{G}_k(\varvec{n}^m)|_{I_j}=\gamma (\varvec{n}^m_j)I_2-\varvec{n}_j^m (\varvec{\xi }_j^m)^T+\varvec{\xi }_j^m(\varvec{n}^m_j)^T+k(\varvec{n}^m_j)\,\varvec{n}_j^m(\varvec{n}_j^m)^T, \end{aligned}$$

(2.37)

and

$$\begin{aligned} \varvec{n}^{m+\frac{1}{2}}:=-\frac{1}{2}\frac{1}{|\partial _\rho \varvec{X}^m|}(\partial _\rho \varvec{X}^m+\partial _\rho \varvec{X}^{m+1})^\perp . \end{aligned}$$

(2.38)

The SP-PFEM (2.36) is fully-implicit, and can be numerically solved by Newton’s iterative method. Here we adapt the iterative solver, originally proposed for isotropic surface diffusion in [8], to solve our SP-PFEM (2.36) of anisotropic surface diffusion. The only change is we replace their identity matrix $I_2$ with our surface energy matrix $\varvec{G}_k(\varvec{n}^m)$. We observe that most variables, especially the integral domain $\Gamma ^m$, are treated explicitly, the non-linearity arises from $\varvec{n}^{m+\frac{1}{2}}$. If $\varvec{n}^{m+\frac{1}{2}}$ is replaced by $\varvec{n}^{m}$, then (2.36) becomes semi-implicit. However, the clever approximation $\varvec{n}^{m+\frac{1}{2}}$ is critical in preserving the area conservation, and the semi-implicit PFEM can only preserve the energy dissipation property.

2.6 Main results

Denote the enclosed area and the total energy of the closed line segments $\Gamma ^m$ as $A^m$ and $W^m$, respectively, which are given by

$$\begin{aligned} A^m&=\frac{1}{2}\sum _{j=1}^N(x^m(\rho _j)-x^m(\rho _{j-1})(y^m(\rho _j) +y^m(\rho _{j-1})), \end{aligned}$$

(2.39a)

$$\begin{aligned} W^m&=\sum _{j=1}^N|\varvec{h}_j^m|\gamma (\varvec{n}^m_j). \end{aligned}$$

(2.39b)

Introduce two auxiliary functions $P_{\alpha }(\varvec{n}, \hat{\varvec{n}}), Q(\varvec{n}, \hat{\varvec{n}})$ as

$$\begin{aligned}&P_{\alpha }(\varvec{n}, \hat{\varvec{n}}):=2\sqrt{(\gamma (\varvec{n}) +\alpha (\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2)\gamma (\varvec{n})},\quad \forall \varvec{n}\in \mathbb {S}^1, \quad \alpha \ge 0, \end{aligned}$$

(2.40a)

$$\begin{aligned}&Q(\varvec{n}, \hat{\varvec{n}}):=\gamma (\hat{\varvec{n}})+\gamma (\varvec{n})(\varvec{n}\cdot \hat{\varvec{n}})-(\varvec{\xi }\cdot \varvec{n}^\perp )(\hat{\varvec{n}}\cdot \varvec{n}^\perp ), \quad \forall \varvec{n}, \hat{\varvec{n}}\in \mathbb {S}^1, \end{aligned}$$

(2.40b)

we define the minimal stabilizing function $k_0(\varvec{n})$ as (its existence will be given in the next section)

$$\begin{aligned} k_0(\varvec{n}):=\inf \{\alpha \ge 0: P_{\alpha }(\varvec{n}, \hat{\varvec{n}})-Q(\varvec{n}, \hat{\varvec{n}})\ge 0, \ \forall \hat{\varvec{n}}\in \mathbb {S}^1\}, \quad \forall \varvec{n}\in \mathbb {S}^1. \end{aligned}$$

(2.41)

For the SP-PFEM (2.36), we have

Theorem 2.1

(Structure-preserving) Under the simple and mild condition (1.6) on the anisotropic surface energy $\gamma (\varvec{n})$ and taking the stabilizing function $k(\varvec{n})\ge k_0(\varvec{n})$ for $\varvec{n}\in \mathbb {S}^1$, then the SP-PFEM (2.36) preserves the two geometric properties—mass conservation and energy dissipation—in the discrete level, i.e.,

$$\begin{aligned} A^{m+1}\equiv A^0,\qquad W^{m+1}\le W^m\le \ldots \le W^0, \qquad \forall m\ge 0. \end{aligned}$$

(2.42)

The area conservation part is a direct result of [8, Theorem 2.1]. And the energy dissipation will be proved in the next section.

Remark 2.3

The construction of the two auxiliary functions $P_{\alpha }(\varvec{n}, \hat{\varvec{n}}), Q(\varvec{n}, \hat{\varvec{n}})$ are inspired by [35]. Indeed, as elaborated in Remark 2.1, [35] is a special case $\alpha =0$ of this paper, and its energy stability result was established by showing that $Q(\varvec{n}, \hat{\varvec{n}})-2\gamma (\varvec{n})\le 0=P_{0}(\varvec{n}, \hat{\varvec{n}})-2\gamma (\varvec{n})$.

3 Energy dissipation of the SP-PFEM (2.36)

In this section, we first show if $\gamma (\varvec{n})$ satisfies (1.6), the minimal stabilizing function $k_0(\varvec{n})$ defined in (2.41) is well-defined, thus we can always choose a nonnegative stabilizing function $k(\varvec{n})\ge k_0(\varvec{n})$ for $\varvec{n}\in \mathbb {S}^1$. After that, we will use $k_0(\varvec{n})$ to give the proof of the unconditional energy stability part of the main theorem 2.1.

3.1 Existence of the minimal stabilizing function $k_0(\varvec{n})$ defined in (2.41)

From the definition of $k_0(\varvec{n})$ in (2.41), we observe that if $(\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2> 0$, then intuitively for sufficiently large $\alpha$, we know the $P_{\alpha }(\varvec{n}, \hat{\varvec{n}})\ge Q(\varvec{n}, \hat{\varvec{n}})$. But this approach will fail when $(\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2=0$, and this can happen if $\hat{\varvec{n}}=\pm \varvec{n}$, which suggests us to treat the two cases $\hat{\varvec{n}}\cdot \varvec{n}\ge 0$ and $\hat{\varvec{n}}\cdot \varvec{n}\le 0$ separately. To simplify the notations, we introduce a compact set $\mathbb {S}^1_{\varvec{n}}$ as

$$\begin{aligned} \mathbb {S}^1_{\varvec{n}}:=\{\hat{\varvec{n}}\in \mathbb {S}^1| \varvec{n}\cdot \hat{\varvec{n}}\ge 0\}, \quad \varvec{n}\in \mathbb {S}^1. \end{aligned}$$

(3.1)

Then $\varvec{n}\cdot \hat{\varvec{n}}\ge 0\iff \hat{\varvec{n}}\in \mathbb {S}^1_{\varvec{n}}$, and $\varvec{n}\cdot \hat{\varvec{n}}\le 0\iff \hat{\varvec{n}}\in \mathbb {S}^1_{-\varvec{n}}$.

Theorem 3.1

The $k_0(\varvec{n})$ defined as (2.41) is bounded if the condition (1.6) on $\gamma (\varvec{n})$ is satisfied.

Proof

First we consider the case $\hat{\varvec{n}}\in \mathbb {S}^1_{\varvec{n}}$. Since $\gamma (\varvec{n})$ satisfies (1.6), we know $\gamma (\varvec{p})\in C^2(\mathbb {R}^2{\setminus } \{\varvec{0}\})$. Thus there exists a constant $C_1>0$, such that

$$\begin{aligned} \left\| \textbf{H}_{\gamma }(\varvec{p})\right\| _2\le C_1, \quad \forall \frac{1}{2}\le |\varvec{p}|^2\le 1, \end{aligned}$$

(3.2)

here $\textbf{H}_{\gamma }$ is the Hessian matrix of $\gamma (\varvec{p})$, and $\left\| \cdot \right\| _2$ denotes the 2-norm.

