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BIT Numerical Mathematics

, Volume 55, Issue 3, pp 625–646 | Cite as

Spline element method for Monge–Ampère equations

  • Gerard AwanouEmail author
Article

Abstract

We analyze the convergence of an iterative method for solving the nonlinear system resulting from a natural discretization of the Monge–Ampère equation with smooth approximations. We make the assumption, supported by numerical experiments for the two dimensional problem, that the discrete problem has a convex solution. The method we analyze is the discrete version of Newton’s method in the vanishing moment methodology. Numerical experiments are given in the framework of the spline element method.

Keywords

Iterative methods Monge–Ampère C1 conforming approximations 

Mathematics Subject Classification

65N30 35J25 

Notes

Acknowledgments

The author would like to thank the referees for a careful reading of the paper and their suggestions which lead to a better paper. This work began when the author was supported in part by a 2009–2013 Sloan Foundation Fellowship and continued while the author was in residence at the Mathematical Sciences Research Institute (MSRI) in Berkeley, California, Fall 2013. The MSRI receives major funding from the National Science Foundation under Grant No. 0932078 000. The author was partially supported by NSF DMS Grant No. 1319640.

References

  1. 1.
    Awanou, G.: Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: classical solutions. IMA J Numer Anal (2014). doi: 10.1093/imanum/dru028
  2. 2.
    Awanou, G.: Energy methods in 3D spline approximations of the Navier-Stokes equations, Ph.D. Dissertation, University of Georgia (2003)Google Scholar
  3. 3.
    Awanou, G.: Robustness of a spline element method with constraints. J. Sci. Comput. 36, 421–432 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Awanou, G.: On standard finite difference discretizations of the elliptic Monge–Ampère equation. (2014). http://arxiv.org/abs/1311.2812
  5. 5.
    Awanou, G.: Pseudo transient continuation and time marching methods for Monge–Ampère type equations (2014). http://arxiv.org/pdf/1301.5891v4
  6. 6.
    Awanou, G.: Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: Aleksandrov solutions (2014). http://arxiv.org/abs/1310.4568
  7. 7.
    Awanou, G.: Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: mixed methods (2014). http://arxiv.org/abs/1406.5666
  8. 8.
    Awanou, G., Lai, M.-J.: Trivariate spline approximations of 3D Navier-Stokes equations. Math. Comp. 74, 585–601 (2005). (electronic)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Awanou, G., Lai, M.-J., Wenston, P.: The multivariate spline method for scattered data fitting and numerical solution of partial differential equations, in Wavelets and splines: Athens. Mod. Methods Math. 2006, 24–74 (2005)MathSciNetGoogle Scholar
  10. 10.
    Awanou, G., Li, H.: Error analysis of a mixed finite element method for the Monge–Ampère equation. Int. J. Num. Anal. Model. 11, 745–761 (2014)MathSciNetGoogle Scholar
  11. 11.
    Awanou, G.M., Lai, M.J.: On convergence rate of the augmented Lagrangian algorithm for nonsymmetric saddle point problems. Appl. Numer. Math. 54, 122–134 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20, pp. 179–192 (1972/73)Google Scholar
  13. 13.
    Baramidze, V., Lai, M.-J.: Spherical spline solution to a PDE on the sphere, in Wavelets and splines: Athens. Mod. Methods Math. 2006, 75–92 (2005)Google Scholar
  14. 14.
    Benamou, J.-D., Froese, B.D., Oberman, A.M.: Two numerical methods for the elliptic Monge–Ampère equation, M2AN Math. Model. Numer. Anal. 44, 737–758 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Böhmer, K.: On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46, 1212–1249 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Böhmer, K.: Numerical methods for nonlinear elliptic differential equations: a synopsis. Oxford University Press, Oxford (2010)CrossRefGoogle Scholar
  17. 17.
    Bouchiba, M., Belgacem, F.B.: Numerical solution of Monge–Ampere equation. Math. Balkanica (N.S.) 20, 369–378 (2006)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Brenner, S.C., Gudi, T., Neilan, M., Sung, L.-Y.: \(C^0\) penalty methods for the fully nonlinear Monge–Ampère equation. Math. Comp. 80, 1979–1995 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Brenner, S.C., Neilan, M.: Finite element approximations of the three dimensional Monge–Ampère equation. ESAIM Math. Model. Numer. Anal. 46, 979–1001 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Texts in Applied Mathematics, vol. 15, 2nd edn. Springer, New York (2002)CrossRefGoogle Scholar
  21. 21.
    Dahmen, W.: Convexity and Bernstein-Bézier polynomials, in Curves and surfaces (Chamonix-Mont-Blanc, 1990), pp. 107–134. Academic Press, Boston (1991)Google Scholar
  22. 22.
    Davydov, O., Saeed, A.: Numerical solution of fully nonlinear elliptic equations by Böhmer’s method. J. Comput. Appl. Math. 254, 43–54 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    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, 779–784 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    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, 887–892 (2004)Google Scholar
