On algorithms with good mesh properties for problems with moving boundaries based on the Harmonic Map Heat Flow and the DeTurck trick
The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 141-176.

In this paper, we present a general approach to obtain numerical schemes with good mesh properties for problems with moving boundaries, that is for evolving submanifolds with boundaries. This includes moving domains and surfaces with boundaries. Our approach is based on a variant of the so-called the DeTurck trick. By reparametrizing the evolution of the submanifold via solutions to the harmonic map heat flow of manifolds with boundary, we obtain a new velocity field for the motion of the submanifold. Moving the vertices of the computational mesh according to this velocity field automatically leads to computational meshes of high quality both for the submanifold and its boundary. Using the ALE-method in [16], this idea can be easily built into algorithms for the computation of physical problems with moving boundaries.

Published online:
DOI: 10.5802/smai-jcm.12
Classification: 65M50, 65M60, 35R01, 35R35
Keywords: Moving boundary, surface finite elements, mesh improvement, harmonic map heat flow, DeTurck trick

Charles M. Elliott 1; Hans Fritz 2

1 Mathematics Institute, University of Warwick, Coventry, UK
2 Department of Mathematics, University of Regensburg, Regensburg, Germany
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Charles M. Elliott; Hans Fritz. On algorithms with good mesh properties for problems with moving boundaries based on the Harmonic Map Heat Flow and the DeTurck trick. The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 141-176. doi : 10.5802/smai-jcm.12. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.12/

[1] C. Baker The mean curvature flow of submanifolds of high codimension, Australian National University (2010) (Ph. D. Thesis http://www.arxiv.org/abs/1104.4409v1)

[2] A. Bonito; R. Nochetto; M. S. Pauletti Geometrically consistent mesh modification, SIAM J. Numer. Anal., Volume 48 (2010), pp. 1877-1899 | DOI | MR | Zbl

[3] C. J. Budd; W. Huang; R. D. Russell Adaptivity with moving grids, Acta Numerica, Volume 18 (2009), pp. 111-241 | DOI | MR | Zbl

[4] B. Chow; P. Lu; L. Ni Hamilton’s Ricci Flow, Graduate Studies in Mathematics, AMS Science Press, 2006 | Zbl

[5] U. Clarenz; G. Dziuk Numerical methods for conformally parametrized surfaces, CPDw04 - Interphase 2003: Numerical Methods for Free Boundary Problems (2003) (http://www.newton.ac.uk/webseminars/pg+ws/2003/cpd/cpdw04/0415/dziuk)

[6] U. Clarenz; N. Litke; M. Rumpf Axioms and variational problems in surface parameterization, Computer Aided Geometric Design, Volume 21 (2004), pp. 727-749 | DOI | MR | Zbl

[7] K. Deckelnick; G. Dziuk; C. M. Elliott Computation of geometric partial differential equations and mean curvature flow, Acta Numerica, Volume 14 (2005), pp. 139-232 | DOI | MR | Zbl

[8] D. M. DeTurck Deforming metrics in the direction of their Ricci tensor, Journal of Differential Geometry, Volume 18 (1983) no. 11, pp. 157-162 | DOI | MR | Zbl

[9] A. S. Dvinsky Adaptive grid generation from harmonic maps on Riemannian manifolds, J. of Comp. Phys., Volume 95 (1991), pp. 450-476 | DOI | MR | Zbl

[10] G. Dziuk An algorithm for evolutionary surfaces, Numerische Mathematik, Volume 58 (1991) no. 1, pp. 603-611 | DOI | MR | Zbl

[11] G. Dziuk; C. M. Elliott Finite element methods for surface PDEs, Acta Numerica, Volume 22 (2013), pp. 289-396 | DOI | MR | Zbl

[12] J. Eells; J. H. Sampson Harmonic mappings of Riemannian manifolds, Amer. J. Math., Volume 86 (1964), pp. 109-160 | DOI | MR | Zbl

[13] J. Eells; J. C. Wood Restrictions on harmonic maps of surfaces, Topology, Volume 15 (1976), pp. 263-266 | DOI | MR | Zbl

[14] C. M. Elliott; H. Fritz On Approximations of the Curve Shortening Flow and of the Mean Curvature Flow based on the DeTurck trick, IMA Journal of Numerical Analysis (2016) (doi: 10.1093/imanum/drw020) | Zbl