By mean value theorem, we know for all $\hat{\varvec{n}}\in \mathbb {S}^1_{\varvec{n}}$, there exists a constant $0<c<1$ such that

$$\begin{aligned} \gamma (\hat{\varvec{n}})=\gamma (\varvec{n})+\varvec{\xi }\cdot (\hat{\varvec{n}}-\varvec{n})+ \frac{1}{2}(\hat{\varvec{n}}-\varvec{n})^T\, \textbf{H}_\gamma \left( c\varvec{n}+(1-c)\hat{\varvec{n}}\right) \, (\hat{\varvec{n}}-\varvec{n}). \end{aligned}$$

(3.3)

It is easy to see that $|c\varvec{n}+(1-c)\hat{\varvec{n}}|\le 1$ and $|c\varvec{n}+(1-c)\hat{\varvec{n}}|^2\ge c^2+(1-c)^2\ge \frac{1}{2}$. Hence we know

$$\begin{aligned} \gamma (\hat{\varvec{n}})\le \gamma (\varvec{n})+\varvec{\xi }\cdot (\hat{\varvec{n}}-\varvec{n})+\frac{C_1}{2} |\hat{\varvec{n}}-\varvec{n}|^2. \end{aligned}$$

(3.4)

And notice that $\varvec{\xi }\cdot \varvec{n}=\gamma (\varvec{n})$, we can then get the following estimate of $Q(\varvec{n}, \hat{\varvec{n}})$:

$$\begin{aligned}&Q(\varvec{n}, \hat{\varvec{n}})-2\gamma (\varvec{n})\nonumber \\&\le \left( \gamma (\varvec{n})+\varvec{\xi }\cdot (\hat{\varvec{n}}-\varvec{n})+\frac{C_1}{2}|\hat{\varvec{n}}-\varvec{n}|^2\right) -\Bigl (\varvec{\xi }\cdot \hat{\varvec{n}}-(\varvec{\xi }\cdot \varvec{n})(\hat{\varvec{n}}\cdot \varvec{n})\Bigr )\nonumber \\&\qquad +\gamma (\varvec{n})(\varvec{n}\cdot \hat{\varvec{n}})-2\gamma (\varvec{n})\nonumber \\&=2\gamma (\varvec{n})((\varvec{n}\cdot \hat{\varvec{n}})-1)+\frac{C_1}{2}|\hat{\varvec{n}}-\varvec{n}|^2\nonumber \\&=\left( \frac{C_1}{2}-\gamma (\varvec{n})\right) |\hat{\varvec{n}}-\varvec{n}|^2. \end{aligned}$$

(3.5)

Here we use the fact $\varvec{\xi }\cdot \hat{\varvec{n}}=(\varvec{\xi }\cdot \varvec{n})(\hat{\varvec{n}}\cdot \varvec{n})+(\varvec{\xi }\cdot \varvec{n}^\perp )(\hat{\varvec{n}}\cdot \varvec{n}^\perp )$ and $|\hat{\varvec{n}}-\varvec{n}|^2=2-2\varvec{n}\cdot \hat{\varvec{n}}$.

On the other hand, using the fact $(\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2=1-(\hat{\varvec{n}}\cdot \varvec{n})^2=\frac{|\hat{\varvec{n}}-\varvec{n}|^2}{2}(1+\varvec{n}\cdot \hat{\varvec{n}})$, we know that for $\alpha >\gamma (\varvec{n})$, it holds

$$\begin{aligned} P_{\alpha }(\varvec{n}, \hat{\varvec{n}})-2\gamma (\varvec{n})&= \frac{2\alpha (\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2}{\sqrt{1+ \frac{\alpha }{\gamma (\varvec{n})}(\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2}+1}\nonumber \\&\ge \frac{2\alpha (\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2}{2\sqrt{1+\frac{\alpha }{\gamma (\varvec{n})}}} =\frac{\alpha (1+\hat{\varvec{n}}\cdot \varvec{n})}{2\sqrt{1+\frac{\alpha }{\gamma (\varvec{n})}}} |\hat{\varvec{n}}-\varvec{n}|^2\nonumber \\&\ge \frac{\sqrt{\gamma (\varvec{n})(\alpha -\gamma (\varvec{n}))}}{2}|\hat{\varvec{n}}-\varvec{n}|^2. \end{aligned}$$

(3.6)

Combining (3.5) and (3.6), we know that for $\alpha \ge \alpha _1:= \frac{(C_1-2\gamma (\varvec{n}))^2+\gamma ^2(\varvec{n})}{\gamma (\varvec{n})}<\infty$, it holds $P_{\alpha }(\varvec{n}, \hat{\varvec{n}})\ge Q(\varvec{n}, \hat{\varvec{n}}), \forall \hat{\varvec{n}}\in \mathbb {S}^1_{\varvec{n}}$.

For the case $\hat{\varvec{n}}\in \mathbb {S}^1_{-\varvec{n}}$, by (1.6), when $\hat{\varvec{n}}=-\varvec{n}$, we know that

$$\begin{aligned} Q(\varvec{n}, -\varvec{n})&=\gamma (-\varvec{n})+\gamma (\varvec{n}) (\varvec{n}\cdot (-\varvec{n}))-(\varvec{\xi }\cdot \varvec{n}^\perp )(-\varvec{n}\cdot \varvec{n}^\perp )\nonumber \\&<3\gamma (\varvec{n})-\gamma (\varvec{n})=2\gamma (\varvec{n}). \end{aligned}$$

(3.7)

Thus for $\alpha =0:=\alpha _{-\varvec{n}}<\infty$, we know $P_0(\varvec{n}, -\varvec{n})=2\gamma (\varvec{n})>Q(\varvec{n}, -\varvec{n})$. By continuity of $P_{\alpha }$ and Q, there exits an open set $\mathcal {O}_{-\varvec{n}, 0}\subset \mathbb {S}^1$ such that $-\varvec{n}\in \mathcal {O}_{-\varvec{n}, 0}$, and for all $\hat{\varvec{n}}\in \mathcal {O}_{-\varvec{n}, 0}$ and $\alpha \ge 0$, we have $P_{\alpha }(\varvec{n}, \hat{\varvec{n}})-Q(\varvec{n}, \hat{\varvec{n}})\ge 0$. For a given $\varvec{p}\in \mathbb {S}^1_{-\varvec{n}}, \varvec{p}\ne -\varvec{n}$, we know that $(\varvec{p}\cdot \varvec{n}^\perp )^2>0$. Therefore we can choose a sufficiently large but finite $\alpha _{\varvec{p}}<\infty$, such that $P_{\alpha _{\varvec{p}}}(\varvec{n},\varvec{p})-Q(\varvec{n}, \varvec{p})> 0$. And by the same argument, there exists an open set $\mathcal {O}_{\varvec{p}, \alpha _{\varvec{p}}}\subset \mathbb {S}^1$, such that $\varvec{p}\in \mathcal {O}_{\varvec{p}, \alpha _{\varvec{p}}}$ and $P_{\alpha }(\varvec{n}, \hat{\varvec{n}})-Q(\varvec{n}, \hat{\varvec{n}})\ge 0, \forall \hat{\varvec{n}}\in \mathcal {O}_{\varvec{p}, \alpha _{\varvec{p}}}, \alpha \ge \alpha _{\varvec{p}}$. And we obtain an open cover for $\mathbb {S}_{-\varvec{n}}^1$ as

$$\begin{aligned} \mathbb {S}^1_{-\varvec{n}}\subset \bigcup \limits _{\varvec{p}\in \mathbb {S}^1_{-\varvec{n}}} \mathcal {O}_{\varvec{p}, \alpha _{\varvec{p}}}. \end{aligned}$$

(3.8)

Since $\mathbb {S}^1_{-\varvec{n}}$ is compact, by open cover theorem, there is a finite set of vectors $\varvec{p}_1, \ldots , \varvec{p}_M \in \mathbb {S}^1_{-\varvec{n}}$, such that

$$\begin{aligned} \mathbb {S}^1_{-\varvec{n}}\subset \bigcup \limits _{i=1}^M \mathcal {O}_{\varvec{p}_i, \alpha _{\varvec{p}_i}}. \end{aligned}$$

