Advertisement

Journal of Scientific Computing

, Volume 79, Issue 1, pp 1–47 | Cite as

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

  • Roland Glowinski
  • Hao LiuEmail author
  • Shingyu Leung
  • Jianliang Qian
Article
  • 332 Downloads

Abstract

We discuss in this article a novel method for the numerical solution of the two-dimensional elliptic Monge–Ampère equation. Our methodology relies on the combination of a time-discretization by operator-splitting with a mixed finite element based space approximation where one employs the same finite-dimensional spaces to approximate the unknown function and its three second order derivatives. A key ingredient of our approach is the reformulation of the Monge–Ampère equation as a nonlinear elliptic equation in divergence form, involving the cofactor matrix of the Hessian of the unknown function. With the above elliptic equation we associate an initial value problem that we discretize by operator-splitting. To enforce the pointwise positivity of the approximate Hessian we employ a hard thresholding based projection method. As shown by our numerical experiments, the resulting methodology is robust and can handle a large variety of triangulations ranging from uniform on rectangles to unstructured on domains with curved boundaries. For those cases where the solution is smooth and isotropic enough, we suggest also a two-stage method to improve the computational efficiency, the second stage being reminiscent of a Newton-like method. The methodology discussed in this article is able to handle domains with curved boundaries and unstructured meshes, using piecewise affine continuous approximations, while preserving optimal, or nearly optimal, convergence orders for the approximation error.

Keywords

Fully nonlinear elliptic partial differential equations Monge–Ampère equations Operator-splitting method Finite element approximations Mixed finite element methods Tychonoff regularization Variational crimes 

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers of this article for most helpful comments and suggestions. The work of S. Leung is partially supported by the Hong Kong RGC Grants 16303114 and 16309316. The work of J. Qian is partially supported by NSF Grants 1522249 and 1614566.

