Foundations of Computational Mathematics

, Volume 16, Issue 5, pp 1115–1150 | Cite as

A Laplace Operator on Semi-Discrete Surfaces



This paper studies a Laplace operator on semi-discrete surfaces. A semi-discrete surface is represented by a mapping into three-dimensional Euclidean space possessing one discrete variable and one continuous variable. It can be seen as a limit case of a quadrilateral mesh, or as a semi-discretization of a smooth surface. Laplace operators on both smooth and discrete surfaces have been an object of interest for a long time, also from the viewpoint of applications. There are a wealth of geometric objects available immediately once a Laplacian is defined, e.g., the mean curvature normal. We define our semi-discrete Laplace operator to be the limit of a discrete Laplacian on a quadrilateral mesh, which converges to the semi-discrete surface. The main result of this paper is that this limit exists under very mild regularity assumptions. Moreover, we show that the semi-discrete Laplace operator inherits several important properties from its discrete counterpart, like symmetry, positive semi-definiteness, and linear precision. We also prove consistency of the semi-discrete Laplacian, meaning that it converges pointwise to the Laplace–Beltrami operator, when the semi-discrete surface converges to a smooth one. This result particularly implies consistency of the corresponding discrete scheme.


Laplace operator Semi-discrete surfaces Quadrilateral meshes Consistency 

Mathematics Subject Classification

Primary 53B20 Secondary 53A05 41A25 



The author would like to thank J. Wallner for fruitful discussions and comments. This research was supported by the DFG-Collaborative Research Center, TRR 109, “Discretization in Geometry and Dynamics” and the Doctoral Program “Discrete Mathematics” through grants I 706-N26 and W1230 of the Austrian Science Fund (FWF). We further acknowledge support from NAWI Graz and are grateful to the anonymous reviewers for their remarks and suggestions.


  1. 1.
    M. Alexa and M. Wardetzky. Discrete Laplacians on General Polygonal Meshes. ACM Trans. Graph., 30(4):1–10, 2011.CrossRefGoogle Scholar
  2. 2.
    M. Belkin, J. Sun, and Y. Wang. Discrete Laplace Operator on Meshed Surfaces. In Proc. of the 24th ACM Symposium on Computational Geometry, pages 278–287, 2008.Google Scholar
  3. 3.
    A. Bobenko and B. Springborn. A Discrete Laplace-Beltrami Operator for Simplicial Surfaces. Discrete & Computational Geometry, 38(4):740–756, Dec. 2007.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    A. Bobenko and Y. Suris. Discrete Differential Geometry: Integrable Structure. Graduate studies in mathematics. American Math. Soc., 2008.Google Scholar
  5. 5.
    T. H. Colding and W. P. Minicozzi. A Course in Minimal Surfaces, volume 121 of Graduate Studies in Mathematics. American Math. Soc., 2011.Google Scholar
  6. 6.
    M. Desbrun, M. Meyer, P. Schröder, and A. H. Barr. Implicit Fairing of Irregular Meshes Using Diffusion and Curvature Flow. In SIGGRAPH ’99, pages 317–324, New York, 1999. ACM.Google Scholar
  7. 7.
    R. J. Duffin. Distributed and lumped networks. J. Math. Mech., 8:793–826, 1959.MathSciNetMATHGoogle Scholar
  8. 8.
    G. Dziuk. Finite elements for the Beltrami operator on arbitrary surfaces. In Partial differential equations and calculus of variations, pages 142–155. Springer, Berlin, 1988.Google Scholar
  9. 9.
    O. Karpenkov and J. Wallner. On offsets and curvatures for discrete and semidiscrete surfaces. Beitr. Algebra Geom., 55(1):207–228, 2014.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    D. Liu, G. Xu, and Q. Zhang. A discrete scheme of Laplace-Beltrami operator and its convergence over quadrilateral meshes. Comput. Math. Appl., 55(6):1081–1093, 2008.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    R. H. MacNeal. The Solution of Partial Differential Equations by Means of Electrical Networks. PhD thesis, California Institute of Technology, 1949.Google Scholar
  12. 12.
    C. Müller and J. Wallner. Semi-discrete isothermic surfaces. Results Math., 63(3-4):1395–1407, 2013.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    U. Pinkall and K. Polthier. Computing discrete minimal surfaces and their conjugates. Experiment. Math., 2(1):15–36, 1993.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    S. Rosenberg. The Laplacian on a Riemannian manifold, volume 31 of London Mathematical Society Student Texts. CUP, Cambridge, 1997.Google Scholar
  15. 15.
    M. Wardetzky. Convergence of the cotangent formula: an overview. In Discrete differential geometry, volume 38 of Oberwolfach Semin., pages 275–286. Birkhäuser, Basel, 2008.Google Scholar
  16. 16.
    M. Wardetzky, S. Mathur, F. Kälberer, and E. Grinspun. Discrete Laplace operators: No free lunch. In Eurographics Symposium on Geometry Processing, pages 33–37. Eurographics Association, 2007.Google Scholar
  17. 17.
    Y. Xiong, G. Li, and G. Han. Mean Laplace-Beltrami Operator for Quadrilateral Meshes. T. Edutainment, 5:189–201, 2011.CrossRefGoogle Scholar
  18. 18.
    G. Xu. Discrete Laplace-Beltrami operators and their convergence. Computer Aided Geometric Design, 21(8):767–784, 2004.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Institute of GeometryGraz University of TechnologyGrazAustria

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