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.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
G. Bai and B. Li. Erratum: Convergence of Dziuk’s semidiscrete finite element method for mean curvature flow of closed surfaces with high-order finite elements. SIAM J. Numer. Anal., 61(3):1609–1612, 2023.
E. Bänsch, P. Morin, and R. H. Nochetto. A finite element method for surface diffusion: The parametric case. J. Comput. Phys., 203:321–343, 2005.
W. Bao, W. Jiang, Y. Wang, and Q. Zhao. A parametric finite element method for solid-state dewetting problems with anisotropic surface energies. J. Comput. Phys., 330:380–400, 2017.
W. Bao, W. Jiang, and Q. Zhao. A parametric finite element method for solid-state dewetting problems in three dimensions. SIAM J. Sci. Comput., 42:B327–B352, 2020.
J. Barrett, K. Deckelnick, and R. Nürnberg. A finite element error analysis for axisymmetric mean curvature flow. IMA J. Numer. Anal., 41(3):1641–1667, 2021.
J. W. Barrett, K. Deckelnick, and V. Styles. Numerical analysis for a system coupling curve evolution to reaction diffusion on the curve. SIAM J. Numer. Anal., 55(2):1080–1100, 2017.
J. W. Barrett, H. Garcke, and R. Nürnberg. A parametric finite element method for fourth order geometric evolution equations. J. Comput. Phys., 222:441–467, 2007.
J. W. Barrett, H. Garcke, and R. Nürnberg. On the parametric finite element approximation of evolving hypersurfaces in \(\mathbb{R}^3\). J. Comput. Phys., 227:4281–4307, 2008.
J. W. Barrett, H. Garcke, and R. Nürnberg. Parametric approximation of willmore flow and related geometric evolution equations. SIAM Journal on Scientific Computing, 31(1):225–253, 2008.
J. W. Barrett, H. Garcke, and R. Nürnberg. Parametric finite element approximations of curvature-driven interface evolutions. In Handbook of numerical analysis, volume 21, pages 275–423. Elsevier, 2020.
S. Bartels. A simple scheme for the approximation of the elastic flow of inextensible curves. IMA J. Numer. Anal., 33:1115–1125, 2013.
S. Bartels, R. Müller, and C. Ortner. Robust a priori and a posteriori error analysis for the approximation of Allen–Cahn and Ginzburg–Landau equations past topological changes. SIAM J. Numer. Anal., 49:110–134, 2011.
T. Binz and B. Kovács. A convergent finite element algorithm for generalized mean curvature flows of closed surfaces. IMA J. Numer. Anal., 42(3):2545–2588, 2021.
A. Bonito, R. H. Nochetto, and M. S. Pauletti. Parametric FEM for geometric biomembranes. J. Comput. Phys., 229:3171–3188, 2010.
K. Deckelnick. Error bounds for a difference scheme approximating viscosity solutions of mean curvature flow. Interfaces Free Bound., 2:117–142, 2000.
K. Deckelnick and G. Dziuk. Convergence of a finite element method for non-parametric mean curvature flow. Numer. Math., 72:197–222, 1995.
K. Deckelnick and G. Dziuk. On the approximation of the curve shortening flow. In Calculus of variations, applications and computations (Pont-à-Mousson, 1994), volume 326 of Pitman Res. Notes Math. Ser., pages 100–108. Longman Sci. Tech., Harlow, 1995.
K. Deckelnick and G. Dziuk. Error analysis of a finite element method for the Willmore flow of graphs. Interfaces Free Bound., 8:21–46, 2006.
K. Deckelnick and G. Dziuk. Error analysis for the elastic flow of parametrized curves. Math. Comp., 78:645–671, 2009.
K. Deckelnick and R. Nürnberg. Error analysis for a finite difference scheme for axisymmetric mean curvature flow of genus-0 surfaces. SIAM J. Numer. Anal., 59(5):2698–2721, 2021.
K. Deckelnick and V. Styles. Finite element error analysis for a system coupling surface evolution to diffusion on the surface. Interfaces Free Bound., 24:63–93, 2022.
A. Demlow. Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal., 47(2):805–827, 2009.
G. Dziuk. An algorithm for evolutionary surfaces. Numer. Math., 58:603–611, 1990.
G. Dziuk. Convergence of a semi-discrete scheme for the curve shortening flow. Math. Models Methods Appl. Sci., 4:589–606, 1994.
G. Dziuk. Computational parametric Willmore flow. Numer. Math., 111:55–80, 2008.
G. Dziuk and C. M. Elliott. Finite elements on evolving surfaces. IMA J. Numer. Anal., 27:262–292, 2007.
G. Dziuk and C. M. Elliott. A fully discrete evolving surface finite element method. SIAM J. Numer. Anal., 50:2677–2694, 2012.
G. Dziuk, D. Kröner, and T. Müller. Scalar conservation laws on moving hypersurfaces. Interfaces Free Bound., 15(2):203–236, 2013.