(3.9)

If we take $\alpha _2:=\max \limits _{1\le i\le M} \alpha _{\varvec{p}_i}<\infty$, we have $P_{\alpha }(\varvec{n}, \hat{\varvec{n}})-Q(\varvec{n}, \hat{\varvec{n}})\ge 0, \forall \hat{\varvec{n}}\in \mathbb {S}^1_{-\varvec{n}}, \alpha \ge \alpha _2$, hence

$$\begin{aligned} \infty >\max (\alpha _1, \alpha _2)\in \left\{ \alpha \ge 0: P_{\alpha }(\varvec{n}, \hat{\varvec{n}})-Q(\varvec{n}, \hat{\varvec{n}})\ge 0,\ \forall \hat{\varvec{n}}\in \mathbb {S}^1\right\} . \end{aligned}$$

(3.10)

Which means $k_0(\varvec{n})=\inf \left\{ \alpha \ge 0: P_{\alpha }(\varvec{n}, \hat{\varvec{n}})-Q(\varvec{n}, \hat{\varvec{n}})\ge 0, \forall \hat{\varvec{n}}\in \mathbb {S}^1\right\} <\infty$. $\square$

Remark 3.1

The $\hat{\varvec{n}}\in \mathbb {S}^1_{\varvec{n}}$ part only requires inequality (3.4), thus condition $\gamma (\varvec{p})\in C^2(\mathbb {R}^2{\setminus } \{\varvec{0}\})$ can be relaxed to $\gamma (\varvec{p})$ is piecewise $C^2(\mathbb {R}^2\setminus \{\varvec{0}\})$. And the condition $\gamma (-\varvec{n})<3\gamma (\varvec{n})$ is to ensure the existence of $\mathcal {O}_{-\varvec{n}, 0}$, which suggests we may find a larger $\alpha _{-\varvec{n}}>0$ such that $\mathcal {O}_{-\varvec{n}, \alpha _{-\varvec{n}}}$ exists for $\gamma (\varvec{n})$ satisfies $\gamma (-\varvec{n})\le 3\gamma (\varvec{n})$. And by the same argument, we can show such $\mathcal {O}_{-\varvec{n}, \alpha _{-\varvec{n}}}$ exists if and only if $\varvec{\xi }(\varvec{n}_i)=\gamma (\varvec{n}_i)\varvec{n}_i \iff \gamma (-\varvec{n}_i)=3\gamma (\varvec{n}_i)$.

By the existence of $k_0(\varvec{n})$, once the $\gamma (\varvec{n})$ is given, the minimal stabilizing function $k_0(\varvec{n})$ is then determined, i.e. there is a map from $\gamma (\varvec{n})$ to $k_0(\varvec{n})$. Similar to the proof when $\gamma (\varvec{n})$ is symmetric, i.e. $\gamma (-\varvec{n})=\gamma (\varvec{n})$ in [2, 8], we can show such map is a sub-linear with respect to $\gamma (\varvec{n})$ when it satisfies (1.6).

Lemma 3.1

(Positive homogeneity and subadditivity) Let $\gamma _1(\varvec{n}), \gamma _2(\varvec{n})$ and $\gamma _3(\varvec{n})$ be three functions satisfying (1.6) with minimal stabilizing functions $k_0^{(1)}(\varvec{n}), k_0^{(2)}(\varvec{n})$ and $k_0^{(3)}(\varvec{n})$, respectively, then we have

(i) if $\gamma _2(\varvec{n})=c \gamma _1(\varvec{n})$, where $c>0$ is a positive number, then $k_0^{(2)}(\varvec{n})=ck_0^{(1)}(\varvec{n})$; and

(ii) if $\gamma _3(\varvec{n})=\gamma _1(\varvec{n})+\gamma _2(\varvec{n})$, then $k_0^{(3)}(\varvec{n})\le k_0^{(1)}(\varvec{n})+k_0^{(2)}(\varvec{n})$.

Proof

Let $\varvec{\xi }^{(i)}, P_{\alpha }^{(i)}(\varvec{n}, \hat{\varvec{n}}), Q^{(i)}(\varvec{n}, \hat{\varvec{n}}), k_0^{(i)}(\varvec{n})$ be the $\varvec{\xi }$-vector, the auxiliary function $\varvec{P}_{\alpha }(\varvec{n}, \hat{\varvec{n}}), Q(\varvec{n}, \hat{\varvec{n}})$ and the minimal stabilizing function $k_0(\varvec{n})$ for $\gamma _i(\varvec{n}), i=1,2,3$, respectively.

(i) If $\gamma _2(\varvec{n})=c\gamma _1(\varvec{n})$ with $c>0$, then it is easy to see that

$$\begin{aligned} \varvec{\xi }^{(2)}=c\, \varvec{\xi }^{(1)}, \quad P_{c\,\alpha }^{(2)}(\varvec{n}, \hat{\varvec{n}})=c\,P_{\alpha }^{(1)}(\varvec{n}, \hat{\varvec{n}}), \quad Q^{(2)}(\varvec{n}, \hat{\varvec{n}})=c\,Q^{(1)}(\varvec{n}, \hat{\varvec{n}}). \end{aligned}$$

This, together with the definition of $k_0^{(1)}(\varvec{n})$ yields that

$$\begin{aligned} P_{c\,k_0^{(1)}(\varvec{n})}^{(2)}(\varvec{n}, \hat{\varvec{n}})-Q^{(2)}(\varvec{n}, \hat{\varvec{n}})=c\left( P_{k_0^{(1)}(\varvec{n})}^{(1)}(\varvec{n}, \hat{\varvec{n}})-Q^{(1)}(\varvec{n}, \hat{\varvec{n}})\right) \ge 0. \end{aligned}$$

Which indicates $k_0^{(2)}(\varvec{n})\le c\,k_0^{(1)}(\varvec{n})$. Similarly, $\gamma _1(\varvec{n})=\frac{1}{c}\gamma _2(\varvec{n})$ implies that

$k_0^{(1)}(\varvec{n})\le \frac{1}{c}\,k_0^{(2)}(\varvec{n})$. Therefore, we obtain $k_0^{(2)}(\varvec{n})=c\,k_0^{(1)}(\varvec{n})$.

(ii) If $\gamma _3(\varvec{n})=\gamma _1(\varvec{n})+\gamma _2(\varvec{n})$, then from (2.40b) it is easy to see $\varvec{\xi }^{(3)}=\varvec{\xi }^{(1)}+\varvec{\xi }^{(2)}$ and $Q^{(3)}(\varvec{n}, \hat{\varvec{n}})=Q^{(1)}(\varvec{n}, \hat{\varvec{n}})+Q^{(2)}(\varvec{n}, \hat{\varvec{n}})$.

