A meshfree method for solving the Monge–Ampère equation

  • Klaus Böhmer
  • Robert SchabackEmail author
Original Paper


This paper solves the two-dimensional Dirichlet problem for the Monge-Ampère equation by a strong meshless collocation technique that uses a polynomial trial space and collocation in the domain and on the boundary. Convergence rates may be up to exponential, depending on the smoothness of the true solution, and this is demonstrated numerically and proven theoretically, applying a sufficiently fine collocation discretization. A much more thorough investigation of meshless methods for fully nonlinear problems is in preparation.


Collocation Fully nonlinear PDE Monge–Ampère Nonlinear optimizer MATLAB implementation Convergence Error analysis Error estimates 

Mathematics Subject Classification (2010)

35J36 65D99 65N12 65N35 


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The authors thank the referees for several suggestions improving the paper, in particular the presentation of the numerical results.


  1. 1.
    Awanou, G.: Spline element method for Monge-Ampere equations. B.I.T Num. Analysis 55, 625–646 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Awanou, G.: On standard finite difference discretizations of the elliptic Monge-Ampere equation. J. Sci. Comput. 69, 892–904 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benamou, J.-D., Froese, B.D., Oberman, A.M.: Two numerical methods for the elliptic Monge-Ampère equation. ESAIM: Mathematical Modelling and Numerical Analysis 44(4), 737–758 (2010)CrossRefGoogle Scholar
  4. 4.
    Böhmer, K.: On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 3, 1212–1249 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Böhmer, K.: Numerical Methods for Nonlinear Elliptic Differential Equations, a Synopsis. Oxford University Press, Oxford (2010)CrossRefGoogle Scholar
  6. 6.
    Böhmer, K., Schaback, R.: A nonlinear discretization theory. J. Comput. Appl. Math. 254, 204–219 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Böhmer, K., Schaback, R.: Nonlinear discretization theory applied to meshfree methods and the Monge-Ampère equation. Fachbereich Mathematik und Informatik, Philipps–Universitȧt Marburg in preparation (2016)Google Scholar
  8. 8.
    Braess, D.: Finite Elements. Theory, Fast Solvers and Applications in Solid Mechanics, 2nd edn. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  9. 9.
    Brenner, S. C., Gudi, T., Neilan, M., Sung, L.Y.: C0 penalty methods for the fully nonlinear Monge-Ampère equation. Math. Comput. 80, 1979–1995 (2011)CrossRefGoogle Scholar
  10. 10.
    Davydov, O.: Smooth finite elements and stable splitting. Fachbereich Mathematik und Informatik, Philipps–Universität Marburg (2007)Google Scholar
  11. 11.
    Davydov, O., Saeed, A.: Numerical solution of fully nonlinear elliptic equations by Böhmer’s method. J. Comp. Appl.Math. 254, 43–54 (2013)CrossRefGoogle Scholar
  12. 12.
    Fasshauer, G., McCourt, M.: Kernel-Based Approximation Methods using MATLAB, Volume 19 of Interdisciplinary Mathematical Sciences. World Scientific, Singapore (2015)zbMATHGoogle Scholar
  13. 13.
    Feng, X., Neilan, M.: Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal. 47, 1226–1250 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Froese, B.D.: Meshfree finite difference approximations for functions of the eigenvalues of the Hessian. Numer. Math. 138(1), 75–99 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Froese, B.D., Oberman, A.M.: Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher. SIAM J. Numer. Anal. 49(4), 1692–1714 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Froese, B.D., Oberman, A.M.: Convergent filtered schemes for the Monge–Ampère partial differential equation. SIAM J. Numer. Anal. 51(1), 423–444 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Li, Q., Liu, Z.Y.: Solving the 2-D elliptic Monge-Ampère equation by a Kansa’s method. Acta Mathematicae Applicatae Sinica, English Series 33(2), 269–276 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Liu, J., Froese, B.D., Oberman, A.M., Xiao, M.Q.: A multigrid scheme for 3D Monge-Ampère equations. Int. J. Comput. Math. 94(9), 1850–1866 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Liu, Z.Y., He, Y.: Cascadic meshfree method for the elliptic Monge-Ampère equation. Engineering Analysis with Boundary Elements 37(7), 990–996 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Liu, Z.Y., He, Y.: An iterative meshfree method for the elliptic Monge-Ampère equation in 2D. Numer. Methods Partial Differential Equations 30(5), 1507–1517 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Oberman, A.: Wide stencil finite difference schemes for the elliptic Monge-Ampère equations and functions of the eigenvalues of the Hessian. Discrete Contin. Dyn. Syst. Ser. B 10(1), 221–238 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schaback, R.: Unsymmetric meshless methods for operator equations. Numer. Math. 114, 629–651 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Schaback, R.: All well–posed problems have uniformly stable and convergent discretizations. Numer. Math. 132, 597–630 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikUniversität Marburg, Arbeitsgruppe NumerikLahnbergeGermany
  2. 2.Institut für Numerische und Angewandte MathematikUniversität GöttingenGöttingenGermany

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