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A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation

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In this paper, we study finite element approximations of the viscosity solution of the fully nonlinear Monge-Ampère equation, det(D 2 u) = f (> 0) using the well-known nonconforming Morley element. Our approach is based on the vanishing moment method, which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and Feng (J Sci Comput 38(1):74–98, 2009). The vanishing moment method approximates the Monge-Ampère equation by the fourth order quasilinear equation \({-\epsilon\Delta^2u^\epsilon + {\rm det}(D^2u^\epsilon) = f}\) with appropriate boundary conditions. We develop a finite element scheme using the n-dimensional Morley element introduced in Wang and Xu (Numer Math 103:155–169, 2006) to approximate the regularized fourth order problem in two and three dimensions, and then derive optimal order error estimates.

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References

  1. Aleksandrov A.D.: Certain estimates for the Dirichlet problem. Soviet Math. Dokl. 1, 1151–1154 (1961)

    MathSciNet  Google Scholar 

  2. Benamou J., Brenier Y.: Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem. SIAM J. Appl. Math. 58(5), 1450–1461 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  5. Caffarelli, L.A., Cabré, X.: Fully nonlinear elliptic equations. American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Providence (1995)

  6. Caffarelli L.A., Milman M.: Properties of the solutions of the linearized Monge-Ampère equation. Am. J. Math. 119(2), 423–465 (1997)

    Article  MATH  Google Scholar 

  7. Caffarelli, L.A., Milman, M.: Monge Ampère equation: applications to geometry and optimization. Contemporary Mathematics. American Mathematical Society, Providence (1999)

  8. Cheng S.Y., Yau S.T.: On the regularity of the Monge-Ampère equation det(∂2 u/∂x i x j ) = F(x, u). Commun. Pure Appl. Math. 30(1), 41–68 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  10. Crandall M.G., Lions P.-L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 277(1), 1–42 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  11. Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dean E.J., Glowinski R.: Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type. Comput. Methods Appl. Mech. Eng. 195(13–16), 1344–1386 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)

  14. Feng, X.: Convergence of the vanishing moment method for the Monge-Ampère equations in two spatial dimensions (2009, submitted)

  15. Feng X., Neilan M.: Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations. J. Sci. Comput. 38(1), 74–98 (2009)

    Article  MathSciNet  Google Scholar 

  16. 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)

    Article  MathSciNet  Google Scholar 

  17. Feng, X., Neilan, M.: Analysis of Galerkin methods for the fully nonlinear Monge-Ampère equation (2009, submitted)

  18. Feng X., Neilan M.: A modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equation. SIAM J. Numer. Anal. 47(4), 2952–2981 (2009)

    Article  MathSciNet  Google Scholar 

  19. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition

  20. Gutierrez C.E.: The Monge-Ampère equation. Progress in Nonlinear Differential Equations and Their Applications, vol. 44. Birkhauser, Boston (2001)

    Google Scholar 

  21. Ishii H.: On uniqueness and existence of viscosity solutions of fully nonlinear second order PDE’s. Commun. Pure Appl. Math. 42, 14–45 (1989)

    Article  Google Scholar 

  22. Jensen R.: The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Rational Mech. Anal. 101, 1–27 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  23. Loeper, G.: A fully non-linear version of the incompressible Euler equations: the semi-geostrophic system. http://arxiv.org/abs/math/0504138v1

  24. Morley L.S.D.: The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19, 149–169 (1968)

    Google Scholar 

  25. Neilan, M.: Numerical methods for second order fully nonlinear partial differential equations. Ph.D. thesis, The University of Tennessee, Knoxville (2009)

  26. Shi Z.C.: On the error estimate of Morley element. Numer. Math. Sin. 12(2), 113–118 (1990)

    MATH  Google Scholar 

  27. Wang M., Xu J.: The Morley element for fourth order elliptic equations in any dimension. Numer. Math. 103, 155–169 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Neilan, M. A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation. Numer. Math. 115, 371–394 (2010). https://doi.org/10.1007/s00211-009-0283-x

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