A New Approach to the Analysis of Parametric Finite Element Approximations to Mean Curvature Flow
Springer Science and Business Media LLC - Trang 1-65 - 2023
Tóm tắt
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.
Tài liệu tham khảo
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.