A New Approach to the Analysis of Parametric Finite Element Approximations to Mean Curvature Flow

Abstract

Parametric finite element methods have achieved great success in approximating the evolution of surfaces under various different geometric flows, including mean curvature flow, Willmore flow, surface diffusion, and so on. However, the convergence of Dziuk’s parametric finite element method, as well as many other widely used parametric finite element methods for these geometric flows, remains open. In this article, we introduce a new approach and a corresponding new framework for the analysis of parametric finite element approximations to surface evolution under geometric flows, by estimating the projected distance from the numerically computed surface to the exact surface, rather than estimating the distance between particle trajectories of the two surfaces as in the currently available numerical analyses. The new framework can recover some hidden geometric structures in geometric flows, such as the full $H^1$ parabolicity in mean curvature flow, which is used to prove the convergence of Dziuk’s parametric finite element method with finite elements of degree $k \ge 3$ for surfaces in the three-dimensional space. The new framework introduced in this article also provides a foundational mathematical tool for analyzing other geometric flows and other parametric finite element methods with artificial tangential motions to improve the mesh quality.

Appendix: Optimal Approximation Properties of the Interpolated Surfaces

The quantities $\kappa _l$ and $\kappa _{*,l}$ defined in (3.1) characterize the shape regularity, quasi-uniformity and optimal approximation properties of the interpolated surface ${{\hat{\Gamma }}_{h,*}^m}$. In this appendix, we show that $\kappa _l$ and $\kappa _{*,l}$ have an upper bound which may depend on the exact solution and T, but is independent of $\tau$, h and l. In order to make the argument clear, we denote by $C_{\kappa _l}$ and $C_0$ some generic constants which are dependent and independent of $\kappa _l$, respectively.

1.1 A.1. Boundedness of Discrete Flow Maps in the $\mathbf{W^{k-1,\infty }}$ and $\mathbf{H^{k}}$ Norms: Part I

In terms of the notation in Sect. 3.2, for a curved triangle $K^0\subset \Gamma _h^0$ we denote by $K_\textrm{f}^0$ the unique flat triangle with the same three vertices as $K^0$, and consider the piecewise flat triangular surface

$$\begin{aligned} \Gamma _{h,\textrm{f}}^0=\bigcup _{K^0\subset \Gamma _h^0} K_\textrm{f}^0. \end{aligned}$$

We still denote by $\hat{X}_{h,*}^m:\Gamma _{h,\textrm{f}}^0\rightarrow {{\hat{\Gamma }}_{h,*}^m}$ the unique piecewise polynomial of degree k (with the nodal vector $\hat{\textbf{x}}_*^m$ as before) which parametrizes ${{\hat{\Gamma }}_{h,*}^m}$, and consider the following decomposition:

$$\begin{aligned} \hat{X}_{h,*}^{q+1} = \hat{X}_{h,*}^0 + \sum _{m=0}^q (\hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^m) . \end{aligned}$$

Using the triangle inequality and the good quality of initial triangulation at $t=0$, as shown in (2.1), we have

$$\begin{aligned} \Vert \hat{X}_{h,*}^{q+1} \Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)}&\le C_0 + \sum _{m=0}^q \Vert \hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^m \Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)} \quad \text{ for }\,\,\, 1\le j\le k-1 , \end{aligned}$$

(A.1)

$$\begin{aligned} \Vert \hat{X}_{h,*}^{q+1} \Vert _{H^{j}_h(\Gamma _{h,\mathrm f}^0)}&\le C_0 + \sum _{m=0}^q \Vert \hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^m \Vert _{H^{j}_h(\Gamma _{h,\mathrm f}^0)} \quad \text{ for }\,\,\, 1\le j\le k , \end{aligned}$$

(A.2)

where $\Vert \cdot \Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)}$ and $\Vert \cdot \Vert _{H^{j}_h(\Gamma _{h,\mathrm f}^0)}$ denote the piecewise $W^{j,\infty }$ norm and piecewise $H^j$ norm, respectively, on the piecewise flat triangular surface $\Gamma _{h,\mathrm f}^0$; see the definition of these piecewise Sobolev norms in Sect. 3.2.

In the next section, we shall prove the following two results under the condition $\tau \le c h^k$ and $h\le h_{\kappa _m}$ (where $h_{\kappa _m}$ is some constant depending on $\kappa _m$):

$$\begin{aligned}&\Vert \hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m} \Vert _{W^{j,\infty }_h(\Gamma _ {h,\mathrm f}^0)} \nonumber \\&\le C_0 \tau [1 + (j-1)\Vert \hat{X}_{h, *}^{m}\Vert _{W^{j-1,\infty }_h(\Gamma _{h,\mathrm f}^0)}^j] + C_0\tau \Vert \hat{X}_{h, *}^{m}\Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)} \nonumber \\&\quad \, + C_0 h^{k-j} \tau + C_{\kappa _m} (1+\kappa _{*,m}) h^{k-j-1} \tau \nonumber \\&\quad \, + C_{\kappa _m} h^{-j-1} \big ( \Vert {T^m_{*}}(e_{h}^{m+1} - \hat{e}_{h}^{m}) \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} +h\Vert e_{h}^{m+1} - \hat{e}_{h}^{m} \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} \big ) \end{aligned}$$

