A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two

Alexandre Caboussat; Roland Glowinski; Danny C. Sorensen

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 3, page 780-810
  • ISSN: 1292-8119

Abstract

top
We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge − Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson − Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the computer implementation of our least-squares/relaxation methodology. Domains with curved boundaries are easily accommodated. Numerical experiments show the convergence of the computed solutions to their continuous counterparts when such solutions exist. On the other hand, when classical solutions do not exist, our methodology produces solutions in a least-squares sense.

How to cite

top

Caboussat, Alexandre, Glowinski, Roland, and Sorensen, Danny C.. "A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two." ESAIM: Control, Optimisation and Calculus of Variations 19.3 (2013): 780-810. <http://eudml.org/doc/272752>.

@article{Caboussat2013,
abstract = {We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge − Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson − Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the computer implementation of our least-squares/relaxation methodology. Domains with curved boundaries are easily accommodated. Numerical experiments show the convergence of the computed solutions to their continuous counterparts when such solutions exist. On the other hand, when classical solutions do not exist, our methodology produces solutions in a least-squares sense.},
author = {Caboussat, Alexandre, Glowinski, Roland, Sorensen, Danny C.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Monge − ampère equation; least-squares method; biharmonic problem; conjugate gradient method; quadratic constraint minimization; mixed finite element methods; Dirichlet problem; elliptic Monge-Ampère equation; numerical experiments; convergence},
language = {eng},
number = {3},
pages = {780-810},
publisher = {EDP-Sciences},
title = {A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two},
url = {http://eudml.org/doc/272752},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Caboussat, Alexandre
AU - Glowinski, Roland
AU - Sorensen, Danny C.
TI - A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 3
SP - 780
EP - 810
AB - We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge − Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson − Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the computer implementation of our least-squares/relaxation methodology. Domains with curved boundaries are easily accommodated. Numerical experiments show the convergence of the computed solutions to their continuous counterparts when such solutions exist. On the other hand, when classical solutions do not exist, our methodology produces solutions in a least-squares sense.
LA - eng
KW - Monge − ampère equation; least-squares method; biharmonic problem; conjugate gradient method; quadratic constraint minimization; mixed finite element methods; Dirichlet problem; elliptic Monge-Ampère equation; numerical experiments; convergence
UR - http://eudml.org/doc/272752
ER -

