A convexity-preserving and perimeter-decreasing parametric finite element method for the area-preserving curve shortening flow. (English) Zbl 1522.65174
Summary: We propose and analyze a semidiscrete parametric finite element scheme for solving the area-preserving curve shortening flow. The scheme is based on Dziuk’s approach [G. Dziuk, SIAM J. Numer. Anal. 36, No. 6, 1808–1830 (1999; Zbl 0942.65112)] for the anisotropic curve shortening flow. We prove that the scheme preserves two fundamental geometric structures of the flow with an initially convex curve: (i) the convexity-preserving property, and (ii) the perimeter-decreasing property. To the best of our knowledge, the convexity-preserving property of numerical schemes which approximate the flow is rigorously proved for the first time. Furthermore, the error estimate of the semidiscrete scheme is established, and numerical results are provided to demonstrate the structure-preserving properties as well as the accuracy of the scheme.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35K55 | Nonlinear parabolic equations |
Keywords:
area-preserving curve shortening flow; parametric finite element method; error estimate; convexity-preserving; perimeter-decreasingCitations:
Zbl 0942.65112References:
| [1] | Almgren, F., Taylor, J. E., and Wang, L., Curvature-driven flows: A variational approach, SIAM J. Control Optim., 31 (1993), pp. 387-438, doi:10.1137/0331020. · Zbl 0783.35002 |
| [2] | Andrews, B., Chow, B., Guenther, C., and Langford, M., Extrinsic Geometric Flows, , AMS, Providence, RI, 2020. · Zbl 1475.53002 |
| [3] | Bao, W., Garcke, H., Nürnberg, R., and Zhao, Q., Volume-preserving parametric finite element methods for axisymmetric geometric evolution equations, J. Comput. Phys., 460 (2022), 111180. · Zbl 07525159 |
| [4] | Bao, W. and Zhao, Q., A structure-preserving parametric finite element method for surface diffusion, SIAM J. Numer. Anal., 59 (2021), pp. 2775-2799, doi:10.1137/21M1406751. · Zbl 1489.65136 |
| [5] | Barrett, J. W., Deckelnick, K., and Styles, V., Numerical analysis for a system coupling curve evolution to reaction diffusion on the curve, SIAM J. Numer. Anal., 55 (2017), pp. 1080-1100, doi:10.1137/16M1083682. · Zbl 1365.65218 |
| [6] | Barrett, J. W., Garcke, H., and Nürnberg, R., On the variational approximation of combined second and fourth order geometric evolution equations, SIAM J. Sci. Comput., 29 (2007), pp. 1006-1041, doi:10.1137/060653974. · Zbl 1148.65074 |
| [7] | Barrett, J. W., Garcke, H., and Nürnberg, R., A parametric finite element method for fourth order geometric evolution equations, J. Comput. Phys., 222 (2007), pp. 441-467. · Zbl 1112.65093 |
| [8] | Barrett, J. W., Garcke, H., and Nürnberg, R., On the parametric finite element approximation of evolving hypersurfaces in \({\mathbb R}^3\), J. Comput. Phys., 227 (2008), pp. 4281-4307. · Zbl 1145.65068 |
| [9] | Barrett, J. W., Garcke, H., and Nürnberg, R., Numerical approximation of anisotropic geometric evolution equations in the plane, IMA J. Numer. Anal., 28 (2008), pp. 292-330. · Zbl 1145.65069 |
| [10] | Barrett, J. W., Garcke, H., and Nürnberg, R., Variational discretization of axisymmetric curvature flows, Numer. Math., 141 (2019), pp. 791-837. · Zbl 1419.65051 |
| [11] | Barrett, J. W., Garcke, H., and Nürnberg, R., Parametric finite element method approximations of curvature driven interface evolutions, in Handbook of Numerical Analysis 21, Bonito, A. and Nochetto, R. H., eds., North-Holland, Amsterdam, 2020, pp. 275-423. · Zbl 1455.35185 |
| [12] | Brenner, S. C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer, New York, 2008. · Zbl 1135.65042 |
| [13] | Bronsard, L. and Stoth, B., Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 (1997), pp. 769-807, doi:10.1137/S0036141094279279. · Zbl 0874.35009 |
| [14] | Chow, B. and Glickenstein, D., Semidiscrete geometric flows of polygons, Amer. Math. Monthly, 114 (2007), pp. 316-328. · Zbl 1171.52009 |
| [15] | Deckelnick, K. and Dziuk, G., On the approximation of the curve shortening flow, in Calculus of Variations, Applications and Computations (Pont-à-Mousson, 1994), , Longman Sci. Tech., Harlow, UK, 1995, pp. 100-108. · Zbl 0830.65096 |
| [16] | Deckelnick, K., Dziuk, G., and Elliott, C. M., Computation of geometric partial differential equations and mean curvature flow, Acta Numer., 14 (2005), pp. 139-232. · Zbl 1113.65097 |
| [17] | do Carmo, M. P., Differential Geometry of Curves and Surfaces, Dover, Mineola, NY, 2016. · Zbl 1352.53002 |
| [18] | Dziuk, G., An algorithm for evolutionary surfaces, Numer. Math., 58 (1991), pp. 603-611. · Zbl 0714.65092 |