(A.3)

and

$$\begin{aligned}&\Vert \hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m} \Vert _{H^{j}_h(\Gamma _{h,\mathrm f}^0)} \nonumber \\&\le C_0 \tau [1 + (j-1)(j-2)\Vert \hat{X}_{h, *}^{m}\Vert _{W^{j-2,\infty }_h(\Gamma _{h,\mathrm f}^0)}^j\nonumber \\&\quad + (j-1)\Vert \hat{X}_{h, *}^{m}\Vert _{W^{1,\infty }(\Gamma _{h,\mathrm f}^0)}\Vert \hat{X}_{h, *}^{m}\Vert _{H^{j-1,\infty }_h(\Gamma _{h,\mathrm f}^0)}^j] \nonumber \\&\quad \, + C_0j \tau \Vert \hat{X}_{h, *}^{m}\Vert _{H^{j}_h(\Gamma _{h,\mathrm f}^0)} + C_0 h^{k-j} \tau + C_{\kappa _m} (1+\kappa _{*,m}) h^{k-j} \tau \nonumber \\&\quad \, + C_{\kappa _m} h^{-j} \big ( \Vert {T^m_{*}}(e_{h}^{m+1} - \hat{e}_{h}^{m}) \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} +h\Vert e_{h}^{m+1} - \hat{e}_{h}^{m} \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} \big ) . \end{aligned}$$

(A.4)

From (5.28), (5.52) and (5.74) we see that, by applying the inverse inequality,

$$\begin{aligned}&h^{-j-1}\Vert {T^m_{*}}({e_h^{m+1}}- {\hat{e}_h^m}) \Vert _{L^{2}({{\hat{\Gamma }}_{h,*}^m})} +h^{-j}\Vert {e_h^{m+1}}- {\hat{e}_h^m}\Vert _{L^{2}({{\hat{\Gamma }}_{h,*}^m})} \nonumber \\&\le C_{\kappa _m} h^{-j-1}\tau \Vert \nabla _{{{\hat{\Gamma }}_{h,*}^m}} {\hat{e}_h^m}\Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} + C_{\kappa _m} h^{-j-3} \tau \Vert \nabla _{{{\hat{\Gamma }}_{h,*}^m}} {\hat{e}_h^m}\Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})}^2 \nonumber \\ {}&\quad + C_{\kappa _m} h^{-j-1}\tau (1+\kappa _{*,m}) h^{k} . \end{aligned}$$

(A.5)

Then, substituting this result into (A.3) and choosing $j=1$, we derive the following estimate for $0\le q\le l$:

$$\begin{aligned}&\sum _{m=0}^{q} \Vert \hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m} \Vert _{W^{1,\infty }(\Gamma _{h,\mathrm f}^0)} \\&\le C_0+ \sum _{m=0}^{q} C_0 \tau \Vert \hat{X}_{h, *}^{m}\Vert _{W^{1,\infty }(\Gamma _{h,\mathrm f}^0)} +C_0h^{k-1} + \sum _{m=0}^{q} C_{\kappa _m} \tau (1+\kappa _{*,m}) h^{k-2}\\&\quad \, + \sum _{m=0}^{q} C_{\kappa _m} \Big [ h^{-2}\tau \Vert \nabla _{{\hat{\Gamma }}_{h,*}^m} \hat{e}_h^m \Vert _{L^2({\hat{\Gamma }}_{h,*}^m)} + h^{-4} \tau \Vert \nabla _{{\hat{\Gamma }}_{h,*}^m} \hat{e} _h^m \Vert _{L^2({\hat{\Gamma }}_{h,*}^m)}^2 \Big ] \\&\le C_0 + C_{\kappa _{l}} (1+\kappa _{*,l}) h^{k-2} + C_{\kappa _{l}} (1+\kappa _{*,l})^2 h^{2k-4} + \sum _{m=0}^{q} C_0 \tau \Vert \hat{X}_{h, *}^{m}\Vert _{W^{1,\infty }(\Gamma _{h,\mathrm f}^0)} . \end{aligned}$$

Here we have used the error estimate in (5.74) with $\tau \le c h^k$ in the last inequality. Since $k\ge 3$, for sufficiently small mesh size $h\le h_{\kappa _{l},\kappa _{*,l}}$ (with some constant which depends on $\kappa _{l}$ and $\kappa _{*,l}$), substituting the last inequality into (A.1) and taking the square yield the following result:

$$\begin{aligned} \Vert \hat{X}_{h, *}^{q+1} \Vert _{W^{1,\infty }(\Gamma _{h,\mathrm f}^0)}^2 \le C_0 + \sum _{m=0}^{q} C_0 \tau \Vert \hat{X}_{h, *}^{m}\Vert _{W^{1,\infty }(\Gamma _{h,\mathrm f}^0)}^2 \quad \text{ for }\,\,\, 0\le q\le l. \end{aligned}$$

(A.6)