References

top
  1. [1] A.D. Aleksandrov, Uniqueness conditions and estimates for the solution of the Dirichlet problem. Amer. Math. Soc. Trans.68 (1968) 89–119. Zbl0177.36802
  2. [2] J.D. Benamou, B.D. Froese and A.M. Oberman, Two numerical methods for the elliptic Monge − Ampère equation. ESAIM: M2AN 44 (2010) 737–758. Zbl1192.65138MR2683581
  3. [3] M. Bernadou, P.L. George, A. Hassim, P. Joly, P. Laug, A. Perronet, E. Saltel, D. Steer, G. Vanderborck and M. Vidrascu, Modulef, a modular library of finite elements. Technical report, INRIA (1988). Zbl0594.65080
  4. [4] P.B. Bochev and M.D. Gunzburger, Least-Squares Finite Element Methods. Springer-Verlag, New York (2009). Zbl1168.65067MR2490235
  5. [5] K. Boehmer, On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal.46 (2008) 1212–1249. Zbl1166.35322MR2390991
  6. [6] S.C. Brenner, T. Gudi, M. Neilan and L.-Y. Sung, c0 penalty methods for the fully nonlinear Monge − Ampere equation. Math. Comput. 80 (2011) 1979–1995. Zbl1228.65220MR2813346
  7. [7] S.C. Brenner and M. Neilan, Finite element approximations of the three dimensional Monge − Ampère equation. ESAIM: M2AN 46 (2012) 979–1001. Zbl1272.65088MR2916369
  8. [8] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, NewYork (1991). Zbl0788.73002MR1115205
  9. [9] A. Caboussat and R. Glowinski, Regularization methods for the numerical solution of the divergence equation ∇·u = f. J. Comput. Math. 30 (2012) 354–380. Zbl1274.65178MR2965988
  10. [10] X. Cabré, Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete Contin. Dyn. Systems8 (2002) 289–302. Zbl1003.35053MR1897687
  11. [11] L.A. Caffarelli, Nonlinear elliptic theory and the Monge − Ampère equation, in Proc. of the International Congress of Mathematicians. Higher Education Press, Beijing (2002) 179–187. Zbl1040.35018MR1989184
  12. [12] L.A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence, RI (1995). Zbl0834.35002MR1351007
  13. [13] L.A. Caffarelli and R. Glowinski, Numerical solution of the Dirichlet problem for a Pucci equation in dimension two. Application to homogenization. J. Numer. Math. 16 (2008) 185–216. Zbl1155.65097MR2484327
  14. [14] L.A. Caffarelli, S.A. Kochenkgin and V.I. Olicker, On the numerical solution of reflector design with given far field scattering data, in Monge − Ampère Equation: Application to Geometry and Optimization, American Mathematical Society, Providence, RI (1999) 13–32. Zbl0917.65104MR1660740
  15. [15] M.G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc.27 (1992) 1–67. Zbl0755.35015MR1118699
  16. [16] E.J. Dean and R. Glowinski, Numerical solution of the two-dimensional elliptic Monge − Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach. C. R. Acad. Sci. Paris, Ser. I 336 (2003) 779–784. Zbl1028.65120MR1989280
  17. [17] E.J. Dean and R. Glowinski, Numerical solution of the two-dimensional elliptic Monge − Ampère equation with Dirichlet boundary conditions: a least-squares approach. C. R. Acad. Sci. Paris, Ser. I 339 (2004) 887–892. Zbl1063.65121MR2111728
  18. [18] E.J. Dean and R. Glowinski, Numerical solution of a two-dimensional elliptic Pucci’s equation with Dirichlet boundary conditions: a least-squares approach. C. R. Acad. Sci. Paris, Ser. I 341 (2005) 374–380. Zbl1081.65543MR2169156
  19. [19] E.J. Dean and R. Glowinski, An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge − Ampère equation in two dimensions. Electronic Transactions in Numerical Analysis 22 (2006) 71–96. Zbl1112.65119MR2208483
  20. [20] E.J. Dean and R. Glowinski, Numerical methods for fully nonlinear elliptic equations of the Monge − Ampère type. Comput. Meth. Appl. Mech. Engrg. 195 (2006) 1344–1386. Zbl1119.65116MR2203972
  21. [21] E.J. Dean and R. Glowinski, On the numerical solution of the elliptic Monge − Ampère equation in dimension two: A least-squares approach, in Partial Differential Equations: Modeling and Numerical Simulation, vol. 16 of Comput. Methods Appl. Sci., edited by R. Glowinski and P. Neittaanmäki. Springer (2008) 43–63. Zbl1152.65479MR2484684
  22. [22] E.J. Dean, R. Glowinski and T.W. Pan, Operator-splitting methods and applications to the direct numerical simulation of particulate flow and to the solution of the elliptic Monge − Ampère equation. in Control and Boundary Analysis, edited by J.P. Zolésio J. Cagnol, CRC Boca Raton, FLA (2005) 1–27. Zbl1188.65115MR2144346
  23. [23] E.J. Dean, R. Glowinski and D. Trevas, An approximate factorization/least squares solution method for a mixed finite element approximation of the Cahn-Hilliard equation. Jpn J. Ind. Appl. Math.13 (1996) 495–517. Zbl0874.65073MR1415067
  24. [24] 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. Zbl1195.65170MR2485451