On the other hand, for any $a, b>0$, by using the Cauchy inequality and noting (2.40a), we derive that

$$\begin{aligned}&P_{a+b}^{(3)}(\varvec{n}, \hat{\varvec{n}})\nonumber \\&=2\sqrt{\left[ \left( \gamma _1(\varvec{n})+a(\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2\right) +\left( \gamma _2(\varvec{n})+b(\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2\right) \right] \left[ (\gamma _1(\varvec{n}))+(\gamma _2(\varvec{n}))\right] }\nonumber \\&\ge 2\sqrt{\left( \sqrt{\left( \gamma _1(\varvec{n})+a(\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2\right) \gamma _1(\varvec{n})}+\sqrt{\left( \gamma _2(\varvec{n})+b(\hat{\varvec{n}}\cdot \varvec{n}^\perp )^2\right) \gamma _2(\varvec{n})}\right) ^2}\nonumber \\&=P^{(1)}_a(\varvec{n}, \hat{\varvec{n}})+P_b^{(2)}(\varvec{n}, \hat{\varvec{n}}). \end{aligned}$$

(3.11)

Taking $a=k_0^{(1)}(\varvec{n})>0, b=k_0^{(2)}(\varvec{n})>0$ in (3.11) and using the definition of $k_0(\varvec{n})$, we find that

$$\begin{aligned}&P_{k_0^{(1)}(\varvec{n})+k_0^{(2)}(\varvec{n})}^{(3)}(\varvec{n}, \hat{\varvec{n}})-Q^{(3)}(\varvec{n}, \hat{\varvec{n}})\\&\ge P_{k_0^{(1)}(\varvec{n})}^{(1)}(\varvec{n}, \hat{\varvec{n}})+P_{k_0^{(2)}(\varvec{n})}^{(2)}(\varvec{n}, \hat{\varvec{n}})-Q^{(1)}(\varvec{n}, \hat{\varvec{n}})-Q^{(2)}(\varvec{n}, \hat{\varvec{n}})\ge 0, \end{aligned}$$

which indicates $k_0^{(3)}(\varvec{n})\le k_0^{(1)}(\varvec{n})+k_0^{(2)}(\varvec{n})$.$\square$

3.2 Proof of the energy dissipation in (2.42)

To prove the main result, we first need the following lemma:

Lemma 3.2

Suppose $\varvec{h}, \hat{\varvec{h}}$ are two non-zero vectors in $\mathbb {R}^2$ and $\varvec{n}=-\frac{\varvec{h}^\perp }{|\varvec{h}|}, \hat{\varvec{n}}=-\frac{\hat{\varvec{h}}^\perp }{|\hat{\varvec{h}}|}$ to be the corresponding unit normal vectors. Then for any $k(\varvec{n})\ge k_0(\varvec{n})$, the following inequality holds

$$\begin{aligned} \frac{1}{|\varvec{h}|}\left( \varvec{G}_k(\varvec{n}) \hat{\varvec{h}}\right) \cdot (\hat{\varvec{h}}-\varvec{h})\ge |\hat{\varvec{h}}|\gamma (\hat{\varvec{n}})-|\varvec{h}|\,\gamma (\varvec{n}). \end{aligned}$$

(3.12)

Proof

By applying the definition of $\varvec{G}_k(\varvec{n})$ in (2.7) and $P_{\alpha }(\varvec{n}, \hat{\varvec{n}})$ in (2.40a), and noticing $\hat{\varvec{h}}=\hat{\varvec{n}}^\perp |\hat{\varvec{h}}|, \varvec{n}\cdot \hat{\varvec{n}}^\perp =-\hat{\varvec{n}}\cdot \varvec{n}^\perp$, the term $\frac{1}{|\varvec{h}|}\left( \varvec{G}_k(\varvec{n})\hat{\varvec{h}}\right) \cdot \hat{\varvec{h}}$ can be simplified as

$$\begin{aligned}&\frac{1}{|\varvec{h}|}\left( \varvec{G}_k(\varvec{n})\hat{\varvec{h}}\right) \cdot \hat{\varvec{h}}\nonumber \\&=\frac{1}{|\varvec{h}|}\left[ \left( (\gamma (\varvec{n})I_2+k(\varvec{n}) \varvec{n}(\varvec{n})^T)+(\varvec{\xi } (\varvec{n})^T-\varvec{n}(\varvec{\xi })^T) \right) \hat{\varvec{h}}\right] \cdot \hat{\varvec{h}}\nonumber \\&=\frac{1}{|\varvec{h}|}\left[ \left( \gamma (\varvec{n})I_2+k(\varvec{n}) \varvec{n}(\varvec{n})^T \right) \hat{\varvec{h}}\right] \cdot \hat{\varvec{h}}\nonumber \\&=\frac{1}{|\varvec{h}|}\left( \gamma (\varvec{n})|\hat{\varvec{h}}|^2+k(\varvec{n})(\varvec{n}\cdot \hat{\varvec{h}})^2\right) \nonumber \\&=\frac{1}{|\varvec{h}|}\left( \gamma (\varvec{n})|\hat{\varvec{h}}|^2+k(\varvec{n})(\hat{\varvec{n}}\cdot (\varvec{n})^\perp )^2|\hat{\varvec{h}}|^2\right) =\frac{|\hat{\varvec{h}}|^2}{4|\varvec{h}|\gamma (\varvec{n})}P^2_{k(\varvec{n})}(\varvec{n}, \hat{\varvec{n}}). \end{aligned}$$

(3.13)

Similarly, by applying the definition of $\varvec{G}_k(\varvec{n})$ in (2.7) and $Q(\varvec{n}, \hat{\varvec{n}})$ in (2.40b), and noticing $\varvec{h}=\varvec{n}^\perp |\varvec{h}|, \hat{\varvec{h}}=\hat{\varvec{n}}^\perp |\hat{\varvec{h}}|,\varvec{h}\cdot \hat{\varvec{h}}= \varvec{n}\cdot \hat{\varvec{n}}|\varvec{h}|\,|\hat{\varvec{h}}|, \varvec{n}\cdot \hat{\varvec{n}}^\perp =-\hat{\varvec{n}}\cdot \varvec{n}^\perp$, the term $\frac{1}{|\varvec{h}|}\left( \varvec{G}_k(\varvec{n})\hat{\varvec{h}}\right) \cdot \varvec{h}$ can be simplified as

$$\begin{aligned}&\frac{1}{|\varvec{h}|}\left( \varvec{G}_k(\varvec{n})\hat{\varvec{h}}\right) \cdot \varvec{h}=\frac{1}{|\varvec{h}|}\left( \varvec{G}_k^T(\varvec{n})\varvec{h}\right) \cdot \hat{\varvec{h}}\nonumber \\&=\frac{1}{|\varvec{h}|}\left[ \left( (\gamma (\varvec{n})I_2+\varvec{n} (\varvec{\xi })^T)+(-\varvec{\xi }(\varvec{n})^T +k(\varvec{n})\varvec{n}(\varvec{n})^T)\right) \varvec{h}\right] \cdot \hat{\varvec{h}}\nonumber \\&=\frac{1}{|\varvec{h}|}\left[ \left( \gamma (\varvec{n})I_2+\varvec{n} (\varvec{\xi })^T\right) \varvec{h}\right] \cdot \hat{\varvec{h}}\nonumber \\&=\frac{1}{|\varvec{h}|}\left[ \gamma (\varvec{n})(\varvec{h}\cdot \hat{\varvec{h}})+(\varvec{\xi }\cdot \varvec{h})(\varvec{n}\cdot \hat{\varvec{h}})\right] \nonumber \\&=|\hat{\varvec{h}}|\,\left( \gamma (\varvec{n})(\varvec{n}\cdot \hat{\varvec{n}})+(\varvec{\xi }\cdot \varvec{n}^\perp )(\varvec{n}\cdot \hat{\varvec{n}}^\perp )\right) \nonumber \\&=|\hat{\varvec{h}}|\,\left( \gamma (\varvec{n})(\varvec{n}\cdot \hat{\varvec{n}})-(\varvec{\xi }\cdot \varvec{n}^\perp )(\hat{\varvec{n}}\cdot \varvec{n}^\perp )\right) =|\hat{\varvec{h}}|\,\left( Q(\varvec{n}, \hat{\varvec{n}})-\gamma (\hat{\varvec{n}})\right) . \end{aligned}$$

(3.14)

Finally, combining the definition of $k_0(\varvec{n})$ (2.41), (3.13), (3.14), and the fact $\frac{a^2}{4|\varvec{h}|\gamma (\varvec{n})}\ge ab-|\varvec{h}|\gamma (\varvec{n})b^2$ yields that

$$\begin{aligned} \frac{1}{|\varvec{h}|}\left( \varvec{G}_k(\varvec{n}) \hat{\varvec{h}}\right) \cdot (\hat{\varvec{h}}-\varvec{h})&=\frac{|\hat{\varvec{h}}|^2}{4|\varvec{h}|\gamma (\varvec{n})}P^2_{k(\varvec{n})}(\varvec{n}, \hat{\varvec{n}})-|\hat{\varvec{h}}|\,\left( Q(\varvec{n}, \hat{\varvec{n}})-\gamma (\hat{\varvec{n}})\right) \nonumber \\&\ge |\hat{\varvec{h}}|P_{k(\varvec{n})}-|\varvec{h}|\gamma (\varvec{n})-|\hat{\varvec{h}}|\,\left( P_{k(\varvec{n})}(\varvec{n}, \hat{\varvec{n}})-\gamma (\hat{\varvec{n}})\right) \nonumber \\&=|\hat{\varvec{h}}|\gamma (\hat{\varvec{n}})-|\varvec{h}|\,\gamma (\varvec{n}), \end{aligned}$$

(3.15)

which validates (3.12).$\square$

Now we can prove the energy dissipation part in our main result Theorem 2.1.