K. Ecker. Regularity theory for mean curvature flow. Springer, 2012.
C. Elliott, H. Garcke, and B. Kovács. Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces. Numer. Math., 151:873–925, 2022.
C. M. Elliott and H. Fritz. On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick. IMA J. Numer. Anal., 37:543–603, 2017.
X. Feng and Y. Li. Analysis of symmetric interior penalty discontinuous Galerkin methods for the Allen–Cahn equation and the mean curvature flow. IMA J. Numer. Anal., 35:1622–1651, 2015.
X. Feng and A. Prohl. Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numer. Math., 94:33–65, 2003.
D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Springer, Berlin, Germany, 2001.
J. Hu and B. Li. Evolving finite element methods with an artificial tangential velocity for mean curvature flow and willmore flow. Numer. Math., 152:127–181, 2022.
B. Kovács. High-order evolving surface finite element method for parabolic problems on evolving surfaces. IMA J. Numer. Anal., 38(1):430–459, 2018.
B. Kovács, B. Li, and C. Lubich. A convergent evolving finite element algorithm for mean curvature flow of closed surfaces. Numer. Math., 143:797–853, 2019.
B. Kovács, B. Li, and C. Lubich. A convergent evolving finite element algorithm for Willmore flow of closed surfaces. Numer. Math., 149:595–643, 2021.
B. Kovács, B. Li, C. Lubich, and C. A. P. Guerra. Convergence of finite elements on an evolving surface driven by diffusion on the surface. Numer. Math., 137:643–689, 2017.
B. Li. Convergence of Dziuk’s linearly implicit parametric finite element method for curve shortening flow. SIAM J. Numer. Anal., 58:2315–2333, 2020.
B. Li. Convergence of Dziuk’s semidiscrete finite element method for mean curvature flow of closed surfaces with high-order finite elements. SIAM J. Numer. Anal., 59:1592–1617, 2021.
C. Mantegazza. Lecture Notes on Mean Curvature Flow. . Basel AG, 2012.
A. Mierswa. Error estimates for a finite difference approximation of mean curvature flow for surfaces of torus type, PhD Thesis, Otto-von-Guericke-Universität, Magdeburg, 2020.
B. White. Evolution of curves and surfaces by mean curvature. Proceedings of the International Congress of Mathematicians, 1:525–538, 2002.
C. Ye and J. Cui. Convergence of Dziuk’s fully discrete linearly implicit scheme for curve shortening flow. SIAM J. Numer. Anal., 59:2823–2842, 2021.
Q. Zhao, W. Jiang, and W. Bao. A parametric finite element method for solid-state dewetting problems in three dimensions. SIAM J. Sci. Comput., 42:B327–B352, 2020.
Acknowledgements
The work in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU15303022), and an internal grant of The Hong Kong Polytechnic University (Project ID: P0038843).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Rob Stevenson.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Optimal Approximation Properties of the Interpolated Surfaces
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
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:
Using the triangle inequality and the good quality of initial triangulation at \(t=0\), as shown in (2.1), we have
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 \(\kappa _m\)):
and
From (5.28), (5.52) and (5.74) we see that, by applying the inverse inequality,
Then, substituting this result into (A.3) and choosing \(j=1\), we derive the following estimate for \(0\le q\le l\):
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:
Then, by applying the discrete Grönwall’s inequality and taking the square root, we obtain
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
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:
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 \(j\le k-2\):
Then, by applying the discrete Grönwall’s inequality and taking the square root, we obtain
This proves (A.8) for \(1\le j\le k-2\). Analogously, by using (A.4), we can prove the following result for \(1\le j\le k-1\):
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 \(h\le h_{\kappa _{l},\kappa _{*,l}}\):
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
By applying Grönwall’s inequality and taking the square root, we obtain
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
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:
and passing to finite element functions, it holds that
for some finite element function \(\rho _h\) such that by the triangle inequality
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 \(O(|T_*^m (\hat{X}_{h,*}^{m + 1} -\hat{X}_{h,*}^{m})|)\).
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:
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
The first term on the right-hand side of (A.18) can be estimated by using (A.16) as follows:
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.,
In the case \(j=0\) we get from (A.18) and (A.19) the following result:
where the last inequality follows from the estimate in (5.39), which implies that
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:
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
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
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.,
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
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:
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
By substituting (A.28) into (A.23) we obtain the following result for \(j\ge 1\):
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.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bai, G., Li, B. A New Approach to the Analysis of Parametric Finite Element Approximations to Mean Curvature Flow. Found Comput Math (2023). https://doi.org/10.1007/s10208-023-09622-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10208-023-09622-x
Keywords
- Geometric flow
- Mean curvature flow
- Parametric finite element method
- Stability, convergence
- Particle trajectory
- Projected distance
- \(H^1\) parabolicity