A Finite Element/Operator-Splitting Method for the Numerical Solution of the Three Dimensional Monge–Ampère Equation

  • Hao LiuEmail author
  • Roland Glowinski
  • Shingyu Leung
  • Jianliang Qian


In the present article we extend to the three-dimensional elliptic Monge–Ampère equation the method discussed in Glowinski et al. (J Sci Comput 79:1–47, 2019) for the numerical solution of its two-dimensional variant. As in Glowinski et al. (2019) we take advantage of an equivalent divergence formulation of the Monge–Ampère equation, involving the cofactor matrix of the Hessian of the solution. We associate with the above divergence formulation an initial value problem, well suited to time discretization by operator splitting and space approximation by low order mixed finite element methods. An important ingredient of our methodology is forcing the positive semi-definiteness of the approximate Hessian by a hard thresholding eigenvalue projection. The resulting method is robust and easy to implement. It can handle problems with smooth and non-smooth solutions on domains with curved boundary. Using piecewise affine approximations for the solution and its six second-order derivatives, one can achieve second-order convergence rates for problems with smooth solutions.



The work of R. Glowinski was partially supported by the Hong Kong Kennedy Wong foundation. The work of S. Leung was partially supported by the Hong Kong RGC Grants 16302819 and 16309316. The work of J. Qian was partially supported by NSF.


  1. 1.
    Awanou, G.: Pseudo transient continuation and time marching methods for Monge–Ampère type equations. Adv. Comput. Math. 41(4), 907–935 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bakelman, I.J.: Convex Analysis and Nonlinear Geometric Elliptic Equations. Springer, Berlin (2012)zbMATHGoogle Scholar
  3. 3.
    Benamou, J., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Böhmer, K.: On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46(3), 1212–1249 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brenner, S., Gudi, T., Neilan, M., Sung, L.: \({C}^0\) penalty methods for the fully nonlinear Monge–Ampère equation. Math. Comput. 80(276), 1979–1995 (2011)zbMATHGoogle Scholar
  6. 6.
    Brenner, S.C., Neilan, M.: Finite element approximations of the three dimensional Monge–Ampère equation. ESAIM Math. Modell. Numer. Anal. 46(5), 979–1001 (2012)zbMATHCrossRefGoogle Scholar
  7. 7.
    Caboussat, A., Glowinski, R., Gourzoulidis, D.: A least-squares/relaxation method for the numerical solution of the three-dimensional elliptic Monge–Ampère equation. J. Sci. Comput. 77(1), 53–78 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Caboussat, A., Glowinski, R., Sorensen, D.C.: A least-squares method for the numerical solution of the Dirichlet problem for the elliptic Monge–Ampère equation in dimension two. ESAIM Control Optim. Calc. Var. 19(3), 780–810 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Caffarelli, L.A.: Interior \({W}^{2, p}\) estimates for solutions of the Monge–Ampère equation. Ann. Math. 131(1), 135–150 (1990)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations, vol. 43. American Mathematical Society, Providence (1995)zbMATHGoogle Scholar
  11. 11.
    Caffarelli, L.A., Milman, M.: Monge–Ampère equation: applications to geometry and optimization. In: NSF-CBMS Conference on the Monge–Ampère Equation, Applications to Geometry and Optimization, July 9–13, 1997, Florida Atlantic University, vol. 226. American Mathematical Soc. (1999)Google Scholar
  12. 12.
    Ciarlet, P.G.: Linear and Nonlinear Functional Analysis with Applications, vol. 130. SIAM, Philadelphia (2013)zbMATHGoogle Scholar
  13. 13.
    De Philippis, G., Figalli, A.: Sobolev regularity for Monge–Ampère type equations. SIAM J. Math. Anal. 45(3), 1812–1824 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Feng, X., Glowinski, R., Neilan, M.: Recent developments in numerical methods for fully nonlinear second order partial differential equations. SIAM Rev. 55(2), 205–267 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Feng, X., Neilan, M.: Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. J. Sci. Comput. 38(1), 74–98 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Froese, B.D.: A numerical method for the elliptic Monge–Ampère equation with transport boundary conditions. SIAM J. Sci. Comput. 34(3), A1432–A1459 (2012)zbMATHCrossRefGoogle Scholar
  17. 17.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Froese, B.D., Oberman, A.M.: Fast finite difference solvers for singular solutions of the elliptic Monge–Ampère equation. J. Comput. Phys. 230(3), 818–834 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2015)zbMATHGoogle Scholar
  20. 20.
    Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984). (2nd printing: 2008)zbMATHCrossRefGoogle Scholar
  21. 21.
    Glowinski, R.: Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems. SIAM, Philadelphia (2015)zbMATHCrossRefGoogle Scholar
  22. 22.
    Glowinski, R., Liu, H., Leung, S., Qian, J.: A finite element/operator-splitting method for the numerical solution of the two dimensional elliptic Monge–Ampère equation. J. Sci. Comput. 79(1), 1–47 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Glowinski, R., Osher, S.J., Yin, W.: Splitting Methods in Communication, Imaging, Science, and Engineering. Springer, Berlin (2017)zbMATHGoogle Scholar
  24. 24.
    Kazdan, J.L.: Prescribing the curvature of a Riemannian manifold. In: Conference Board of the Mathematical Sciences (1985)Google Scholar
  25. 25.
    Ming, W., Xu, J.: The Morley element for fourth order elliptic equations in any dimensions. Numer. Math. 103(1), 155–169 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Mirebeau, J.M.: Discretization of the 3D Monge–Ampère operator, between wide stencils and power diagrams. ESAIM Math. Modell. Numer. Anal. 49(5), 1511–1523 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Mohammadi, B.: Optimal transport, shape optimization and global minimization. C. R. Math. 344(9), 591–596 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Neilan, M.: A nonconforming Morley finite element method for the fully nonlinear Monge–Ampère equation. Numer. Math. 115(3), 371–394 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Oberman, A.M.: Wide stencil finite difference schemes for the elliptic Monge–Ampère equation and functions of the eigenvalues of the Hessian. Discrete Contin. Dyn. Syst. Ser. B 10(1), 221–238 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Sorensen, D.C., Glowinski, R.: A quadratically constrained minimization problem arising from PDE of Monge–Ampère type. Numer. Algorithms 53(1), 53–66 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of MathematicsUniversity of HoustonHoustonUSA
  3. 3.Department of MathematicsThe Hong Kong Baptist UniversityKowloon TongPeople’s Republic of China
  4. 4.Department of MathematicsThe Hong Kong University of Science and TechnologyKowloonPeople’s Republic of China
  5. 5.Department of MathematicsMichigan State UniversityEast LansingUSA

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