References

  1. 1.
    Awanou, G.: Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: classical solutions. IMA J. Numer. Anal. 35(3), 1150–1166 (2014)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bakelman, I.J.: Convex Analysis and Nonlinear Geometric Elliptic Equations. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benamou, J.D., Froese, B.D., Oberman, A.M.: Two numerical methods for the elliptic Monge–Ampère equation. ESAIM Math. Model. Numer. Anal. 44(4), 737–758 (2010)CrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  6. 6.
    Brenner, S.C., Neilan, M.: Finite element approximations of the three dimensional Monge–Ampère equation. ESAIM Math. Model. Numer. Anal. 46(5), 979–1001 (2012)MathSciNetCrossRefzbMATHGoogle 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, 1–26 (2018)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Caffarelli, L.A., Milman, M.: Monge–Ampère Equation: Applications to Geometry and Optimization: NSF-CBMS Conference on the Monge–Ampère Equation, Applications to Geometry and Optimization, July 9–13, 1997, Florida Atlantic University, Volume 226. American Mathematical Society, Providence (1999)Google Scholar
  10. 10.
    Caffarelli, L.A., Cabre, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence (1995)CrossRefzbMATHGoogle Scholar
  11. 11.
    Dean, E.J., Glowinski, R.: Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach. C. R. Math. Acad. Sci. Paris 336(9), 779–784 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dean, E.J., Glowinski, R.: Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: a least-squares approach. C. R. Math. Acad. Sci. Paris. 339(12), 887–892 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dean, E.J., Glowinski, R.: An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge–Ampère equation in two dimensions. Electron. Trans. Numer. Anal. 22, 71–96 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Dean, E.J., Glowinski, R.: Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type. Comput. Methods Appl. Mech. Eng. 195(13), 1344–1386 (2006)CrossRefzbMATHGoogle Scholar
  15. 15.
    Dean, E.J., Glowinski, R.: On the numerical solution of the elliptic Monge–Ampère equation in dimension two: a least-squares approach. In: Glowinski, R., Neittaanmaki, P. (eds.) Partial Differential Equations, pp. 43–63. Springer, Dordrecht (2008)CrossRefGoogle Scholar
  16. 16.
    Dean, E.J., Glowinski, R., Pan, T.W.: Operator-splitting methods and applications to the direct numerical simulation of particulate flow and to the solution of the elliptic Monge–Ampère equation. In: Cagnol, J., Zoésio, J.P. (eds.) Control Boundary Analysis, pp. 1–27. CRC, Boca Raton (2005)Google Scholar
  17. 17.
    D’Onofrio, L., Giannetti, F., Greco, L.: On weak Hessian determinants. In: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, vol. 16(3), pp. 159–169 (2005)Google Scholar
  18. 18.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    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(2), 1226–1250 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Froese, B.D.: Meshfree finite difference approximations for functions of the eigenvalues of the Hessian. Numer. Math. 138(1), 75–99 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  26. 26.
    Glowinski, R.: Numerical Nethods for Nonlinear Variational Problems. Springer, New York (1984). (2nd printing: 2008)CrossRefGoogle Scholar
  27. 27.
    Glowinski, R.: Finite element methods for incompressible viscous flow. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. IX, pp. 3–1176. North-Holland, Amsterdam (2003)Google Scholar
  28. 28.
    Glowinski, R.: Numerical methods for fully nonlinear elliptic equations. In: 6th International congress on industrial and applied mathermatics, ICIAM, vol. 7, pp. 155–192 (2009)Google Scholar
  29. 29.
    Glowinski, R.: Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems. SIAM, Philadelphia (2015)CrossRefzbMATHGoogle Scholar
  30. 30.
    Glowinski, R., Osher, S., Yin, W. (eds.): Splitting Methods in Communication, Imaging, Science, and Engineering. Springer, Berlin (2016)zbMATHGoogle Scholar
  31. 31.
    Gutiérrez, C.E.: The Monge–Ampère Equation. Birkhaüser, Basel (2001)CrossRefzbMATHGoogle Scholar
  32. 32.
    Hamfeldt, B.F., Salvador, T.: Higher-order adaptive finite difference methods for fully nonlinear elliptic equations. J. Sci. Comput. 75(3), 1282–1306 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lindsey, M., Rubinstein, Y.A.: Optimal transport via a Monge–Ampère optimization problem. SIAM J. Math. Anal. 49(4), 3073–3124 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Loeper, G., Rapetti, F.: Numerical solution of the Monge–Ampère equation by a Newton algorithm. C. R. Acad. Sci. Paris Ser. I 340, 319–324 (2005)CrossRefzbMATHGoogle Scholar
  35. 35.
    Mohammadi, B.: Optimal transport, shape optimization and global minimization. C. R. Math. 344(9), 591–596 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Nesterov, Y.: A method of solving a convex programming problem with convergence rate \({O}(1/k^2)\). Sov. Math. Dokl. 27(2), 372–376 (1983)zbMATHGoogle Scholar
  37. 37.
    Nochetto, R., Ntogkas, D., Zhang, W.: Two-scale method for the Monge–Ampère equation: Convergence to the viscosity solution. In: Mathematics of Computation (2018).  https://doi.org/10.1093/imanum/dry026
  38. 38.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Oliker, V.I., Prussner, L.D.: On the numerical solution of the equation \(\frac{\partial ^2 z}{\partial x^2}\frac{\partial ^2 z}{\partial y^2}-\left(\frac{\partial ^2 z}{\partial x \partial y}\right)^2=f\) and its discretizations, i. Numer. Math. 54(3), 271–293 (1989)CrossRefGoogle Scholar
  40. 40.
    Persson, P.O., Strang, G.: A simple mesh generator in MATLAB. SIAM Rev. 46(2), 329–345 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964)CrossRefGoogle Scholar
  42. 42.
    Savin, O.: The obstacle problem for Monge-Ampère equation. Calc. Var. Partial Differ. Equ. 22(3), 303–320 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Schaeffer, H., Hou, T.Y.: An accelerated method for nonlinear elliptic PDE. J. Sci. Comput. 69(2), 556–580 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Strang, G., Fix, G.J.: An Analysis of The Finite Element Method, vol. 212. Prentice-Hall, Englewood Cliffs (1973)zbMATHGoogle Scholar
  45. 45.
    Su, W., Boyd, S., Candes, E.: A differential equation for modeling Nesterov’s accelerated gradient method: theory and insights. J. Mach. Learn. Res. 17, 1–43 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018
Corrected publication October/2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Department of MathematicsThe Hong Kong Baptist UniversityKowloon TongHong Kong
  3. 3.Department of MathematicsHong Kong University of Science and TechnologyClear Water BayHong Kong
  4. 4.Department of MathematicsMichigan State UniversityEast LansingUSA

Personalised recommendations