Then, by applying the discrete Grönwall’s inequality and taking the square root, we obtain

$$\begin{aligned} \max _{0\le q\le l} \Vert \hat{X}_{h, *}^{q+1} \Vert _{W^{1,\infty }(\Gamma _{h,\mathrm f}^0)} \le C_0 . \end{aligned}$$

(A.7)

Now, by using mathematical induction, we shall prove that if $j\le k-2$ and $\max \limits _{0\le q\le l} \Vert \hat{X}_{h, *}^{q+1} \Vert _{W_h^{j-1,\infty }(\Gamma _{h,\mathrm f}^0)}\le C_0$ then

$$\begin{aligned} \max _{0\le q\le l} \Vert \hat{X}_{h, *}^{q+1} \Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)} \le C_0. \end{aligned}$$

In fact, if $\max \limits _{0\le q\le l} \Vert \hat{X}_{h, *}^{q+1} \Vert _{W^{j-1,\infty }(\Gamma _{h,\mathrm f}^0)}\le C_0$ then summing up (A.3) for $m=0,\ldots ,q$ and using (A.5) yield the following result:

$$\begin{aligned}&\sum _{m=0}^{q} \Vert \hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m} \Vert _{W^{j,\infty }_h(\Gamma _h^0)} \\&\le C_0(1+h^{k-j}) + \sum _{m=0}^{q} C_0 \tau \Vert \hat{X}_{h, *}^{m}\Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)} + \sum _{m=0}^{q} C_{\kappa _m} h^{k-j-1}\tau (1+\kappa _{*,m}) \\&\quad \, + \sum _{m=0}^{q} C_{\kappa _m} \Big [ h^{-j-1}\tau \Vert \nabla _{{\hat{\Gamma }}_{h,*}^m} \hat{e}_h^m \Vert _{L^2({\hat{\Gamma }}_{h,*}^m)} + h^{-j-3} \tau \Vert \nabla _{{\hat{\Gamma }}_{h,*}^m} \hat{e} _h^m \Vert _{L^2({\hat{\Gamma }}_{h,*}^m)}^2 \Big ]\\&\le C_0(1+h^{k-j}) + \sum _{m=0}^{q} C_0 \tau \Vert \hat{X}_{h, *}^{m}\Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)} \\&\quad \, + C_{\kappa _{l}} \Big (1+\sum _{m=0}^q \tau \kappa _{*,m}^2\Big )^{\frac{1}{2}} h^{k-j-1} + C_{\kappa _{l}} (1+\kappa _{*,l})^2 h^{2k-j-3} , \end{aligned}$$

where we have used the error estimate in (5.74) in the last inequality. Since $k\ge 3$, for sufficiently small mesh size $h\le h_{\kappa _{l},\kappa _{*,l}}$ (with some constant which depends on $\kappa _{l}$ and $\kappa _{*,l}$), substituting the last inequality into (A.1) and taking the square yield the following result for $j\le k-2$:

$$\begin{aligned} \Vert \hat{X}_{h, *}^{q+1}\Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)}^2&\le C_0+ \sum _{m=0}^{q} C_0 \tau \Vert \hat{X}_{h, *}^{m}\Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)}^2 \quad \text{ for }\,\,\, 0\le q\le l. \end{aligned}$$

Then, by applying the discrete Grönwall’s inequality and taking the square root, we obtain

$$\begin{aligned} \max _{0\le q\le l} \Vert \hat{X}_{h, *}^{q+1} \Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)} \le C_0 . \end{aligned}$$

(A.8)

This proves (A.8) for $1\le j\le k-2$. Analogously, by using (A.4), we can prove the following result for $1\le j\le k-1$:

$$\begin{aligned} \max _{0\le q\le l} \Vert \hat{X}_{h, *}^{q+1} \Vert _{H^{j}_h(\Gamma _{h,\mathrm f}^0)} \le C_0 . \end{aligned}$$

(A.9)

Similar estimate for $\max \limits _{0\le q\le l}\Vert (\hat{X}_{h,*}^{q+1})^{-1} \Vert _{W^{1,\infty }_h({\hat{\Gamma }}_{h,*}^j)}$ can also be proved and omitted here. This proves that if $\tau \le c h^k$ and $h\le h_{\kappa _{l},\kappa _{*,l}}$ then $\kappa _{l+1}\le C_0$ in view of the definition in (3.1).

Therefore, we can replace $C_{\kappa _m}$ by $C_0$ in (A.3)–(A.4) and obtain the following results for $0\le q\le l$ in the same way as above, under the conditions $\tau \le c h^k$ and $h\le h_{\kappa _{l},\kappa _{*,l}}$:

$$\begin{aligned} \Vert \hat{X}_{h, *}^{q+1}\Vert _{W^{k-1,\infty }_h(\Gamma _{h,\mathrm f}^0)}^2&\le C_0+\sum _{m=0}^{q} C_0 \tau \kappa _{*,m} ^2 + \sum _{m=0}^{q} C_0 \tau \Vert \hat{X}_{h, *}^{m}\Vert _{W^{k-1,\infty }_h(\Gamma _{h,\mathrm f}^0)}^2 ,\end{aligned}$$