Proof

The key point of the proof is to establish the following energy estimation

$$\begin{aligned} \Bigl ( \varvec{G}_k(\varvec{n}^m)\partial _s \varvec{X}^{m+1},\partial _s(\varvec{X}^{m+1} -\varvec{X}^m)\Bigr )_{\Gamma ^m}^h\ge W^{m+1}-W^{m},\quad m\ge 0. \end{aligned}$$

(3.16)

For any $1\le j\le N$, take $\varvec{h}=\varvec{h}_j^m, \hat{\varvec{h}}=\varvec{h}_j^{m+1}$ in Lemma 3.2, we know that $\varvec{n}=-\frac{\varvec{h}^\perp }{|\varvec{h}|}=\varvec{n}^m_j, \hat{\varvec{n}}=\varvec{n}_j^{m+1}$, and the following inequality holds

$$\begin{aligned} \frac{1}{|\varvec{h}_j^m|}\left( \varvec{G}_k(\varvec{n}_j^m) \varvec{h}_j^{m+1}\right) \cdot (\varvec{h}_j^{m+1}-\varvec{h}_j^m)\ge |\varvec{h}_j^{m+1}|\gamma (\varvec{n}_j^{m+1})-|\varvec{h}_j^m|\,\gamma (\varvec{n}_j^m). \end{aligned}$$

(3.17)

Taking the summation over $1\le j\le N$ for (3.17) and notice (2.28) and (2.39b), we have

$$\begin{aligned}&\Bigl ( \varvec{G}_k(\varvec{n}^m)\partial _s \varvec{X}^{m+1},\partial _s(\varvec{X}^{m+1}-\varvec{X}^m)\Bigr )_{\Gamma ^m}^h\\&\quad =\sum _{j=1}^N\left[ |\varvec{h}_j^m|\left( \varvec{G}_k(\varvec{n}_j^m)\frac{\varvec{h}_j^{m+1}}{|\varvec{h}_j^m|}\right) \cdot \frac{\varvec{h}_j^{m+1}-\varvec{h}_j^m}{|\varvec{h}_j^m|}\right] \\&\quad =\sum _{j=1}^N\left[ \frac{1}{|\varvec{h}_j^m|}\left( \varvec{G}_k(\varvec{n}_j^m)\varvec{h}_j^{m+1}\right) \cdot (\varvec{h}_j^{m+1}-\varvec{h}_j^m)\right] \\&\quad \ge \sum _{j=1}^N\left[ |\varvec{h}_j^{m+1}|\gamma (\varvec{n}_j^{m+1})-|\varvec{h}_j^m|\,\gamma (\varvec{n}_j^m)\right] \\&\quad =W^{m+1}-W^m, \qquad \forall m=0, 1,\ldots \end{aligned}$$

which proves the energy estimation in (3.16).

Finally, take $\varphi ^h=\mu ^{m+1}$ in (2.36a) and $\varvec{\omega }^h=\varvec{X}^{m+1}-\varvec{X}^m$ in (2.36b), we have

$$\begin{aligned} 0&\ge -\tau \Bigl ( \partial _s\mu ^{m+1},\partial _s\mu ^{m+1}\Bigr )_{\Gamma ^m}^h\\&=\Bigl (\varvec{n}^{m+\frac{1}{2}}\cdot \left( \varvec{X}^{m+1}-\varvec{X}^m\right) ,\mu ^{m+1}\Bigr )_{\Gamma ^m}^h\\&=\Bigl ( \varvec{G}_k(\varvec{n}^m)\partial _s \varvec{X}^{m+1},\partial _s(\varvec{X}^{m+1}-\varvec{X}^m)\Bigr )_{\Gamma ^m}^h\ge W^{m+1}-W^m, \end{aligned}$$

this is true for any m; therefore, the energy $W^m$ decreases monotonically, and the proof is completed. $\square$

Remark 3.2

The condition $\gamma (\varvec{p})\in C^2(\mathbb {R}^2{\setminus } \{\varvec{0}\})$ in (1.6) is natural, but $\gamma (-\varvec{n})<3\gamma (\varvec{n})$ looks quite complicated and seems not very sharp. However, the proof shows the condition $\gamma (-\varvec{n})<3\gamma (\varvec{n})$ is indeed natural! To see this, inequality (3.12) in Lemma 3.2 is essential in showing the energy estimate (3.16). And if we take $\hat{\varvec{h}}=-\varvec{h}$ in Lemma 3.2, then $\hat{\varvec{n}}=-\varvec{n}$, and the inequality (3.12) becomes

$$\begin{aligned} 2\gamma (\varvec{n})\,|\varvec{h}|\ge |\varvec{h}|\,\gamma (-\varvec{n})-\gamma (\varvec{n})\, |\varvec{h}| \iff \gamma (-\varvec{n})|\varvec{h}|\le 3\gamma (\varvec{n})|\varvec{h}|, \end{aligned}$$

(3.18)

which means $\gamma (-\varvec{n})\le 3\gamma (\varvec{n})$. Our sufficient condition for energy stability, as stated in (1.6), replaces $\ge$ with >, thus it is both natural and almost necessary.

4 Extension to other anisotropic geometric flows

In fact, the energy stable condition on $\gamma (\varvec{n})$ in (1.6), the definition of $\varvec{G}_k(\varvec{n})$ in (2.7), the alternative expression for $\mu$ in (2.9), and the definition of $k_0(\varvec{n})$ in (2.41) are independent of the anisotropic surface diffusion flow. Thus these definitions and even the proof of energy stability can be directly extended to other anisotropic geometric flows.

4.1 Anisotropic curvature flow

Similar to (2.12), for the anisotropic curvature flow in (1.3), we have a conservative geometric PDE as

$$\begin{aligned}&\varvec{n}\cdot \partial _t \varvec{X}=-\mu , \end{aligned}$$

(4.1a)

$$\begin{aligned}&\mu \varvec{n}=-\partial _s(\varvec{G}_k(\varvec{n})\partial _s \varvec{X}). \end{aligned}$$

(4.1b)

Suppose the initial curve $\varvec{X}(\mathbb {T}, 0)=(x(\mathbb {T}, 0), y(\mathbb {T}, 0))^T:=\Gamma _0\in [H^1(\mathbb {T})]^2$ and the initial weighted curvature $\mu (\cdot , 0):=\mu _0(\cdot )\in H^1(\mathbb {T})$. Based on the conservative form (4.1), the variational formulation for anisotropic curvature flow is as follows: For any $t>0$, find the solution $(\varvec{X}(\cdot , t), \mu (\cdot , t))\in [H^1(\mathbb {T})]^2\times H^1(\mathbb {T})$ satisfying

$$\begin{aligned}&\Bigl (\varvec{n}\cdot \partial _{t}\varvec{X}, \varphi \Bigr )_{\Gamma (t)}+\Bigl (\mu , \varphi \Bigr )_{\Gamma (t)}=0,\qquad \forall \varphi \in H^1(\mathbb {T}), \end{aligned}$$

