Finite element approximations of the three dimensional Monge-Ampère equation

Susanne Cecelia Brenner; Michael Neilan

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 5, page 979-1001
  • ISSN: 0764-583X

Abstract

top
In this paper, we construct and analyze finite element methods for the three dimensional Monge-Ampère equation. We derive methods using the Lagrange finite element space such that the resulting discrete linearizations are symmetric and stable. With this in hand, we then prove the well-posedness of the method, as well as derive quasi-optimal error estimates. We also present some numerical experiments that back up the theoretical findings.

How to cite

top

Brenner, Susanne Cecelia, and Neilan, Michael. "Finite element approximations of the three dimensional Monge-Ampère equation." ESAIM: Mathematical Modelling and Numerical Analysis 46.5 (2012): 979-1001. <http://eudml.org/doc/276379>.

@article{Brenner2012,
abstract = {In this paper, we construct and analyze finite element methods for the three dimensional Monge-Ampère equation. We derive methods using the Lagrange finite element space such that the resulting discrete linearizations are symmetric and stable. With this in hand, we then prove the well-posedness of the method, as well as derive quasi-optimal error estimates. We also present some numerical experiments that back up the theoretical findings.},
author = {Brenner, Susanne Cecelia, Neilan, Michael},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Monge-Ampère equation; three dimensions; finite element method; convergence analysis; finite element; Lagrange finite element space; well-posedness; numerical results},
language = {eng},
month = {2},
number = {5},
pages = {979-1001},
publisher = {EDP Sciences},
title = {Finite element approximations of the three dimensional Monge-Ampère equation},
url = {http://eudml.org/doc/276379},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Brenner, Susanne Cecelia
AU - Neilan, Michael
TI - Finite element approximations of the three dimensional Monge-Ampère equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 5
SP - 979
EP - 1001
AB - In this paper, we construct and analyze finite element methods for the three dimensional Monge-Ampère equation. We derive methods using the Lagrange finite element space such that the resulting discrete linearizations are symmetric and stable. With this in hand, we then prove the well-posedness of the method, as well as derive quasi-optimal error estimates. We also present some numerical experiments that back up the theoretical findings.
LA - eng
KW - Monge-Ampère equation; three dimensions; finite element method; convergence analysis; finite element; Lagrange finite element space; well-posedness; numerical results
UR - http://eudml.org/doc/276379
ER -

References

top
  1. G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equtions. Asymptotic Anal.4 (1991) 271–283.  
  2. C. Bernardi, Optimal finite element interpolation on curved domains. SIAM J. Numer. Anal.26 (1989) 1212–1240.  
  3. K. Böhmer, On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal.46 (2008) 1212–1249.  
  4. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3th edition. Springer (2008).  
  5. S.C. Brenner, T. Gudi, M. Neilan and L.-Y. Sung, 𝒞0 penalty methods for the fully nonlinear Monge-Ampère equation. Math. Comput.80 (2011) 1979–1995.  
  6. L.A. Caffarelli and C.E. Gutiérrez, Properties of the solutions of the linearized Monge-Ampère equation. Amer. J. Math.119 (1997) 423–465.  
  7. L.A. Caffarelli and M. Milman, Monge-Ampère Equation : Applications to Geometry and Optimization. Amer. Math. Soc. Providence, RI (1999).  
  8. L.A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-Ampère equation. Comm. Pure Appl. Math.37 (1984) 369–402.  
  9. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).  
  10. M.G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc.27 (1992) 1–67.  
  11. E.J. Dean and R. Glowinski, Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type. Comput. Methods Appl. Mech. Engrg.195 (2006) 1344–1386.  
  12. G.L. Delzanno, L. Chacón, J.M. Finn, Y. Chung and G. Lapenta, An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization. J. Comput. Phys.227 (2008) 9841–9864.  
  13. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics. Providence, RI. Amer. Math. Soc.19 (1998).  
  14. X. Feng and M. Neilan, Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations. J. Sci. Comput.38 (2009) 74–98.  
  15. X. Feng and M. Neilan, Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal.47 (2009) 1226–1250.  
  16. B.D. Froese and A.M. Oberman, Convergent finite difference solvers for viscosity solutions of the ellptic Monge-Ampère equation in dimensions two and higher. SIAM J. Numer. Anal.49 (2011) 1692–1714.  
  17. B.D. Froese and A.M. Oberman, Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation. J. Comput. Phys.230 (2011) 818–834.  
  18. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001).  
  19. P. Grisvard, Elliptic Problems on Nonsmooth Domains. Pitman Publishing Inc. (1985).  
  20. C.E. Gutiérrez, The Monge-Ampère Equation, Progress in Nonlinear Differential Equations and Their Applications44. Birkhauser, Boston, MA (2001).  
  21. T. Muir, A Treatise on the Theory of Determinants. Dover Publications Inc., New York (1960).  
  22. M. Neilan, A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation. Numer. Math.115 (2010) 371–394.  
  23. M. Neilan, A unified analysis of some finite element methods for the Monge-Ampère equation. Submitted.  
  24. J.A. Nitsche, Über ein Variationspirinzip zur Lösung Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unteworfen sind. Abh. Math. Sem. Univ. Hamburg36 (1971) 9–15.  
  25. A.M. Oberman, Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete Contin. Dyn. Syst. Ser. B10 (2008) 221–238.  
  26. D.C. Sorensen and R. Glowinski, A quadratically constrained minimization problem arising from PDE of Monge-Ampère type. Numer. Algorithm53 (2010) 53–66.  
  27. N.S. Trudinger and X.-J. Wang, The Monge-Ampère equation and its geometric applications, Handbook of Geometric Analysis I. International Press (2008) 467–524.  
  28. C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics. Providence, RI. Amer. Math. Soc.58 (2003).  
  29. A. Ženíšek, Polynomial approximation on tetrahedrons in the finite element method. J. Approx. Theory7 (1973) 334–351.  
  30. V. Zheligovsky, O. Podvigina and U. Frisch, The Monge-Ampère equation : various forms and numerical solutions. J. Comput. Phys.229 (2010) 5043–5061.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.