(A.10)

$$\begin{aligned} \Vert \hat{X}_{h, *}^{l+1}\Vert _{H^{k}_h(\Gamma _{h,\mathrm f}^0)}^2&\le C_0+\sum _{m=0}^{q} C_0 \tau \kappa _{*,m} ^2 + \sum _{m=0}^{q} C_0 \tau \Vert \hat{X}_{h, *}^{m}\Vert _{H^{k}_h(\Gamma _{h,\mathrm f}^0)}^2 . \end{aligned}$$

(A.11)

In regard to the definition of $\kappa _{*,m}$ in (3.1), we can replace $\kappa _{*,m}$ by $\Vert \hat{X}_{h, *}^{m}\Vert _{W^{k-1,\infty }_h(\Gamma _{h,\mathrm f}^0)}+\Vert \hat{X}_{h, *}^{m}\Vert _{H^{k}_h(\Gamma _{h,\mathrm f}^0)}$ and then sum up the two inequalities above. This yields that

$$\begin{aligned}&\Vert \hat{X}_{h, *}^{q+1}\Vert _{H^{k}_h(\Gamma _{h,\mathrm f}^0)}^2 +\Vert \hat{X}_{h, *}^{q+1}\Vert _{W^{k-1,\infty }_h(\Gamma _{h,\mathrm f}^0)}^2 \nonumber \\&\le C_0 + \sum _{m=0}^{q} C_0 \tau (\Vert \hat{X}_{h, *}^{m}\Vert _{H^{k}_h(\Gamma _{h,\mathrm f}^0)}^2 +\Vert \hat{X}_{h, *}^{m}\Vert _{W^{k-1,\infty }_h(\Gamma _{h,\mathrm f}^0)}^2) . \end{aligned}$$

(A.12)

By applying Grönwall’s inequality and taking the square root, we obtain

$$\begin{aligned} \max _{0\le q\le l} (\Vert \hat{X}_{h, *}^{q+1}\Vert _{H^{k}_h(\Gamma _{h,\mathrm f}^0)} +\Vert \hat{X}_{h, *}^{q+1}\Vert _{W^{k-1,\infty }_h(\Gamma _{h,\mathrm f}^0)} ) \le C_0 . \end{aligned}$$

(A.13)

This proves that $\kappa _{*,l+1}\le C_0$ in view of its definition in (3.1). For this constant $C_0$ (which is independent of l) we get the following result: If $\kappa _l\le C_0$ and $\kappa _{*,l}\le C_0$, and $\tau \le c h^k$ with $h\le h_{C_0,C_0}$, then

$$\begin{aligned} \kappa _{l+1}\le C_0 \quad \text{ and }\quad \kappa _{*,l+1} \le C_0 . \end{aligned}$$

(A.14)

This proves (A.14) by mathematical induction under the conditions $\tau \le c h^k$ and $h\le h_{C_0,C_0}$. As a result, the quantities $\kappa _l$ and $\kappa _{*,l}$ defined in (3.1) are uniformly bounded with respect to $\tau$, h and l under the required conditions on the stepsize and mesh size.

1.2 A.2. Boundedness of Discrete Flow Maps in the $\mathbf{W^{k-1,\infty }}$ and $\mathbf{H^{k}}$ Norms: Part II

In this appendix we prove (A.3)–(A.4), which are used in Appendix A.1 to prove (A.14).

Note that the nodal vectors ${\hat{\textbf{x}}}_*^{m}$ and ${\hat{\textbf{x}}}_*^{m + 1}$ are defined as the distance projection of $\textbf{x}^m$ and $\textbf{x}^{m+1}$ onto the smooth surfaces $\Gamma ^m$ and $\Gamma ^{m+1}$. Therefore, $\hat{e}_{h}^m = X_{h}^{m} - \hat{X}_{h,*}^{m}$ and $\hat{e}_{h}^{m+1} = X_{h}^{m+1} - \hat{X}_{h,*}^{m+1}$ are in the directions of ${n^m_{*}}$ and $n_*^{m+1}$ at the nodes, respectively. From the geometric relation in Fig. 4 we observe the following vector decomposition at the j-th node:

$$\begin{aligned} {N^m_{*}}(\hat{x}_{j,*}^{m + 1} -\hat{x}_{j,*}^{m}) = x_{j,*}^{m + 1} -\hat{x}_{j,*}^{m} + \rho _h|_{{j\hbox {-th node}}} , \end{aligned}$$

and passing to finite element functions, it holds that

$$\begin{aligned} {N^m_{*}}(\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m}) = (X^{m + 1} - \textrm{id})\circ a^m + \rho _h \quad \text{ at } \text{ the } \text{ nodes } \text{ of }\, {{\hat{\Gamma }}_{h,*}^m}\end{aligned}$$

(A.15)

for some finite element function $\rho _h$ such that by the triangle inequality

$$\begin{aligned} |\rho _h| \le C_0 \tau ^2 + C_0 |T_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})|^2 \quad \text{ at } \text{ the } \text{ nodes } , \end{aligned}$$

(A.16)

where $C_0\tau ^2$ arises from the quadratic term in the Taylor expansion of the exact flow, which measures the deviation of $X_{h, *}^{m+1}$ away from the normal direction, while $|T_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})|^2$ measures the difference of lengths in the normal direction, as shown in Fig. 4. The latter is essentially the product of $|T_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})|$ (the length of one side of a right triangle) and the tangent of an angle whose amplitude is of order $O(|T_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})|)$.