(4.2a)

$$\begin{aligned}&\Bigl (\mu \varvec{n},\varvec{\omega }\Bigr )_{\Gamma (t)}-\Bigl ( \varvec{G}_k(\varvec{n})\partial _s \varvec{X},\partial _s\varvec{\omega }\Bigr )_{\Gamma (t)}= 0,\quad \forall \varvec{\omega }\in [H^1(\mathbb {T})]^2. \end{aligned}$$

(4.2b)

And the SP-PFEM for the anisotropic curvature flow (4.2) is as follows: Suppose the initial approximation $\Gamma ^0=\varvec{X}^0(\mathbb {T})\in [\mathbb {K}^h]^2$ is given by $\varvec{X}^0(\rho _j)=\varvec{X}_0(\rho _j), \forall 0\le j\le N$, then for any $m=0, 1, 2, \ldots$, find the solution $(\varvec{X}^m(\cdot ), \mu ^m(\cdot ))\in [\mathbb {K}^h]^2\times \mathbb {K}^h$, such that

$$\begin{aligned}&\Bigl (\varvec{n}^{m+\frac{1}{2}}\cdot \frac{\varvec{X}^{m+1}-\varvec{X}^m}{\tau }, \varphi ^h\Bigr )_{\Gamma ^m}^h+\Bigl (\mu ^{m+1}, \varphi ^h\Bigr )_{\Gamma ^m}^h=0,\quad \forall \varphi ^h\in \mathbb {K}^h, \end{aligned}$$

(4.3a)

$$\begin{aligned}&\Bigl (\mu ^{m+1}\varvec{n}^{m+\frac{1}{2}},\varvec{\omega }^h\Bigr )_{\Gamma ^m}^h-\Bigl ( \varvec{G}_k(\varvec{n}^m)\partial _s \varvec{X}^{m+1},\partial _s\varvec{\omega }^h\Bigr )_{\Gamma ^m}^h=0, \quad \forall \varvec{\omega }^h\in [\mathbb {K}^h]^2. \end{aligned}$$

(4.3b)

We refer $\varvec{n}^{m+\frac{1}{2}}$ to (2.38)

For the SP-PFEM (4.3), we have

Theorem 4.1

(Structure-preserving) Suppose $\gamma (\varvec{n})$ satisfies (1.6) and take a stabilizing function $k(\varvec{n})\ge k_0(\varvec{n})$, then the SP-PFEM (4.3) preserves area decay rate and energy dissipation, i.e.,

$$\begin{aligned} \frac{A^{m+1}-A^m}{\tau }=-\Bigl (\mu ^{m+1},1\Bigr )_{\Gamma ^m}^h, \quad W^{m+1}\le W^m\le \ldots \le W^0, \qquad \forall m\ge 0. \end{aligned}$$

(4.4)

Proof

From [8, Theorem 2.1], we know that

$$\begin{aligned} A^{m+1}-A^m=\Bigl (\varvec{n}^{m+\frac{1}{2}}\cdot (\varvec{X}^{m+1}-\varvec{X}^m),1\Bigr )_{\Gamma ^m}^h. \end{aligned}$$

(4.5)

Thus, by taking $\varphi ^h\equiv 1\in \mathbb {K}^h$ in (4.3a), we obtain

$$\begin{aligned} \frac{A^{m+1}-A^m}{\tau }=\Bigl (\varvec{n}^{m+\frac{1}{2}}\cdot \frac{\varvec{X}^{m+1}-\varvec{X}^m}{\tau },1\Bigr )_{\Gamma ^m}^h=-\Bigl (\mu ^{m+1},1\Bigr )_{\Gamma ^m}^h, \end{aligned}$$

(4.6)

which is the desired decay rate in (4.4).

For energy dissipation, we have already known that (3.16) is true. Taking $\varphi ^h=\mu ^{m+1}$ in (4.3a) and $\varvec{\omega }^h=\varvec{X}^{m+1}-\varvec{X}^m$ in (4.3b), we know that

$$\begin{aligned} 0&\ge -\tau \Bigl (\mu ^{m+1},\mu ^{m+1}\Bigr )_{\Gamma ^m}^h\\&=\Bigl ( \varvec{G}_k(\varvec{n}^m)\partial _s \varvec{X}^{m+1},\partial _s(\varvec{X}^{m+1}-\varvec{X}^m)\Bigr )_{\Gamma ^m}^h\ge W^{m+1}-W^m, \end{aligned}$$

which proves the claim. $\square$

Remark 4.1

It is worthwhile to compare our SP-PFEM (4.3) with the energy-stable PFEM for anisotropic curvature flow in [10] (often named the BGN scheme) proposed by Barrett, Garcke, and Nürnberg. The BGN scheme requires $\gamma (\varvec{n})$ to be a special form $\gamma _{_{ BGN}}(\varvec{n})=\left( \sum _{l=1}^L\sqrt{(\varvec{n}^TG_l\varvec{n})^r}\right) ^{\frac{1}{r}}$ with $\infty >r\ge 1, L\ge 1$ and $G_l\in \mathbb {R}^{2\times 2}$ positive definite $\forall 1\le l\le L$. Clearly, $\gamma _{_{BGN}}(-\varvec{n})=\gamma _{_{BGN}}(\varvec{n})<3\gamma _{_{BGN}}(\varvec{n})$; thus, it satisfies the energy-stable condition (1.6). Therefore, our (4.3) can handle a broader range of anisotropies compared to the BGN scheme.

4.2 Area-conserved anisotropic curvature flow

Similarly, for the area-conserved anisotropic curvature flow in (1.3), the conservative geometric PDE is given as

$$\begin{aligned}&\varvec{n}\cdot \partial _t \varvec{X}=-\mu +\lambda (t), \end{aligned}$$

(4.7a)

$$\begin{aligned}&\mu \varvec{n}=-\partial _s(\varvec{G}_k(\varvec{n})\partial _s \varvec{X}), \end{aligned}$$

(4.7b)

where $\lambda (t)$ is given as (1.4) by replacing $\lambda$ and $\Gamma$ by $\lambda (t)$ and $\Gamma (t)$, respectively. And the variational formulation can be derived in a similar way.

In order to design a structure-preserving full discretization, we need to properly discretize $\lambda (t)$. Denote ${\lambda ^{m, *}}$ with respect to $\Gamma ^m$ as

$$\begin{aligned} {\lambda ^{m, *}}:=\frac{\left( \mu ^{m+1}, 1\right) _{\Gamma ^m}^h}{\left( 1, 1\right) _{\Gamma ^m}^h}. \end{aligned}$$

(4.8)

By adopting this ${\lambda ^{m, *}}$, the SP-PFEM for the area-conserved anisotropic curvature flow in (1.3) is as follows: Suppose the initial approximation $\Gamma ^0=\varvec{X}^0(\mathbb {T})\in [\mathbb {K}^h]^2$ is given by $\varvec{X}^0(\rho _j)=\varvec{X}_0(\rho _j), \forall 0\le j\le N$; for any $m=0, 1, 2, \ldots$, find the solution $(\varvec{X}^m(\cdot ), \mu ^m(\cdot ))\in [\mathbb {K}^h]^2\times \mathbb {K}^h$, such that

$$\begin{aligned}&\Bigl (\varvec{n}^{m+\frac{1}{2}}\cdot \frac{\varvec{X}^{m+1}-\varvec{X}^m}{\tau }, \varphi ^h\Bigr )_{\Gamma ^m}^h+\Bigl (\mu ^{m+1}-{\lambda ^{m, *}}, \varphi ^h\Bigr )_{\Gamma ^m}^h=0,\quad \forall \varphi ^h\in \mathbb {K}^h, \end{aligned}$$

(4.9a)

$$\begin{aligned}&\Bigl (\mu ^{m+1}\varvec{n}^{m+\frac{1}{2}},\varvec{\omega }^h\Bigr )_{\Gamma ^m}^h-\Bigl ( \varvec{G}_k(\varvec{n}^m)\partial _s \varvec{X}^{m+1},\partial _s\varvec{\omega }^h\Bigr )_{\Gamma ^m}^h=0, \quad \forall \varvec{\omega }^h\in [\mathbb {K}^h]^2. \end{aligned}$$