  25. 25.
    Dean, E.J., Glowinski, R.: Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type. Comput. Methods Appl. Mech. Eng. 195, 1344–1386 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Feng, X., Neilan, M.: Convergence of a fourth-order singular perturbation of the \(n\)-dimensional radially symmetric Monge–Ampère equation. Appl. Anal. 93(8), 1626–1646 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Feng, X., Neilan, M.: Error analysis for mixed finite element approximations of the fully nonlinear Monge–Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal. 47, 1226–1250 (2009)Google Scholar
  28. 28.
    Feng, X., Neilan, M.: A modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations. SIAM J. Numer. Anal. 47, 2952–2981 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Feng, X., Neilan, M.: Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations. J. Sci. Comput. 38, 74–98 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Feng, X., Neilan, M.: Analysis of Galerkin methods for the fully nonlinear Monge–Ampère equation. J. Sci. Comput. 47, 303–327 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Froese, B., Oberman, A.: Convergent finite difference solvers for viscosity solutions of the elliptic Monge–Ampère equation in dimensions two and higher. SIAM J. Numer. Anal. 49, 1692–1714 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Froese, B.D., Oberman, A.M.: Fast finite difference solvers for singular solutions of the elliptic Monge–Ampère equation. J. Comput. Phys. 230, 818–834 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Froese, B.D., Oberman, A.M.: Convergent filtered schemes for the Monge–Ampère partial differential equation. SIAM J. Numer. Anal. 51, 423–444 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Glowinski, R.: Numerical methods for fully nonlinear elliptic equations, in ICIAM 07–6th International Congress on Industrial and Applied Mathematics, pp. 155–192. Eur. Math. Soc., Zürich (2009)Google Scholar
  35. 35.
    Gutiérrez, C.E.: The Monge–Ampère equation, progress in nonlinear differential equations and their applications, vol. 44. Birkhäuser Boston Inc., Boston (2001)Google Scholar
  36. 36.
    Harris, G., Martin, C.: The roots of a polynomial vary continuously as a function of the coefficients. Proc. Am. Math. Soc. 100, 390–392 (1987)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Hu, X.-L., Han, D.-F., Lai, M.-J.: Bivariate splines of various degrees for numerical solution of partial differential equations. SIAM J. Sci. Comput. 29, 1338–1354 (2007). (electronic)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Lai, M.-J., Schumaker, L.L.: Spline functions on triangulations, vol. 110 of Encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  39. 39.
    Lakkis, O., Pryer, T.: A finite element method for nonlinear elliptic problems. SIAM J. Sci. Comput. 35, A2025–A2045 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Mohammadi, B.: Optimal transport, shape optimization and global minimization. C. R. Math. Acad. Sci. Paris 344, 591–596 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Neilan, M.: A nonconforming Morley finite element method for the fully nonlinear Monge–Ampère equation. Numer. Math. 115, 371–394 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Neilan, M.: Finite element methods for fully nonlinear second order PDEs based on a discrete Hessian with applications to the Monge–Ampère equation. J. Comput. Appl. Math. 263, 351–369 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Neilan, M.: Quadratic finite element approximations of the Monge–Ampère equation. J. Sci. Comput. 54, 200–226 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Nilssen, T.K., Tai, X.-C., Winther, R.: A robust nonconforming \(H^2\)-element. Math. Comp. 70, 489–505 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    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, 221–238 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Oliker, V.I., Prussner, L.D.: On the numerical solution of the equation \((\partial ^2z/\partial x^2)(\partial ^2z/\partial y^2)-((\partial ^2z/\partial x\partial y))^2=f\) and its discretizations. I. Numer. Math. 54, 271–293 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Ong, M.E.G.: Uniform refinement of a tetrahedron. SIAM J. Sci. Comput. 15, 1134–1144 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Ostrowski, A.M.: Solution of equations and systems of equations, Pure and Applied Mathematics, vol. IX. Academic Press, New York (1960)Google Scholar
  49. 49.
    Rauch, J., Taylor, B.A.: The Dirichlet problem for the multidimensional Monge–Ampère equation. Rocky Mountain J. Math. 7, 345–364 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Vuong, A.-V., Heinrich, C., Simeon, B.: ISOGAT: a 2D tutorial MATLAB code for isogeometric analysis. Comput. Aided Geom. Design 27, 644–655 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Zheligovsky, V., Podvigina, O., Frisch, U.: The Monge–Ampère equation: various forms and numerical solution. J. Comput. Phys. 229, 5043–5061 (2010)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer Science (M/C 249)University of Illinois at ChicagoChicagoUSA

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