Fig. 4

The geometric relation at the j-th node

Moreover, since $T_*^m (\hat{X}_{h,*}^{m} - X_{h}^{m})=0$ at the nodes and $T_*^{m}N_*^{m} =0$, the following relation holds:

$$\begin{aligned} T_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})&= T_*^m (X_{h}^{m + 1} - X_{h}^{m}) + T_*^m (\hat{X}_{h,*}^{m + 1} - X_{h}^{m + 1}) - T_*^m (\hat{X}_{h,*}^{m} - X_{h}^{m}) \nonumber \\&= T_*^m (X_{h}^{m + 1} - X_{h}^{m}) + T_*^m N_*^{m+1} (\hat{X}_{h,*}^{m + 1} - X_{h}^{m + 1}) \nonumber \\&= T_*^m (X_{h}^{m + 1} - X_{h}^{m}) + T_*^m (N_*^{m+1}-N_*^{m}) \hat{e}_{h}^{m + 1} \quad \text{ at } \text{ the } \text{ nodes } . \end{aligned}$$

(A.17)

In the last equality we have used $\hat{e}_{h}^{m + 1}=\hat{X}_{h,*}^{m + 1} - X_{h}^{m + 1}$. Therefore, by decomposing $\hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m}$ into the normal and tangential components and applying the triangle inequality, we have

$$\begin{aligned}&\Vert \hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m} \Vert _{W^{j,\infty }_h(\Gamma _ {h,\mathrm f}^0)} \nonumber \\&\le \Vert I_h {N^m_{*}}(\hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m}) \Vert _{W^{j,\infty }_h(\Gamma _ {h,\mathrm f}^0)} + \Vert I_h {T^m_{*}}(\hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m}) \Vert _{W^{j,\infty }_h(\Gamma _ {h,\mathrm f}^0)} \nonumber \\&\le \Vert I_h [(X^{m+1} - \textrm{id})\circ a^m\circ \hat{X}_{h, *}^{m} ] + \rho _h \Vert _{W^{j,\infty }_h(\Gamma _ {h,\mathrm f}^0)}\nonumber \\&\quad \,\, \text{(relation }\,(A.15)\hbox { is pulled back to}\, \Gamma _ {h,\mathrm f}^0) \nonumber \\&\quad + C_0h^{-j} \Vert I_h {T^m_{*}}(\hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m}) \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} . \end{aligned}$$

(A.18)

The first term on the right-hand side of (A.18) can be estimated by using (A.16) as follows:

$$\begin{aligned}&\Vert I_h [(X^{m+1} - \textrm{id})\circ a^m\circ \hat{X}_{h, *}^{m} ] + \rho _h \Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)} \nonumber \\&\le C_0 \Vert X^{m+1} - \textrm{id} \Vert _{W^{j,\infty }({\Gamma ^m})} \Big (1+ \sum _{\begin{array}{c} j_1+\cdots +j_i\le j\\ j_1,\ldots ,j_i\ge 1 \end{array}} \Vert \hat{X}_{h, *}^{m}\Vert _{W_h^{j_1,\infty }(\Gamma _{h,\mathrm f}^0)}\cdots \Vert \hat{X}_{h, *}^{m}\Vert _{W_h^{j_i,\infty }(\Gamma _{h,\mathrm f}^0)} \Big ) \nonumber \\&\quad \, + C_0 h^{-j} \Vert \rho _h \Vert _{L^{\infty }(\Gamma _{h,\mathrm f}^0)} \nonumber \\&\le C_0 \tau [1 + j(j-1)\Vert \hat{X}_{h, *}^{m}\Vert _{W_h^{j-1,\infty }(\Gamma _{h,\mathrm f}^0)}^j] + C_0j \tau \Vert \hat{X}_{h, *}^{m}\Vert _{W_h^{j,\infty }(\Gamma _{h,\mathrm f}^0)} \nonumber \\&\quad \, + C_0 h^{-j}(\tau ^2+\Vert I_hT_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})\Vert _{L^\infty (\Gamma _{h,\mathrm f}^0)}^2 ) . \end{aligned}$$

(A.19)

Here we have added a factor $j(j-1)$ in front of $\Vert \hat{X}_{h, *}^{m}\Vert _{W_h^{j-1,\infty }(\Gamma _{h,\mathrm f}^0)}$ to indicate that this term should disappear in the case $j=1$, and we have added a factor j in front of $\Vert \hat{X}_{h, *}^{m}\Vert _{W_h^{j,\infty }(\Gamma _{h,\mathrm f}^0)}$ to indicate that this term should disappear in the case $j=0$.