(4.9b)

For the above SP-PFEM (4.9), we have

Theorem 4.2

(structure-preserving) Suppose $\gamma (\varvec{n})$ satisfies (1.6) and take a finite stabilizing function $k(\varvec{n})\ge k_0(\varvec{n})$, then the SP-PFEM (4.3) is structure-preserving, i.e.,

$$\begin{aligned} A^{m+1}\equiv A^0, \quad W^{m+1}\le W^m\le \ldots \le W^0, \qquad \forall m\ge 0. \end{aligned}$$

(4.10)

Proof

For the area conservation, taking $\varphi ^h\equiv 1$ in (4.9a) yields that

$$\begin{aligned} \Bigl (\varvec{n}^{m+\frac{1}{2}}\cdot (\varvec{X}^{m+1}-\varvec{X}^m),1\Bigr )_{\Gamma ^m}^h&=-\tau \Bigl (\mu ^{m+1}-{\lambda ^{m, *}},1\Bigr )_{\Gamma ^m}^h\\&=-\tau \Bigl (\mu ^{m+1},1\Bigr )_{\Gamma ^m}^h+\tau \frac{\left( \mu ^{m+1}, 1\right) _{\Gamma ^m}^h}{\left( 1, 1\right) _{\Gamma ^m}^h}\Bigl (1, 1\Bigr )_{\Gamma ^m}^h\\&=0,\qquad m\ge 0. \end{aligned}$$

By noting (4.4), we deduce that $A^{m+1}-A^m=0$, which shows area conservation (2.39a).

For energy dissipation, by the Cauchy inequality, we have

$$\begin{aligned} \Bigl ({\lambda ^{m, *}},\mu ^{m+1}\Bigr )_{\Gamma ^m}^h&={\lambda ^{m, *}}\Bigl (1,\mu ^{m+1}\Bigr )_{\Gamma ^m}^h\\&=\frac{1}{\left( 1, 1\right) _{\Gamma ^m}^h}\left( \Bigl (1,\mu ^{m+1}\Bigr )_{\Gamma ^m}^h\right) ^2\\&\le \frac{1}{\left( 1, 1\right) _{\Gamma ^m}^h} \Bigl (1,1\Bigr )_{\Gamma ^m}^h\, \Bigl (\mu ^{m+1},\mu ^{m+1}\Bigr )_{\Gamma ^m}^h\\&=\Bigl (\mu ^{m+1},\mu ^{m+1}\Bigr )_{\Gamma ^m}^h,\qquad m\ge 0. \end{aligned}$$

Taking $\varphi ^h=\mu ^{m+1}$ in (4.9a) and $\varvec{\omega }^h=\varvec{X}^{m+1}-\varvec{X}^m$ in (4.9b), and adopting the energy estimation (3.16) yields that

$$\begin{aligned} W^{m+1}-W^m\le -\tau \Bigl (\mu ^{m+1}-{\lambda ^{m, *}}, \mu ^{m+1}\Bigr )_{\Gamma ^m}^h\le 0,\qquad m\ge 0, \end{aligned}$$

(4.11)

which implies the energy dissipation in (4.10). $\square$

5 Numerical results

In this section, we present numerical experiments to illustrate the high performance of the proposed SP-PFEMs. The implementations and performances of the three SP-PFEMs are very similar. Thus in Sect. 5.1, we only show test results of the SP-PFEM (2.36) for the anisotropic surface diffusion. The morphological evolutions for three anisotropic geometric flows are shown in Sect. 5.2.

To compute the minimal stabilizing function $k_0(\varvec{n})$, we first solve the optimization problem (2.41) for 20 uniformly distributed points $\varvec{n}_j=(-\sin \frac{\pi }{10}j, \cos \frac{\pi }{10} j)^T\in \mathbb {S}^1$ to get $k_0(\varvec{n}_j)$ for $j=1, \ldots , 20$. Then for the general intermediate point $\varvec{n}=(-\sin \theta , \cos \theta )^T\in \mathbb {S}^1$, say $\frac{\pi }{10}j<\theta <\frac{\pi }{10}(j+1)$, we do linear interpolation and set $k_0(\varvec{n})=\frac{10}{\pi }[(\theta -\frac{\pi }{10}j)k_0(\varvec{n}_j)+(\frac{\pi }{10}(j+1)-\theta )k_0(\varvec{n}_{j+1})]$. In Newton’s iteration, the tolerance value $\varepsilon$ is set as $10^{-12}$.

5.1 Results for the anisotropic surface diffusion

Here we provide convergence tests to show the quadratic convergence rate in space and linear convergence rate in time. To this end, the time step $\tau$ is always chosen as $\tau =h^2$ except it is stated otherwise. The distance between two closed curves $\Gamma _1, \Gamma _2$ is given by the manifold distance $M(\Gamma _1, \Gamma _2)$ in [50] as

$$\begin{aligned} M(\Gamma _1, \Gamma _2):=2|\Omega _1\cup \Omega _2|-|\Omega _1|-|\Omega _2|, \end{aligned}$$

(5.1)

where $\Omega _1, \Omega _2$ are the interior regions of $\Gamma _1, \Gamma _2$, respectively, and $|\Omega |$ denotes the area of $\Omega$. Let $\Gamma ^m$ be the numerical approximation of $\Gamma ^h(t=t_m), t_m:=m\tau$ with mesh size h and time step $\tau$, the numerical error is thus given as

$$\begin{aligned} e^h(t)\Big |_{t=t_m}:=M(\Gamma ^m, \Gamma (t_m)). \end{aligned}$$

(5.2)

Fig. 2

Convergence rates of the SP-PFEM (2.36) with $k(\varvec{n})=k_0(\varvec{n})$ for: a anisotropy in Case I at different times $t=0.125, 0.25, 0.5$; and b anisotropy in Case II at time $t=0.5$ with different $\beta$

To numerically test the energy stability, area conservation and good mesh quality, we introduce the following indicators: the normalized energy $\frac{W^h(t)}{W^h(0)}\Big |_{t=t_m}:=\frac{W^m}{W^0}$, the normalized area loss and the mesh ratio

$$\begin{aligned} \frac{\Delta A^h(t)}{A^h(0)}\Big |_{t=t_m}:=\frac{A^m-A^0}{A^0}, \qquad R^h(t)\Big |_{t=t_m}:=\frac{\max \limits _{1\le j\le N} |\varvec{h}_j^m|}{\min \limits _{1\le j\le N} |\varvec{h}_j^m|}. \end{aligned}$$

(5.3)

In the following numerical tests, the initial curve $\Gamma _0$ is given as an ellipse $\Gamma _0=\{(2\cos \theta , \frac{1}{2}\sin \theta )^T, \forall 0\le \theta <2\pi \}$ with major axis 4 and minor axis 1. The exact solution $\Gamma (t)$ is approximated by choosing $k(\varvec{n})=k_0(\varvec{n})$ with $h=2^{-8}$ and $\tau =2^{-16}$ in (2.36). Here are two typical anisotropic surface energies to be taken in our simulations:

• Case I: $\gamma (\varvec{n})=\sqrt{\left( \frac{5}{2}+\frac{3}{2}\text {sgn}(n_1) \right) n_1^2+n_2^2}$ [20];

• Case II: $\gamma (\varvec{n})=1+\beta \cos (3\theta )$ with $\varvec{n}=(\sin \theta , -\cos \theta )^T$ and $|\beta |<1$ [32]. It is weakly anisotropic when $|\beta |<\frac{1}{8}$, and otherwise it is strongly anisotropic.

