Skip to main content

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

Journal of Scientific Computing volume 81, pages 2271–2302 (2019)Cite this article

Abstract

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

References

  1. Awanou, G.: Pseudo transient continuation and time marching methods for Monge–Ampère type equations. Adv. Comput. Math. 41(4), 907–935 (2015)

    MathSciNet  Article  Google Scholar 

  2. Bakelman, I.J.: Convex Analysis and Nonlinear Geometric Elliptic Equations. Springer, Berlin (2012)

    MATH  Google Scholar 

  3. Benamou, J., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)

    MathSciNet  Article  Google Scholar 

  4. Böhmer, K.: On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46(3), 1212–1249 (2008)

    MathSciNet  Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  9. Caffarelli, L.A.: Interior \({W}^{2, p}\) estimates for solutions of the Monge–Ampère equation. Ann. Math. 131(1), 135–150 (1990)

    MathSciNet  Article  Google Scholar 

  10. Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations, vol. 43. American Mathematical Society, Providence (1995)

    MATH  Google Scholar 

  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)

  12. Ciarlet, P.G.: Linear and Nonlinear Functional Analysis with Applications, vol. 130. SIAM, Philadelphia (2013)

    MATH  Google Scholar 

  13. De Philippis, G., Figalli, A.: Sobolev regularity for Monge–Ampère type equations. SIAM J. Math. Anal. 45(3), 1812–1824 (2013)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  19. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2015)

    MATH  Google Scholar 

  20. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984). (2nd printing: 2008)

    Book  Google Scholar 

  21. Glowinski, R.: Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems. SIAM, Philadelphia (2015)

    Book  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  23. Glowinski, R., Osher, S.J., Yin, W.: Splitting Methods in Communication, Imaging, Science, and Engineering. Springer, Berlin (2017)

    MATH  Google Scholar 

  24. Kazdan, J.L.: Prescribing the curvature of a Riemannian manifold. In: Conference Board of the Mathematical Sciences (1985)

  25. Ming, W., Xu, J.: The Morley element for fourth order elliptic equations in any dimensions. Numer. Math. 103(1), 155–169 (2006)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  27. Mohammadi, B.: Optimal transport, shape optimization and global minimization. C. R. Math. 344(9), 591–596 (2007)

    MathSciNet  Article  Google Scholar 

  28. Neilan, M.: A nonconforming Morley finite element method for the fully nonlinear Monge–Ampère equation. Numer. Math. 115(3), 371–394 (2010)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  31. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA, 30332, USA

    Hao Liu

  2. Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX, 77204, USA

    Roland Glowinski

  3. Department of Mathematics, The Hong Kong Baptist University, Kowloon Tong, Hong Kong, People’s Republic of China

    Roland Glowinski

  4. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People’s Republic of China

    Shingyu Leung

  5. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Jianliang Qian

Authors
  1. Hao Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roland Glowinski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shingyu Leung
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jianliang Qian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hao Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Liu, H., Glowinski, R., Leung, S. et al. A Finite Element/Operator-Splitting Method for the Numerical Solution of the Three Dimensional Monge–Ampère Equation. J Sci Comput 81, 2271–2302 (2019). https://doi.org/10.1007/s10915-019-01080-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-019-01080-4