An unconditionally stable threshold dynamics method for the Willmore flow

Abstract

In this paper, we propose a threshold dynamics method for the Willmore flow with a new kernel constructed based on the combination of a Gaussian kernel and a Cosine function. We show the consistency of the method by asymptotic analysis and construct a Lyapunov functional to show the unconditional stability of the proposed method. Compare to previous work, no artificial parameters are required for the construction of the kernel. Numerical experiments including area preservation or perimeter preservation are performed to show the effectiveness of the method.

Appendix A: Proof of (26)

Proof

When $n=0$, we have

$$\begin{aligned}\frac{\partial ^0}{\partial a^0}\left[ \frac{1}{a}\exp \left( -\frac{c^2}{a}\right) \right] =\frac{1}{a}\exp \left( -\frac{c^2}{a}\right) =\left[ \sum _{k=0}^{0}{\frac{{0!}^2}{{k!}^2 (0-k)!}(-1)^{0-k}a^{-0-k-1}c^{2k}}\right] \exp \left( -\frac{c^2}{a}\right) .\end{aligned}$$

Assume that

$$\begin{aligned} \frac{\partial ^n}{\partial a^n}\left[ \frac{1}{a}\exp \left( -\frac{c^2}{a}\right) \right] =\left[ \sum _{k=0}^{n}{\frac{{n!}^2}{{k!}^2 (n-k)!}(-1)^{n-k}a^{-n-k-1}c^{2k}}\right] \exp \left( -\frac{c^2}{a}\right) . \end{aligned}$$

Then, direct calculation yields

$$\begin{aligned} \begin{aligned}&\frac{\partial ^{n+1}}{\partial a^{n+1}}\left[ \frac{1}{a}\exp \left( -\frac{c^2}{a}\right) \right] =\frac{\partial }{\partial a} \frac{\partial ^n}{\partial a^n}\left[ \frac{1}{a}\exp \left( -\frac{c^2}{a}\right) \right] \\&\quad =\frac{\partial }{\partial a} \left[ \sum _{k=0}^{n}{\frac{{n!}^2}{{k!}^2 (n-k)!}(-1)^{n-k}a^{-n-k-1}c^{2k}}\right] \exp \left( -\frac{c^2}{a}\right) \\&\quad =\left[ \sum _{k=0}^{n}{\frac{{n!}^2}{{k!}^2 (n-k)!}(-1)^{n-k}(-n-k-1)a^{-n-k-2}c^{2k}}\right. \\&\qquad \left. +\sum _{k=0}^{n}{\frac{{n!}^2}{{k!}^2 (n-k)!}(-1)^{n-k}a^{-n-k-3}c^{2k+2}}\right] \exp \left( -\frac{c^2}{a}\right) \\&\quad =\left[ \sum _{k=0}^{n}{\frac{{n!}^2}{{k!}^2 (n-k)!}(-1)^{(n+1)-k}((n+1)+k)a^{-(n+1)-k-1}c^{2k}} \right. \\&\qquad + \left. \sum _{k=1}^{n+1}{\frac{{n!}^2}{{(k-1)!}^2 (n-(k-1))!}(-1)^{n-(k-1)}a^{-n-(k-1)-3}c^{2k}}\right] \exp \left( -\frac{c^2}{a}\right) \\&\quad =\left[ \sum _{k=0}^{n}{\frac{((n+1)+k)((n+1)-k)}{(n+1)^2} \frac{{(n+1)!}^2}{{k!}^2 ((n+1)-k)!}(-1)^{(n+1)-k}a^{-(n+1)-k-1}c^{2k}} \right. \\&\qquad + \left. \sum _{k=1}^{n+1}{\frac{k^2}{(n+1)^2} \frac{{(n+1)!}^2}{{k!}^2 ((n+1)-k))!}(-1)^{(n+1)-k}a^{-(n+1)-k-1}c^{2k}} \right] \exp \left( -\frac{c^2}{a}\right) \\&\quad =\left[ \sum _{k=0}^{n+1}{\frac{{(n+1)!}^2}{{k!}^2 ((n+1)-k)!}(-1)^{(n+1)-k}a^{-(n+1)-k-1}c^{2k}}\right] \exp \left( -\frac{c^2}{a}\right) . \end{aligned} \end{aligned}$$

Here, the last equality comes from the comparison between terms for $k=0, \ldots , n+1$ by direct calculation and cancellation. $\square$