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High-Order Algorithms for Laplace–Beltrami Operators and Geometric Invariants over Curved Surfaces

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Abstract

The Laplace–Beltrami operators on curved surfaces play an essential role in the convergence analysis of the numerical simulations of some important geometric partial differential equations which involve this operator. In this note we shall combine the local tangential lifting method with the configuration equation to develop a new effective and convergent generalized finite difference method to approximate the Laplace–Beltrami operators acting on functions over discrete surfaces. The convergence rates of our algorithms of discrete Laplace–Beltrami operators over surfaces is \(O(r^n)\), \(n \ge 1\), where \(r\) represents the size of the mesh of discretization of the surface. The problem of high-order accuracies will also be discussed and used to compute geometric invariants of the underlying surfaces. Some convergence tests and eigenvalue computations on the sphere, tori and a dumbbell are presented.

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References

  1. Augoula, S., Abgrall, R.: High order numerical discretization for Hamilton–Jacobi equations on triangular meshes. J. Sci. Comput. 15(2), 197–229 (2000)

    Article  MATH  Google Scholar 

  2. Chen, S.-G., Wu, J.-Y.: Estimating normal vectors and curvatures by centroid weights. Comput. Aided Geom. Des. 21, 447–458 (2004)

    Article  Google Scholar 

  3. Chen, S.-G., Chi, M.-H., Wu, J.-Y.: Curvature estimation and curvature flow for digital curves. WSEAS Trans. Comput. 5, 804–809 (2006)

    Google Scholar 

  4. Chen, S.-G., Chi, M.-H., Wu, J.-Y.: Boundary and interior derivatives estimation for 2D scattered data points. WSEAS Trans. Comput. 5, 824–829 (2006)

    Google Scholar 

  5. Chen, S.-G., Wu, J.-Y.: Discrete conservation laws on curved surfaces. SIAM J. Sci. Comput. 35(2), A719–A739 (2013)

    Article  MATH  Google Scholar 

  6. Chun, S., Hesthaven, J.S.: High-order accurate thin layer approximations for time-domain electromagnetics. Part I: general metal backed coatings. J. Comput. Appl. Math. 231, 598–611 (2009)

    Article  MATH  Google Scholar 

  7. Clarenz, U., Diewald, U., Rumpf, M.: Anisotropic geometric diffusion in surface processing. In: Proceedings of Viz 2000, IEEE Visualization, pp. 397–405 (2000)

  8. Desbrun, M., Meyer, M., Schroder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: SIGGRAPH99, pp. 317–324 (1999)

  9. do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall International Inc, London (1976)

    MATH  Google Scholar 

  10. do Carmo, M.: Riemannian Geometry. Birkhauser, Boston (1992)

    Book  MATH  Google Scholar 

  11. Goudenege, L., Martin, D., Vial, G.: High order finite element calculations for the Cahn–Hilliard equation. J. Sci. Comput. 52, 294–321 (2012)

    Article  MATH  Google Scholar 

  12. Hesthaven, J.S., Warburton, T.: High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem. R. Soc. Lond. Ser. A 362, 493–524 (2004)

    Article  MATH  Google Scholar 

  13. Macdonald, C.B., Brandman, J., Steven J. Ruuth.: Solving eigenvalue problems on curved surfaces using the Closest Point Method. J. Comput. Phys. 230, 7944–7956 (2011)

  14. Ray, N., Wang, D., Jiao, X., Glimm, J.: High-order numerical integration over discrete surfaces. SIAM J. Numer. Anal. 50(6), 3061–3083 (2012)

    Article  MATH  Google Scholar 

  15. Romeny, B.H.: Geometry Driven Diffusion in Computer Vision. Springer, Boston (1994)

  16. Ruuth, S.J., Merriman, B.: A simple embedding method for solving partial differential equations on surfaces. J. Comput. Phys. 227, 1943–1961 (2008)

    Article  MATH  Google Scholar 

  17. Kluwer Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  18. Shakeri, F., Dehghan, M.: A high order finite volume element method for solving elliptic partial integro-differential equations. Appl. Numer. Math. 65, 105–118 (2013)

    Article  MATH  Google Scholar 

  19. Shi, Y., Lai, R., Krishna, S., Sicotte, N., Dinov, I., Toga, A.W.: Anisotropic Laplace–Beltrami eigenmaps: bridging reeb graphs and skeletons. In: Proceedings MMBIA (2008)

  20. Shi, Y., Lai, R., Kern, K., Sicotte, N., Dinov, I., Toga, A.W.: Harmonic surface mapping with Laplace–Beltrami eigenmaps. In: Proceedings MICCAI (2008)

  21. Strange, G., Fix, G.: An Analysis of the Finite Element Method, 2nd edn. Wellesley-Cambridge press, Wellesley (2008)

    Google Scholar 

  22. Trasdahl, Ø., Rønquist, E.M.: High order numerical approximation of minimal surfaces. J. Comput. Phys. 230, 4795–4810 (2011)

  23. Wu, J.-Y., Chi, M.-H., Chen, S.-G.: A new intrinsic numerical method for PDE on surfaces. Int. J. Comput. Math. 89(1), 54–79 (2012)

    Article  MATH  Google Scholar 

  24. Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)

    Article  MATH  Google Scholar 

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Acknowledgments

This paper is partially supported by MOST, Taiwan.

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Authors and Affiliations

  1. Department of Applied Mathematics, National Chiayi University, Chia-Yi, 600, Taiwan

    Sheng-Gwo Chen

  2. Department of Mathematics, National Chung Cheng University, Chia-Yi, 621, Taiwan

    Mei-Hsiu Chi & Jyh-Yang Wu

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  1. Sheng-Gwo Chen
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Correspondence to Sheng-Gwo Chen.

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Chen, SG., Chi, MH. & Wu, JY. High-Order Algorithms for Laplace–Beltrami Operators and Geometric Invariants over Curved Surfaces. J Sci Comput 65, 839–865 (2015). https://doi.org/10.1007/s10915-015-9986-6

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  • DOI: https://doi.org/10.1007/s10915-015-9986-6

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