Numerische Mathematik

, Volume 141, Issue 1, pp 173–213 | Cite as

Variational convergence of discrete minimal surfaces

  • Henrik SchumacherEmail author
  • Max Wardetzky


Building on and extending tools from variational analysis and relying on certain a priori assumptions, we prove Kuratowski convergence of sets of simplicial area minimizers to minimizers of the smooth Douglas–Plateau problem under simplicial refinement. This convergence is with respect to a topology that is finer than the topology of uniform convergence of both positions and surface normals.

Mathematics Subject Classification

49Q05 53A10 58E12 



The authors would like to thank Heiko von der Mosel for discussions on the regularity of minimal surfaces.


  1. 1.
    Brakke, K.A.: The surface evolver. Exp. Math. 1(2), 141–165 (1992).
  2. 2.
    Courant, R.: Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces. Interscience Publishers Inc., New York (1950). Appendix by M. SchifferzbMATHGoogle Scholar
  3. 3.
    Dal Maso, G.: An Introduction to \(\varGamma \)-Convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston Inc., Boston, MA (1993). Google Scholar
  4. 4.
    Dierkes, U., Hildebrandt, S., Tromba, A.J.: Regularity of Minimal Surfaces, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 340, 2nd edn. Springer, Heidelberg (2010)zbMATHGoogle Scholar
  5. 5.
    Douglas, J.: A method of numerical solution of the problem of Plateau. Ann. Math. (2) 29(1–4), 180–188 (1927). MathSciNetzbMATHGoogle Scholar
  6. 6.
    Douglas, J.: Solution of the problem of Plateau. Trans. Am. Math. Soc. 33(1), 263–321 (1931). MathSciNetzbMATHGoogle Scholar
  7. 7.
    Douglas, J.: The most general form of the problem of Plateau. Am. J. Math. 61, 590–608 (1939)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dziuk, G.: An algorithm for evolutionary surfaces. Numer. Math. 58(6), 603–611 (1991). MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dziuk, G., Hutchinson, J.E.: The discrete Plateau problem: algorithm and numerics. Math. Comput. 68(225), 1–23 (1999). MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dziuk, G., Hutchinson, J.E.: The discrete Plateau problem: convergence results. Math. Comput. 68(226), 519–546 (1999). MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fukui, K., Nakamura, T.: A topological property of Lipschitz mappings. Topolo. Appl. 148(1–3), 143–152 (2005). MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. of Math. (2) 110(3), 439–486 (1979). MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hinze, M.: On the numerical approximation of unstable minimal surfaces with polygonal boundaries. Numer. Math. 73(1), 95–118 (1996). MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lee, J.M.: Introduction to Smooth Manifolds, Graduate Texts in Mathematics. Springer, New York (2003). Google Scholar
  15. 15.
    Micheli, M., Michor, P.W., Mumford, D.: Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds. Izv. Ross. Akad. Nauk. Ser. Mat. 77(3), 109–138 (2013). MathSciNetzbMATHGoogle Scholar
  16. 16.
    Michor, P.W., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Michor, P.W., Mumford, D.: An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal. 23(1), 74–113 (2007). MathSciNetzbMATHGoogle Scholar
  18. 18.
    Moakher, M., Zéraï, M.: The Riemannian geometry of the space of positive-definite matrices and its application to the regularization of positive-definite matrix-valued data. J. Math. Imaging Vis. 40(2), 171–187 (2011). MathSciNetzbMATHGoogle Scholar
  19. 19.
    Munkres, J.R.: Topology, 2nd edn. Prentice Hall Inc., Upper Saddle River (2000)zbMATHGoogle Scholar
  20. 20.
    Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993).
  21. 21.
    Pozzi, P.: The discrete Douglas problem: theory and numerics. Interfaces Free Bound. 6(2), 219–252 (2004). MathSciNetzbMATHGoogle Scholar
  22. 22.
    Pozzi, P.: The discrete Douglas problem: convergence results. IMA J. Numer. Anal. 25(2), 337–378 (2005). MathSciNetzbMATHGoogle Scholar
  23. 23.
    Radó, T.: On the problem of plateau. Ergebnisse der Mathematik und ihrer Grenzgebiete, Julius Springer, Berlin (1933)Google Scholar
  24. 24.
    Rivière, T.: Lipschitz conformal immersions from degenerating Riemann surfaces with \(L^2\)-bounded second fundamental forms. Adv. Calc. Var. 6(1), 1–31 (2013). MathSciNetzbMATHGoogle Scholar
  25. 25.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer, Berlin (1998). Google Scholar
  26. 26.
    Tsuchiya, T.: Discrete solution of the Plateau problem and its convergence. Math. Comput. 49(179), 157–165 (1987). MathSciNetzbMATHGoogle Scholar
  27. 27.
    Wagner, H.J.: A contribution to the numerical approximation of minimal surfaces. Computing 19(1), 35–58 (1977)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Whitehead, J.H.C.: On \(C^1\)-complexes. Ann. Math. 2(41), 809–824 (1940)zbMATHGoogle Scholar
  29. 29.
    Wilson Jr., W.L.: On discrete Dirichlet and Plateau problems. Numer. Math. 3, 359–373 (1961)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for MathematicsRWTH Aachen UniversityAachenGermany
  2. 2.Institute for Numerical and Applied MathematicsGeorg-August-University GöttingenGöttingenGermany

Personalised recommendations