The second term on the right-hand side of (A.18), as well as the last term on the right-hand side of (A.19), can be estimated by using relation (A.17), i.e.,

$$\begin{aligned}&\Vert I_hT_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})\Vert _{L^\infty (\Gamma _{h,\mathrm f}^0)} \nonumber \\&\le \Vert I_h {T^m_{*}}(X_{h}^{m+1} - X_{h}^{m}) \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} + \Vert I_h T_*^m (N_*^{m+1}-N_*^{m}) \hat{e}_{h}^{m + 1} \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} . \end{aligned}$$

(A.20)

In the case $j=0$ we get from (A.18) and (A.19) the following result:

$$\begin{aligned}&\Vert \hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m} \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} \nonumber \\&\le C_0 \tau + C_0\Vert I_hT_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})\Vert _{L^\infty (\Gamma _{h,\mathrm f}^0)}^2 + \Vert I_hT_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})\Vert _{L^\infty (\Gamma _{h,\mathrm f}^0)} \nonumber \\&\le C_0 \tau + C_0\Vert I_hT_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})\Vert _{L^\infty (\Gamma _{h,\mathrm f}^0)} , \end{aligned}$$

(A.21)

where the last inequality follows from the estimate in (5.39), which implies that

$$\begin{aligned} \Vert \hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m} \Vert _{L^\infty (\Gamma _{h,\mathrm f}^0)} + \Vert I_hT_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})\Vert _{L^\infty (\Gamma _{h,\mathrm f}^0)} \le C_{\kappa _l} h^{0.5}\le 1 , \end{aligned}$$

(A.22)

when $h\le h_{\kappa _l}$ (for some constant $h_{\kappa _l}$ which may depend on $\kappa _l$).

In the case $j\ge 1$ we obtain from (A.18)–(A.19) and (A.22) the following result:

$$\begin{aligned}&\Vert \hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m} \Vert _{W^{j,\infty }_h(\Gamma _ {h,\mathrm f}^0)} \nonumber \\&\le C_0 \tau [1 + j(j-1)\Vert \hat{X}_{h, *}^{m}\Vert _{W_h^{j-1,\infty }(\Gamma _{h,\mathrm f}^0)}^j] + C_0j \tau \Vert \hat{X}_{h, *}^{m}\Vert _{W_h^{j,\infty }(\Gamma _{h,\mathrm f}^0)} \nonumber \\&\quad \, + C_0 h^{-j}\tau ^2 + C_0h^{-j} \Vert I_h {T^m_{*}}(\hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m}) \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} . \end{aligned}$$

(A.23)

The first term on the right-hand side of (A.20) can be estimated by using the geometric relation in (3.15), which implies that

$$\begin{aligned}&\Vert I_h {T^m_{*}}(X_{h}^{m+1} - X_{h}^{m}) \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} \\&\le \Vert I_h{T^m_{*}}(e_{h}^{m+1} - \hat{e}_{h}^{m}) \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} + \tau \Vert I_h{T^m_{*}}I_h(H^mn^m - g) \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} \\&= \Vert I_h{T^m_{*}}(e_{h}^{m+1} - \hat{e}_{h}^{m}) \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} + \tau \Vert I_h{T^m_{*}}I_h g \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)}\\&\quad \text{(as }\,{T^m_{*}}I_h(H^mn^m)=0\hbox { at the nodes)}\\&\le \Vert I_h{T^m_{*}}(e_{h}^{m+1} - \hat{e}_{h}^{m}) \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} + C_0 \tau \Vert g \Vert _{L^\infty ({\Gamma ^m})} , \end{aligned}$$

where the last inequality follows from the $L^\infty$ stability of the Lagrange interpolation operator (with respect to the nodal values) on the initial triangulated surface $\Gamma _ {h,\mathrm f}^0$, and the fact that the nodal values of $I_h{T^m_{*}}I_h g$ is bounded by $\Vert g \Vert _{L^\infty ({\Gamma ^m})}$. Then, by applying the inverse inequality to convert the $L^\infty ({{\hat{\Gamma }}_{h,*}^m})$ norm to the $L^2({{\hat{\Gamma }}_{h,*}^m})$ norm (with a constant depending on $\kappa _l$ and independent of $\kappa _{*,l}$), we obtain

$$\begin{aligned}&\Vert I_h {T^m_{*}}(X_{h}^{m+1} - X_{h}^{m}) \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} \nonumber \\&\le C_{\kappa _m} h^{-1}\Vert I_h{T^m_{*}}(e_{h}^{m+1} - \hat{e}_{h}^{m}) \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} + C_0 \tau ^2 \qquad \text{(here }\,(3.14)\hbox { is used)} \nonumber \\&\le C_{\kappa _m} h^{-1} \big ( \Vert {T^m_{*}}(e_{h}^{m+1} - \hat{e}_{h}^{m}) \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} +h\Vert e_{h}^{m+1} - \hat{e}_{h}^{m} \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} \big ) + C_0 \tau ^2 . \end{aligned}$$

(A.24)

In the last inequality, we have used the super-approximation estimates in Lemma 4.4.