  25. [25] X. Feng and M. Neilan, Vanishing moment method and moment solutions of second order fully nonlinear partial differential equations. J. Sci. Comput.38 (2009) 74–98. Zbl1203.65252MR2472219
  26. [26] B.D. Froese and A.M. Oberman, Convergent finite difference solvers for viscosity solutions of the elliptic Monge − Ampère equation in dimensions two and higher. SIAM J. Numer. Anal. 49 (2011) 1692–1715. Zbl1255.65195MR2831067
  27. [27] 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. Zbl1206.65242MR2745457
  28. [28] C. Geuzaine and J.-F. Remacle, Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Eng.79 (2009) 1309–1331. Zbl1176.74181MR2566786
  29. [29] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001). Zbl1042.35002MR1814364
  30. [30] R. Glowinski, Finite Element Methods For Incompressible Viscous Flow, Handbook of Numerical Analysis, edited by P.G. Ciarlet, J.L. Lions. Elsevier, Amsterdam IX (2003) 3–1176. Zbl1040.76001MR2009826
  31. [31] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. 2nd edition, Springer-Verlag, New York, NY (2008). Zbl0536.65054MR2423313
  32. [32] R. Glowinski, Numerical methods for fully nonlinear elliptic equations. in Invited Lectures, 6th Int. Congress on Industrial and Applied Mathematics, Zürich, Switzerland, 16-20 July 2007. EMS (2009) 155–192. Zbl1179.65146MR2588593
  33. [33] R. Glowinski, E.J. Dean, G. Guidoboni, H.L. Juarez and T.W. Pan, Applications of operator-splitting methods to the direct numerical simulation of particulate and free surface flows and to the numerical solution of the two-dimensional Monge − Ampère equation. Jpn J. Ind. Appl. Math. 25 (2008) 1–63. Zbl1141.76043MR2410542
  34. [34] R. Glowinski, J.-L. Lions and J.W. He, Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach. Encyclopedia of Mathematics and its Applications. Cambridge University Press (2008). Zbl0838.93014
  35. [35] R. Glowinski, D. Marini and M. Vidrascu, Finite-element approximations and iterative solutions of a fourth-order elliptic variational inequality. IMA J. Numer. Anal.4 (1984) 127–167. Zbl0544.65043MR744475
  36. [36] R. Glowinski and O. Pironneau, Numerical methods for the first bi-harmonic equation and for the two-dimensional Stokes problem. SIAM Rev.17 (1979) 167–212. Zbl0427.65073MR524511
  37. [37] C.E. Gutiérrez, The Monge − Ampère Equation. Birkhaüser, Boston (2001). Zbl0989.35052
  38. [38] T.J.R. Hughes, L. Franca and M. Balestra, A new finite element formulation for computational fluid dynamics: V. circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolation. Comput. Methods Appl. Mech. Engrg. 59 (1986) 85–100. Zbl0622.76077MR868143
  39. [39] H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Eq.83 (1990) 26–78. Zbl0708.35031MR1031377
  40. [40] G. Loeper and F. Rapetti, Numerical solution of the Monge − Ampère equation by a Newton’s algorithm. C. R. Math. Acad. Sci. Paris 340 (2005) 319–324. Zbl1067.65119MR2121899
  41. [41] B. Mohammadi, Optimal transport, shape optimization and global minimization. C. R. Acad Sci Paris, Ser. I 351 (2007) 591–596. Zbl1115.65075MR2323748
  42. [42] M. Neilan, A nonconforming Morley finite element method for the fully nonlinear Monge − Ampère equation. Numer. Math. 115 (2010) 371–394. Zbl1201.65209MR2640051
  43. [43] A. Oberman, Wide stencil finite difference schemes for the elliptic Monge − Ampère equations and functions of the eigenvalues of the Hessian. Discr. Contin. Dyn. Syst. B 10 (2008) 221–238. Zbl1145.65085MR2399429
  44. [44] V.I. Oliker and L.D. Prussner, On the numerical solution of the equation z x x z y y - z x y 2 = f z xx z yy − z xy 2 = f and its discretization, I. Numer. Math. 54 (1988) 271–293. Zbl0659.65116MR971703
  45. [45] M. Picasso, F. Alauzet, H. Borouchaki and P.-L. George, A numerical study of some Hessian recovery techniques on isotropic and anisotropic meshes. SIAM J. Sci. Comput.33 (2011) 1058–1076. Zbl1232.65147MR2800564
  46. [46] A.V. Pogorelov, Monge − Ampère Equations of Elliptic Type. P. Noordhooff, Ltd, Groningen, Netherlands (1964). Zbl0133.04902
  47. [47] L. Reinhart, On the numerical analysis of the Von Kármán equation: mixed finite element approximation and continuation techniques. Numer. Math.39 (1982) 371–404. Zbl0503.73048MR678742
  48. [48] D.C. Sorensen and R. Glowinski, A quadratically constrained minimization problem arising from PDE of Monge − Ampère type. Numer. Algor. 53 (2010) 53–66. Zbl1187.65073MR2566127
  49. [49] A.N. Tychonoff, The regularization of incorrectly posed problems. Doklady Akad. Nauk. SSSR153 (1963) 42–52. Zbl0183.11601MR162378
  50. [50] V. Zheligovsky, O. Podvigina and U. Frisch, The Monge − Ampère equation: Various forms and numerical solution. J. Comput. Phys. 229 (2010) 5043–5061. Zbl1194.65141MR2643642

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.