| [19] | Dziuk, G., Convergence of a semi-discrete scheme for the curve shortening flow, Math. Models Methods Appl. Sci., 4 (1994), pp. 589-606. · Zbl 0811.65112 |
| [20] | Dziuk, G., Discrete anisotropic curve shortening flow, SIAM J. Numer. Anal., 36 (1999), pp. 1808-1830, doi:10.1137/S0036142998337533. · Zbl 0942.65112 |
| [21] | Elliott, C. M. and Fritz, H., On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick, IMA J. Numer. Anal., 37 (2016), pp. 543-603. · Zbl 1433.65219 |
| [22] | Escher, J. and Simonett, G., The volume preserving mean curvature flow near spheres, Proc. Amer. Math. Soc., 126 (1998), pp. 2789-2796. · Zbl 0909.53043 |
| [23] | Gage, M., On an area-preserving evolution equation for plane curves. Nonlinear problems in geometry, Contemp. Math., 51 (1986), pp. 51-62. · Zbl 0608.53002 |
| [24] | Gruber, P. M., Convex and Discrete Geometry, , Springer, Berlin, 2007. · Zbl 1139.52001 |
| [25] | Huisken, G., The volume preserving mean curvature flow, J. Reine Angew. Math., 382 (1987), pp. 35-48. · Zbl 0621.53007 |
| [26] | Hu, J. and Li, B., Evolving finite element methods with an artificial tangential velocity for mean curvature flow and Willmore flow, Numer. Math., 152 (2022), pp. 127-181. · Zbl 1496.65163 |
| [27] | Italo, C. D., Stefano, F., and Riccardo, M., Area-preserving curve-shortening flows: From phase separation to image processing, Interfaces Free Bound., 4 (2002), pp. 325-343. · Zbl 1021.35129 |
| [28] | Jiang, W. and Li, B., A perimeter-decreasing and area-conserving algorithm for surface diffusion flow of curves, J. Comput. Phys., 443 (2021), 110531. · Zbl 07515430 |
| [29] | Kublik, C., Esedo \({\scriptstyle{\bar{\text{G}}}}\) lu, S., and Fessler, J. A., Algorithms for area preserving flows, SIAM J. Sci. Comput., 33 (2011), pp. 2382-2401, doi:10.1137/100815542. · Zbl 1232.65012 |
| [30] | Kovács, B., Li, B., and Lubich, C., A convergent evolving finite element algorithm for mean curvature flow of closed surfaces, Numer. Math., 143 (2019), pp. 797-853. · Zbl 1427.65250 |
| [31] | Kovács, B., Li, B., and Lubich, C., A convergent evolving finite element algorithm for Willmore flow of closed surfaces, Numer. Math., 149 (2021), pp. 595-643. · Zbl 1496.65165 |
| [32] | Laurain, A. and Walker, S. W., Optimal control of volume-preserving mean curvature flow, J. Comput. Phys., 438 (2021), 110373. · Zbl 07505967 |
| [33] | Li, B., Convergence of Dziuk’s linearly implicit parametric finite element method for curve shortening flow, SIAM J. Numer. Anal., 58 (2020), pp. 2315-2333, doi:10.1137/19M1305483. · Zbl 1452.65240 |
| [34] | Li, B., 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 (2021), pp. 1592-1617, doi:10.1137/20M136935X. · Zbl 1479.65008 |
| [35] | Mayer, U., A numerical scheme for moving boundary problems that are gradient flows for the area functional, European J. Appl. Math., 11 (2000), pp. 61-80. · Zbl 0945.76016 |
| [36] | Mikula, K. and Ševčovič, D., Computational and qualitative aspects of evolution of curves driven by curvature and external force, Comput. Vis. Sci., 6 (2004), pp. 211-225. · Zbl 1522.53008 |
| [37] | Mugnai, L. and Seis, C., On the coarsening rates for attachment-limited kinetics, SIAM J. Math. Anal., 45 (2013), pp. 324-344, doi:10.1137/120865197. · Zbl 1278.35016 |
| [38] | Pei, L. and Li, Y., A Structure-preserving Parametric Finite Element Method for Area-conserved Generalized Mean Curvature Flow, https://arxiv.org/abs/2211.13582v1, 2022. |
| [39] | Pozzi, P. and Stinner, B., Curve shortening flow coupled to lateral diffusion, Numer. Math., 135 (2017), pp. 1171-1205. · Zbl 1369.65111 |
| [40] | Ruuth, S. and Wetton, B. T. R., A simple scheme for volume-preserving motion by mean curvature, J. Sci. Comput., 19 (2003), pp. 373-384. · Zbl 1035.65097 |
| [41] | Sakakibara, K. and Miyatake, Y., A fully discrete curve-shortening polygonal evolution law for moving boundary problems, J. Comput. Phys., 424 (2021), 109857. · Zbl 07508462 |
| [42] | Taylor, J. E. and Cahn, J. W., Linking anisotropic sharp and diffuse surface motion laws via gradient flows, J. Statist. Phys., 77 (1994), pp. 183-197. · Zbl 0844.35044 |
| [43] | Ushijima, T. and Yazaki, S., Convergence of a crystalline approximation for an area-preserving motion, J. Comput. Appl. Math., 166 (2004), pp. 427-452. · Zbl 1052.65082 |
| [44] | Ye, C. and Cui, J., Convergence of Dziuk’s fully discrete linearly implicit scheme for curve shortening flow, SIAM J. Numer. Anal., 59 (2021), pp. 2823-2842, doi:10.1137/21M1391626. · Zbl 1492.65276 |
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