The second term on the right-hand side of (A.20) can be estimated by using the inverse inequality and the expression ${N^m_{*}}= (n_*^{m}\circ \hat{X}_{h,*}^{m}) (n_*^{m}\circ \hat{X}_{h,*}^{m})^\top$ at the nodes, i.e.,

$$\begin{aligned}&\Vert I_h {T^m_{*}}(N_*^{m+1}-{N^m_{*}}) \hat{e}_{h}^{m + 1} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \nonumber \\&\le C_{\kappa _m} \Vert I_h[ ( N_*^{m+1}-{N^m_{*}}) \hat{e}_{h}^{m + 1} ] \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \nonumber \\&\le C_{\kappa _m} \Vert I_h [n_*^{m+1}\circ \hat{X}_{h,*}^{m+1} - n_*^{m}\circ \hat{X}_{h,*}^{m} ]\Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \Vert \hat{e}_{h}^{m + 1} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \nonumber \\&\le C_{\kappa _m} \Vert I_h[n_*^{m+1}\circ \hat{X}_{h,*}^{m+1} - n_*^{m+1}\circ X_{h,*}^{m+1}] \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \Vert \hat{e}_{h}^{m + 1} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \nonumber \\&\quad \, +C_{\kappa _m} \Vert I_h[n_*^{m+1}\circ X_{h,*}^{m+1} - n_*^{m}\circ \hat{X}_{h,*}^{m}] \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \Vert \hat{e}_{h}^{m + 1} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \nonumber \\&\le C_{\kappa _m} (\Vert I_hT_*^{m+1} (\hat{X}_{h,*}^{m+1} - X_{h,*}^{m+1}) \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} +\Vert \hat{X}_{h,*}^{m+1} -\nonumber \\&\quad X_{h,*}^{m+1} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})}^2 +\tau ) \Vert \hat{e}_{h}^{m + 1} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \nonumber \\&\le C_{\kappa _m} (\Vert I_hT_*^{m+1}(\hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^{m}) \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} +\Vert \hat{X}_{h,*}^{m+1}\nonumber \\&\quad - \hat{X}_{h,*}^{m} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})}^2 +\tau ) \Vert \hat{e}_{h}^{m + 1} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} , \end{aligned}$$

(A.25)

where the derivation of the second to last inequality of (A.25) uses the following two arguments:

• (i) We have used the Taylor expansion of $n_*^{m+1}\circ \hat{X}_{h,*}^{m+1} - n_*^{m+1}\circ X_{h,*}^{m+1}$ at $\hat{X}_{h,*}^{m+1}$ up to the quadratic term, with the following observation: Since both $\hat{X}_{h,*}^{m+1}$ and $X_{h,*}^{m+1}$ take values on $\Gamma ^{m+1}$, and the value of $[(\nabla _{\Gamma ^{m+1}} n_*^{m+1}) \circ \hat{X}_{h,*}^{m+1}] (\hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^{m})$ at a node only depends on the value of $T_*^{m+1}(\hat{X}_{h,*}^{m +1} - \hat{X}_{h,*}^{m})$ at the node, it follows that

  $$\begin{aligned} |I_h (\nabla _{\Gamma ^{m+1}} n_*^{m+1}) (\hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^{m}) | \le C_0 |I_hT_*^{m+1}(\hat{X}_{h,*}^{m +1} - \hat{X}_{h,*}^{m}) | \quad \text{ at } \text{ the } \text{ nodes }. \end{aligned}$$

• (ii) The value of $n_*^{m+1}\circ X_{h,*}^{m+1} - n_*^{m}\circ \hat{X}_{h,*}^{m}$ at a node is the change of the normal vector along a particle trajectory of length $O(\tau )$.

The last inequality of (A.25) follows from the triangle inequality and the property $|X_{h,*}^{m+1}-\hat{X}_{h,*}^{m}|=O(\tau )$ at the nodes, because the value of $X_{h,*}^{m+1}-\hat{X}_{h,*}^{m}$ at a node is the distance a particle moves within time period $\tau$.

Then, substituting (A.21) into (A.25) and using the result $\Vert \hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^{m} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \le 1$ shown in (A.22), we get

$$\begin{aligned}&\Vert I_h {T^m_{*}}(N_*^{m+1}-{N^m_{*}}) \hat{e}_{h}^{m + 1} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \nonumber \\&\le C_{\kappa _m} (\Vert I_hT_*^{m+1}(\hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^{m}) \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} +\tau ) \Vert \hat{e}_{h}^{m + 1} \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} \nonumber \\&\le C_{\kappa _m} h^{0.5} \Vert I_hT_*^{m+1}(\hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^{m}) \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} + C_{\kappa _m} \tau h^{-1} \Vert \hat{e}_{h}^{m + 1} \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} \nonumber \\&\le C_{\kappa _m} h^{0.5} \Vert I_hT_*^m(\hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^{m}) \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} + C_{\kappa _m} (1+\kappa _{*,m}) \tau h^{-1}(\tau +h^k) , \end{aligned}$$

