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Two Numerical Methods for the elliptic Monge-Ampère equation

Published online by Cambridge University Press:  23 February 2010

Jean-David Benamou ,
Brittany D. Froese  and
Adam M. Oberman
Jean-David Benamou
Affiliation:
INRIA, Domaine de Voluceau, B.P. 105, 78153 Rocquencourt, France. jean-david.benamou@inria.fr
Brittany D. Froese
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Drive, BC V5A 1S6 Burnaby, Canada. bdf1@sfu.ca; aoberman@sfu.ca
Adam M. Oberman
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Drive, BC V5A 1S6 Burnaby, Canada. bdf1@sfu.ca; aoberman@sfu.ca
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Abstract

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. B10 (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. Paris340 (2005) 319–324]. There are already two methods available [Oliker and Prussner, Numer. Math.54 (1988) 271–293; Oberman, Discrete Contin. Dyn. Syst. Ser. B10 (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.

Keywords

Finite difference schemespartial differential equationsviscosity solutionsMonge-Ampère equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 4 , July 2010 , pp. 737 - 758
Copyright
© EDP Sciences, SMAI, 2010

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References

I. Bakelman, Convex analysis and nonlinear geometric elliptic equations. Springer-Verlag, Germany (1994).
Barles, G. and Souganidis, P.E., Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991) 271–283.
Böhmer, K., On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46 (2008) 1212–1249. CrossRef
L.A. Caffarelli and M. Milman, Eds., Monge Ampère equation: applications to geometry and optimization, Contemporary Mathematics 226. American Mathematical Society, Providence, USA (1999).
Caffarelli, L., Nirenberg, L. and Spruck, J., The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37 (1984) 369–402. CrossRef
Crandall, M.G., Ishii, H. and Lions, P.-L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. CrossRef
Dean, E.J. and Glowinski, R., 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.
Dean, E.J. and Glowinski, R., Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1344–1386. CrossRef
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.
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.
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.
X. Feng and M. Neilan, Galerkin methods for the fully nonlinear Monge-Ampère equation. http://arxiv.org/abs/0712.1240v1 (2007).
Feng, X. and Neilan, M., 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. CrossRef
Feng, X. and Neilan, M., Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. J. Sci. Comput. 38 (2009) 74–98. CrossRef
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.
Glowinski, R., Dean, E.J., Guidoboni, G., Juárez, L.H. and Pan, T.-W., 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. CrossRef
C.E. Gutiérrez, The Monge-Ampère equation, Progress in Nonlinear Differential Equations and their Applications 44. Birkhäuser Boston Inc., Boston, USA (2001).
Loeper, G. and Rapetti, F., Numerical solution of the Monge-Ampère equation by a Newton's algorithm. C. R. Math. Acad. Sci. Paris 340 (2005) 319–324. CrossRef
Oberman, A.M., Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44 (2006) 879–895. CrossRef
Oberman, A.M., Computing the convex envelope using a nonlinear partial differential equation. Math. Models Methods Appl. Sci. 18 (2008) 759–780. CrossRef
Oberman, A.M., 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. CrossRef
Oliker, V.I. and Prussner, L.D., On the numerical solution of the equation (∂2 z/∂x 2)(∂2 z/∂y 2) - (∂2 z/∂x∂y)2 = f and its discretizations, I. Numer. Math. 54 (1988) 271–293. CrossRef