[15] C. M. Elliott; J. R. Ockendon Weak and variational methods for moving boundary problems, Pitman, London, 1982 | Zbl

[16] C. M. Elliott; V. M. Styles An ALE ESFEM for solving PDEs on evolving surfaces, Milan Journal of Mathematics, Volume 80 (2012), pp. 469-501 | DOI | MR | Zbl

[17] C. M. Elliott; C. Venkataraman Error analysis for an ALE evolving surface finite element method, Num. Methods for PDEs, Volume 31 (2015), pp. 459-499 | DOI | MR | Zbl

[18] H. Fritz Isoparametric finite element approximation of Ricci curvature, IMA Journal of Numerical Analysis, Volume 33 (2013) no. 4, pp. 1265-1290 | DOI | MR | Zbl

[19] B. Gustaffson; A. Vasil’ev Conformal and Potential Analysis in Hele-Shaw Cells, Birkhauser Verlag, 2006 (ISBN 3-7643-7703-8)

[20] S. Haker; S. Angenent; A. Tannenbaum; R. Kikinis; G. Sapiro; M. Halle Conformal surface parameterization for texture mapping, IEEE Transactions on Visualization and Computer Graphics, Volume 6 (2000) no. 2, pp. 181-189 | DOI

[21] R. S. Hamilton Harmonic maps of manifolds with boundary, Springer Lecture Notes 471, 1975 | Zbl

[22] R. S. Hamilton Heat equations in geometry (1989) (Lecture notes)

[23] R. S. Hamilton The formation of singularities in the Ricci flow, Surveys in Differential Geometry, Volume 227 (1995), pp. 7-136 | Zbl

[24] C.-J. Heine Curvature reconstruction with linear finite elements (2009) (Private communications)

[25] W. Huang Practical aspects of formulation and solution of moving mesh partial differential equations, J. of Comp. Phys., Volume 171 (2001), pp. 753-775 | DOI | MR | Zbl

[26] W. Huang; R. D. Russell Moving mesh strategy based upon a gradient flow equation for two dimensional problems, SIAM J. Sci. Comput., Volume 20 (1998) no. 3, pp. 998-1015 | DOI

[27] W. Huang; R. D. Russell Adaptive Moving Mesh Methods, Applied Mathematical Sciences Volume 174 (2011) | Zbl

[28] M. Jin; Y. Wang; S.-T. Yau; X. Gu, Proceedings of the Conference on Visualization ’04 (2004), pp. 267-274

[29] J. Jost Ein Existenzbeweis für harmonische Abbildungen, die ein Dirichlet-Problem lösen, mittels der Methode des Wärmeflusses, Manuscripta mathematica, Volume 34 (1981), pp. 17-25 | DOI | Zbl

[30] E. Kelley; E. J. Hinch Numerical simulations of sink flow in the Hele-Shaw cell with small surface tension, Euro. J. Applied Math., Volume 8 (1997), pp. 533-550 | DOI | MR | Zbl

[31] G. Macdonald; J. A. Mackenzie; M. Nolan; R. H. Insall A Computational Method for the Coupled Solution of Reaction-Diffusion Equations on Evolving Domains and Surfaces: Application to a Model of Cell Migration and Chemotaxis (2015) no. 6 (Technical report)

[32] K. Mikula; M. Remešíková; P. Sarkoci; D. Ševčovič Manifold evolution with tangential redistribution of points, SIAM J. Sci. Comput., Volume 36 (2014) no. 4, p. A1384-A1414 | DOI | MR | Zbl

[33] A. Schmidt; K. G. Siebert Design of Adaptive Finite Element Software, Lecture Notes in Computational Science and Engineering 42, Springer, 2005 | Zbl

[34] J. Steinhilber Numerical analysis for harmonic maps between hypersurfaces and grid improvement for computational parametric geometric flows, University of Freiburg (2014) (Ph. D. Thesis http://www.freidok.uni-freiburg.de/volltexte/9537/) | Zbl

[35] A. M. Winslow Numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh, J. of Comp. Phys., Volume 1 (1966) no. 2, pp. 149-172 | DOI | MR | Zbl

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