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, Tome 19 (2013) no. 3, pp. 780-810.

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.

DOI : https://doi.org/10.1051/cocv/2012033
Classification : 65N30,  65K10,  65F30,  49M15,  49K20
Mots clés : Monge − ampère equation, least-squares method, biharmonic problem, conjugate gradient method, quadratic constraint minimization, mixed finite element methods
@article{COCV_2013__19_3_780_0,
author = {Caboussat, Alexandre and Glowinski, Roland and Sorensen, Danny C.},
title = {A least-squares method for the numerical solution of the {Dirichlet} problem for the elliptic monge-amp\ere equation in dimension two},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {780--810},
publisher = {EDP-Sciences},
volume = {19},
number = {3},
year = {2013},
doi = {10.1051/cocv/2012033},
zbl = {1272.65089},
mrnumber = {3092362},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2012033/}
}
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
DA  - 2013///
SP  - 780
EP  - 810
VL  - 19
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2012033/
UR  - https://zbmath.org/?q=an%3A1272.65089
UR  - https://www.ams.org/mathscinet-getitem?mr=3092362
UR  - https://doi.org/10.1051/cocv/2012033
DO  - 10.1051/cocv/2012033
LA  - en
ID  - COCV_2013__19_3_780_0
ER  - 
Caboussat, Alexandre; Glowinski, Roland; 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, Tome 19 (2013) no. 3, pp. 780-810. doi : 10.1051/cocv/2012033. http://www.numdam.org/articles/10.1051/cocv/2012033/`

[1] A.D. Aleksandrov, Uniqueness conditions and estimates for the solution of the Dirichlet problem. Amer. Math. Soc. Trans. 68 (1968) 89-119. | Zbl 0177.36802

[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. | Numdam | MR 2683581 | Zbl 1192.65138

[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). | Zbl 0594.65080

[4] P.B. Bochev and M.D. Gunzburger, Least-Squares Finite Element Methods. Springer-Verlag, New York (2009). | MR 2490235 | Zbl 1168.65067

[5] K. Boehmer, On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46 (2008) 1212-1249. | MR 2390991 | Zbl 1166.35322

[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. | MR 2813346 | Zbl 1228.65220

[7] S.C. Brenner and M. Neilan, Finite element approximations of the three dimensional Monge − Ampère equation. ESAIM: M2AN 46 (2012) 979-1001. | Numdam | MR 2916369 | Zbl 1272.65088

[8] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, NewYork (1991). | MR 1115205 | Zbl 0788.73002

[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. | MR 2965988 | Zbl 1274.65178

[10] X. Cabré, Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete Contin. Dyn. Systems 8 (2002) 289-302. | MR 1897687 | Zbl 1003.35053

[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. | MR 1989184 | Zbl 1040.35018

[12] L.A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations. American Mathematical Society, Providence, RI (1995). | MR 1351007 | Zbl 0834.35002

[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. | MR 2484327 | Zbl 1155.65097

[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. | MR 1660740 | Zbl 0917.65104

[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. | MR 1118699 | Zbl 0755.35015

[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. | MR 1989280 | Zbl 1028.65120

[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. | MR 2111728 | Zbl 1063.65121

[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. | MR 2169156 | Zbl 1081.65543

[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. | MR 2208483 | Zbl 1112.65119

[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. | MR 2203972 | Zbl 1119.65116

[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. | MR 2484684 | Zbl 1152.65479

[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. | MR 2144346 | Zbl 1188.65115

[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. | MR 1415067 | Zbl 0874.65073

[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. | MR 2485451 | Zbl 1195.65170

[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. | MR 2472219 | Zbl 1203.65252

[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. | MR 2831067 | Zbl 1255.65195

[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. | MR 2745457 | Zbl 1206.65242

[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. | MR 2566786 | Zbl 1176.74181

[29] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001). | MR 1814364 | Zbl 1042.35002

[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. | MR 2009826 | Zbl 1040.76001

[31] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. 2nd edition, Springer-Verlag, New York, NY (2008). | MR 2423313 | Zbl 0536.65054

[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. | MR 2588593 | Zbl 1179.65146

[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. | MR 2410542 | Zbl 1141.76043

[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). | Zbl 0838.93014

[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. | MR 744475 | Zbl 0544.65043

[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. | MR 524511 | Zbl 0427.65073

[37] C.E. Gutiérrez, The Monge − Ampère Equation. Birkhaüser, Boston (2001). | Zbl 0989.35052

[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. | MR 868143 | Zbl 0622.76077

[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. | MR 1031377 | Zbl 0708.35031

[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. | MR 2121899 | Zbl 1067.65119

[41] B. Mohammadi, Optimal transport, shape optimization and global minimization. C. R. Acad Sci Paris, Ser. I 351 (2007) 591-596. | MR 2323748 | Zbl 1115.65075

[42] M. Neilan, A nonconforming Morley finite element method for the fully nonlinear Monge − Ampère equation. Numer. Math. 115 (2010) 371-394. | MR 2640051 | Zbl 1201.65209

[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. | MR 2399429 | Zbl 1145.65085

[44] V.I. Oliker and L.D. Prussner, On the numerical solution of the equation | MR 971703 | Zbl 0659.65116

[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. | MR 2800564 | Zbl 1232.65147

[46] A.V. Pogorelov, Monge − Ampère Equations of Elliptic Type. P. Noordhooff, Ltd, Groningen, Netherlands (1964). | Zbl 0133.04902

[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. | MR 678742 | Zbl 0503.73048

[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. | MR 2566127 | Zbl 1187.65073

[49] A.N. Tychonoff, The regularization of incorrectly posed problems. Doklady Akad. Nauk. SSSR 153 (1963) 42-52. | MR 162378 | Zbl 0183.11601

[50] V. Zheligovsky, O. Podvigina and U. Frisch, The Monge − Ampère equation: Various forms and numerical solution. J. Comput. Phys. 229 (2010) 5043-5061. | MR 2643642 | Zbl 1194.65141

Cité par Sources :