Two numerical methods for the elliptic Monge-Ampère equation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 4, p. 737-758

The numerical solution of the elliptic Monge-Ampère Partial Differential Equation has been a subject of increasing interest recently [Glowinski, in 6th International Congress on Industrial and Applied Mathematics, ICIAM 07, Invited Lectures (2009) 155-192; Oliker and Prussner, Numer. Math. 54 (1988) 271-293; Oberman, Discrete Contin. Dyn. Syst. Ser. B 10 (2008) 221-238; Dean and Glowinski, in Partial differential equations, Comput. Methods Appl. Sci. 16 (2008) 43-63; Glowinski et al., Japan J. Indust. Appl. Math. 25 (2008) 1-63; Dean and Glowinski, Electron. Trans. Numer. Anal. 22 (2006) 71-96; Dean and Glowinski, Comput. Methods Appl. Mech. Engrg. 195 (2006) 1344-1386; Dean et al., in Control and boundary analysis, Lect. Notes Pure Appl. Math. 240 (2005) 1-27; Feng and Neilan, SIAM J. Numer. Anal. 47 (2009) 1226-1250; Feng and Neilan, J. Sci. Comput. 38 (2009) 74-98; Feng and Neilan, http://arxiv.org/abs/0712.1240v1; G. Loeper and F. Rapetti, C. R. Math. Acad. Sci. Paris 340 (2005) 319-324]. There are already two methods available [Oliker and Prussner, Numer. Math. 54 (1988) 271-293; Oberman, Discrete Contin. Dyn. Syst. Ser. B 10 (2008) 221-238] which converge even for singular solutions. However, many of the newly proposed methods lack numerical evidence of convergence on singular solutions, or are known to break down in this case. In this article we present and study the performance of two methods. The first method, which is simply the natural finite difference discretization of the equation, is demonstrated to be the best performing method (in terms of convergence and solution time) currently available for generic (possibly singular) problems, in particular when the right hand side touches zero. The second method, which involves the iterative solution of a Poisson equation involving the hessian of the solution, is demonstrated to be the best performing (in terms of solution time) when the solution is regular, which occurs when the right hand side is strictly positive.

DOI : https://doi.org/10.1051/m2an/2010017
Classification:  65N06,  65N12,  65M06,  65M12,  35B50,  35J60,  35R35,  35K65,  49L25
Keywords: finite difference schemes, partial differential equations, viscosity solutions, Monge-Ampère equation
@article{M2AN_2010__44_4_737_0,
     author = {Benamou, Jean-David and Froese, Brittany D. and Oberman, Adam M.},
     title = {Two numerical methods for the elliptic Monge-Amp\`ere equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {4},
     year = {2010},
     pages = {737-758},
     doi = {10.1051/m2an/2010017},
     zbl = {1192.65138},
     mrnumber = {2683581},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_4_737_0}
}
Benamou, Jean-David; Froese, Brittany D.; Oberman, Adam M. Two numerical methods for the elliptic Monge-Ampère equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 4, pp. 737-758. doi : 10.1051/m2an/2010017. http://www.numdam.org/item/M2AN_2010__44_4_737_0/

[1] I. Bakelman, Convex analysis and nonlinear geometric elliptic equations. Springer-Verlag, Germany (1994). | Zbl 0815.35001

[2] G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991) 271-283. | Zbl 0729.65077

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

[4] L.A. Caffarelli and M. Milman, Eds., Monge Ampère equation: applications to geometry and optimization, Contemporary Mathematics 226. American Mathematical Society, Providence, USA (1999). | Zbl 0903.00039

[5] L. 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. | Zbl 0598.35047

[6] 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. (N.S.) 27 (1992) 1-67. | Zbl 0755.35015

[7] 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. Electron. Trans. Numer. Anal. 22 (2006) 71-96. | Zbl 1112.65119

[8] 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. | Zbl 1119.65116

[9] 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, Comput. Methods Appl. Sci. 16, Springer, Dordrecht, The Netherlands (2008) 43-63. | Zbl 1152.65479

[10] 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, Lect. Notes Pure Appl. Math. 240, Chapman & Hall/CRC, Boca Raton, USA (2005) 1-27. | Zbl 1188.65115

[11] L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, in Current developments in mathematics, 1997 (Cambridge, MA), Int. Press, Boston, USA (1999) 65-126. | Zbl 0954.35011

[12] X. Feng and M. Neilan, Galerkin methods for the fully nonlinear Monge-Ampère equation. http://arxiv.org/abs/0712.1240v1 (2007).

[13] 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. | Zbl 1195.65170

[14] X. Feng and M. Neilan, Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. J. Sci. Comput. 38 (2009) 74-98. | Zbl 1203.65252

[15] R. Glowinski, Numerical methods for fully nonlinear elliptic equations, in 6th International Congress on Industrial and Applied Mathematics, ICIAM 07, Invited Lectures, R. Jeltsch and G. Wanner Eds. (2009) 155-192. | Zbl 1179.65146

[16] R. Glowinski, E.J. Dean, G. Guidoboni, L.H. Juárez 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 elliptic Monge-Ampère equation. Japan J. Indust. Appl. Math. 25 (2008) 1-63. | Zbl 1141.76043

[17] C.E. Gutiérrez, The Monge-Ampère equation, Progress in Nonlinear Differential Equations and their Applications 44. Birkhäuser Boston Inc., Boston, USA (2001). | Zbl 0989.35052

[18] 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. | Zbl 1067.65119

[19] A.M. Oberman, Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44 (2006) 879-895. | Zbl 1124.65103

[20] A.M. Oberman, Computing the convex envelope using a nonlinear partial differential equation. Math. Models Methods Appl. Sci. 18 (2008) 759-780. | Zbl 1154.35056

[21] 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. B 10 (2008) 221-238. | Zbl 1145.65085

[22] V.I. Oliker and L.D. Prussner, On the numerical solution of the equation (∂2z/∂x2)(∂2z/∂y2) - (∂2z/∂x∂y)2 = f and its discretizations, I. Numer. Math. 54 (1988) 271-293. | Zbl 0659.65116