(A.26)

where the second to last inequality follows from the estimate $\Vert \hat{e}^{m+1}_{h}\Vert _{L^\infty ({\hat{\Gamma }}_{h,*}^m)}\le C_{\kappa _m} h^{0.5}$ in (5.38) and the inverse inequality, and the last inequality follows from the error estimate of $\Vert \hat{e}_{h}^{m + 1} \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})}$ in (5.74) and from replacing $T_*^{m+1}$ by $T_*^{m}$ with an error of $O(\tau )$ at the nodes. Note that we have replaced $\kappa _l$ and $\kappa _{*,l}$ by $\kappa _m$ and $\kappa _{*,m}$, respectively, when we estimate $\Vert \hat{e}^{m+1}_{h}\Vert _{L^\infty ({\hat{\Gamma }}_{h,*}^m)}$ and $\Vert \hat{e}^{m+1}_{h}\Vert _{L^2({\hat{\Gamma }}_{h,*}^m)}$ using (5.38) and (5.74). This is correct as the estimation of $\hat{e}_{h}^{m + 1}$ only requires using $\kappa _m$ and $\kappa _{*,m}$ instead of $\kappa _l$ and $\kappa _{*,l}$ (unless we want to consider the maximum error among $m=0,\ldots ,l$ as in (5.74)). We also note that the error estimate of $\Vert \hat{e}_{h}^{m + 1} \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})}$ using (5.74) requires the mesh size to satisfy $h\le h_{\kappa _m,\kappa _{*,m}}$ for some constant $h_{\kappa _m,\kappa _{*,m}}$ which may depend on $\kappa _m$ and $\kappa _{*,m}$. Now we can substitute (A.24) and (A.26) into (A.20). This yields the following estimate:

$$\begin{aligned}&\Vert I_hT_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})\Vert _{L^\infty (\Gamma _{h,\mathrm f}^0)} \nonumber \\&\le \Vert I_h {T^m_{*}}(X_{h}^{m+1} - X_{h}^{m}) \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} + \Vert I_h T_*^m (N_*^{m+1}-N_*^{m}) \hat{e}_{h}^{m + 1} \Vert _{L^\infty (\Gamma _ {h,\mathrm f}^0)} \nonumber \\&\le C_{\kappa _m} h^{-1} \big ( \Vert {T^m_{*}}(e_{h}^{m+1} - \hat{e}_{h}^{m}) \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} +h\Vert e_{h}^{m+1} - \hat{e}_{h}^{m} \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} \big ) + C_0 \tau ^2 \nonumber \\&\quad \, + C_{\kappa _m} h^{0.5} \Vert I_hT_*^{m}(\hat{X}_{h,*}^{m+1} - \hat{X}_{h,*}^{m}) \Vert _{L^\infty ({{\hat{\Gamma }}_{h,*}^m})} + C_{\kappa _m} (1+\kappa _{*,m}) \tau h^{-1}(\tau +h^k) . \end{aligned}$$

(A.27)

The second to last term on the right-hand side of (A.27) can be absorbed by the left-hand side by choosing sufficiently small h, say $h\le h_{\kappa _m,\kappa _{*,m}}$ for some constant $h_{\kappa _m,\kappa _{*,m}}$ which may depend on $\kappa _m$ and $\kappa _{*,m}$. Then it holds that

$$\begin{aligned}&\Vert I_hT_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})\Vert _{L^\infty (\Gamma _{h,\mathrm f}^0)} \nonumber \\&\le C_{\kappa _m} h^{-1} \big ( \Vert {T^m_{*}}(e_{h}^{m+1} - \hat{e}_{h}^{m}) \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} +h\Vert e_{h}^{m+1} - \hat{e}_{h}^{m} \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} \big ) \nonumber \\&\quad \, + C_0 \tau ^2 + C_{\kappa _m} (1+\kappa _{*,m}) \tau h^{-1}(\tau +h^k) . \end{aligned}$$

(A.28)

By substituting (A.28) into (A.23) we obtain the following result for $j\ge 1$:

$$\begin{aligned}&\Vert \hat{X}_{h, *}^{m+1} - \hat{X}_{h, *}^{m} \Vert _{W^{j,\infty }_h(\Gamma _ {h,\mathrm f}^0)} \nonumber \\&\le C_0 \tau [1 + (j-1)\Vert \hat{X}_{h, *}^{m}\Vert _{W^{j-1,\infty }_h(\Gamma _{h,\mathrm f}^0)}^j] + C_0 \tau \Vert \hat{X}_{h, *}^{m}\Vert _{W^{j,\infty }_h(\Gamma _{h,\mathrm f}^0)} \nonumber \\&\quad \, + C_0 h^{-j} \tau ^2 + C_{\kappa _m} (1+\kappa _{*,m}) \tau h^{-j-1}(\tau +h^k) \nonumber \\&\quad \, + C_{\kappa _m} h^{-j-1} \big ( \Vert {T^m_{*}}(e_{h}^{m+1} - \hat{e}_{h}^{m}) \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} +h\Vert e_{h}^{m+1} - \hat{e}_{h}^{m} \Vert _{L^2({{\hat{\Gamma }}_{h,*}^m})} \big ) . \end{aligned}$$

(A.29)

This proves the relation in (A.3) under the stepsize condition $\tau \le c h^k$. The proof of (A.4) is similar (only the norm is changed) and omitted.