Fig. 3

Normalized energy of the SP-PFEM (2.36) with $k(\varvec{n})=k_0(\varvec{n})$ and different h for: a anisotropy in Case I; b anisotropy in Case II with $\beta =\frac{1}{3}$

Fig. 4

Normalized area loss (blue dashed line) and iteration number (black line) of the SP-PFEM (2.36) with $k(\varvec{n})=k_0(\varvec{n})$ and $h=2^{-7}, \tau =h^2$ for: a anisotropy in Case I; b anisotropy in Case II with $\beta =\frac{1}{3}$

Fig. 5

Mesh ratio of the SP-PFEM (2.36) with $k(\varvec{n})=k_0(\varvec{n})$ and different h for: a anisotropy in Case I; b anisotropy in Case II with $\beta =\frac{1}{3}$ (color figure online)

Figure 2 presents the convergence rates of the proposed SP-PFEM (2.36) at different times and with different anisotropic strengths $\beta$ for fixed time $t=0.5$. It is apparent from this figure that the second-order convergence in space is independent of anisotropies and computation times, which indicates the convergence rate is very robust.

Fig. 6

Morphological evolutions of an ellipse with major axis 4 and minor axis 1 under anisotropic surface diffusion with four different anisotropic energies: a anisotropy in Case I; b–d anisotropies in case II with $\beta =1/9, 1/7, 1/3$, respectively. The red and blue lines represent the initial curve and the numerical equilibrium, respectively; and the black dashed lines represent the intermediate curves. The mesh size and time step are taken as $h=2^{-7}, \tau =h^2$ (color figure online)

The time evolution of the normalized energy $\frac{W^h(t)}{W^h(0)}$ with different h, the normalized area loss $\frac{\Delta A^h(t)}{A^h(0)}$ and the number of Newton iterations with $h=2^{-7}$, and the mesh ratio $R^h(t)$ with different h are summarised in Figs. 3, 4, 5, respectively.

It can be seen from Figs. 3, 4, 5 that

• (i) The normalized energy is monotonically decreasing when the surface energy density satisfies the energy stable condition (1.6) (cf. Fig. 3);

• (ii) The normalized area loss is at $10^{-15}$ which is almost in the same order as the round-off error (cf. Fig. 3), which confirms the area conservation in practical simulations;

• (iii) Interestingly, the numbers of iterations in Newton’s method are initially 2 and finally 1 (cf. Fig. 4). This finding suggests that although the proposed SP-PFEM (2.36) is full-implicit, but it can be solved very efficiently with a few iterations;

• (iv) In Fig. 5 there is a clear trend of convergence of the mesh ratio $R^h$. Moreover, in contrast to the symmetrized SP-PFEM for symmetric $\gamma (\varvec{n})$ in [4], $R^h$ keeps small even with the strongly anisotropy $\gamma (\varvec{n})=1+\frac{1}{3}\cos 3\theta$.

Fig. 7

Morphological evolutions of an ellipse with major axis 4 and minor axis 1 under anisotropic surface diffusion (left column), area-conserved anisotropic curvature flow (middle column) and anisotropic curvature flow (right column) with the anisotropic surface energy density in Case I at different times. The evolving curves and their enclosed regions are colored blue and black. The mesh size and time step are taken as $h=2^{-7}, \tau =0.001$ (color figure online)

Fig. 8

Morphological evolutions of an ellipse with major axis 4 and minor axis 1 under anisotropic surface diffusion (left column), area-conserved anisotropic curvature flow (middle column) and anisotropic curvature flow (right column) with the anisotropic surface energy density in Case II with $\beta =1/3$ at different times. The evolving curves and their enclosed regions are colored blue and black. The mesh size and time step are taken as $h=2^{-7}, \tau =0.001$

Fig. 9

Area and area decay rate of an ellipse with major axis 4 and minor axis 1 under anisotropic curvature flow by SP-PFEM (4.3) with different anisotropic energy densities: (i) $\gamma (\varvec{n})=1$ (gray line); (ii) $\gamma (\varvec{n})=\sqrt{\left( \frac{5}{2}+\frac{3}{2}\text {sgn}(n_1) \right) n_1^2+n_2^2}$ (blue line) and (iii) $\gamma (\varvec{n})=1+\frac{1}{3}\cos 3\theta$ (green line). The area decay rate is given by the chemical potential and the equilibrium. And the equilibriums are obtained from Fig. 6

5.2 Application for morphological evolutions

Finally, we apply the proposed SP-PFEMs to simulate the morphological evolutions of an ellipse with major axis 4 and minor axis 1 driven by the three anisotropic geometric flows. Figure 6 plots the morphological evolutions of anisotropic surface diffusion for the four different anisotropic energies: (a) anisotropy in case I; (b)–(d) anisotropies in case II with $\beta =1/9, 1/7, 1/3$, respectively. Figures 7 and  8 depict the anisotropic surface diffusion, area-conserved anisotropic curvature flow, and anisotropic curvature flow at different times with anisotropy in case I and in case II with $\beta =1/3$, respectively. Since anisotropic curvature flow will shrink to a point, we also illustrate its area $A^h(t)$ and its area decay rate in Fig. 9.

As shown in Fig. 6b–d, the edges emerge during the evolution and corners become sharper as the strength $\beta$ increases. In fact, if the equilibrium is a polygon, $\gamma (\varvec{n})$ is termed ’crystalline’ [36]. While studies such as [4, 10, 38] mainly focus on the regularization of symmetric crystalline, our SP-PFEM (2.36) can also work for non-symmetric case.

In contrast, from Fig. 6a, we observe that there are no edges or corners in the morphological evolutions with anisotropy in Case I. This suggests that even $\gamma (\varvec{p})=\sqrt{\left( \frac{5}{2}+\frac{3}{2}\text {sgn}(p_1) \right) p_1^2+p_2^2}$ is not a $C^2$-function, it is more like weak anisotropy! From Figs. 7, 8, we can see that the anisotropic surface diffusion and the area-conserved anisotropic curvature flow have the same equilibriums in shapes, while they have different dynamics, i.e., the equilibriums are different in positions, and the anisotropic surface diffusion evolves faster than the area-conserved anisotropic curvature flow.

In isotropic curvature flow, the area decay rate $\frac{dA(t)}{dt}=-\int _{\Gamma } \kappa \,ds = -2\pi$ remains a constant during the whole process [14]. It is important to examine the impact of anisotropic effects on the area decay rate. As illustrated in Fig. 9, the area $A^h(t)$ exhibits a nearly linear decay rate for different anisotropies. However, the area decay rate does not remain at $-2\pi$. Instead, it approaches a different constant, $-\int _{\Gamma } \mu \,ds$, where $\Gamma$ is the equilibrium state of the anisotropic surface diffusion (see Fig. 6). We also observe that the area decay rate may deviate from $-\int _{\Gamma } \mu \,ds$ at the beginning, presenting a remarkable contrast to the constant area decay rate of $-2\pi$ in isotropic curvature flow.

6 Conclusions

By introducing a novel surface energy matrix $\varvec{G}_k(\varvec{n})$ depending on the anisotropic surface energy density $\gamma (\varvec{n})$ and the Cahn-Hoffman $\varvec{\xi }$-vector as well as a nonnegative stabilizing function $k(\varvec{n})$, we proposed conservative geometric partial differential equations for several geometric flows with anisotropic surface energy density $\gamma (\varvec{n})$. We derived their weak formulations and applied PFEM to get their full discretizations. Subsequently, we proved these PFEMs are structure-preserving under a mild condition on $\gamma (\varvec{n})$ with proper choice of the stabilizing function $k(\varvec{n})$. Although our surface energy matrix $\varvec{G}_k(\varvec{n})$ is no longer symmetric, our experiments have shown it maintains a robust second-order convergence rate in space, a linear convergence rate in time, and unconditional energy stability. Specifically, the mesh quality of the proposed SP-PFEM, i.e. the mesh ratio is much smaller, is much better than that in the symmetrized SP-PFEM proposed recently for anisotropic surface diffusion with a symmetric surface energy density [4, 6]. Moreover, our SP-PFEMs work well for the piecewise $C^2$ anisotropy, which is a significant achievement compared with other PFEMs. In the future, we will generalize the surface energy matrix $\varvec{G}_k(\varvec{n})$ to three dimensions (3D) and propose efficient and accurate SP-PFEM for anisotropic geometric flows in 3D.