# Two Numerical Methods for the elliptic Monge-Ampère equation

Jean-David Benamou; Brittany D. Froese; Adam M. Oberman

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 4, page 737-758
- ISSN: 0764-583X

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topBenamou, Jean-David, Froese, Brittany D., and Oberman, Adam M.. "Two Numerical Methods for the elliptic Monge-Ampère equation." ESAIM: Mathematical Modelling and Numerical Analysis 44.4 (2010): 737-758. <http://eudml.org/doc/250759>.

@article{Benamou2010,

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.
},

author = {Benamou, Jean-David, Froese, Brittany D., Oberman, Adam M.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Finite difference schemes; partial differential equations; viscosity solutions; Monge-Ampère equation; finite difference schemes},

language = {eng},

month = {6},

number = {4},

pages = {737-758},

publisher = {EDP Sciences},

title = {Two Numerical Methods for the elliptic Monge-Ampère equation},

url = {http://eudml.org/doc/250759},

volume = {44},

year = {2010},

}

TY - JOUR

AU - Benamou, Jean-David

AU - Froese, Brittany D.

AU - Oberman, Adam M.

TI - Two Numerical Methods for the elliptic Monge-Ampère equation

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/6//

PB - EDP Sciences

VL - 44

IS - 4

SP - 737

EP - 758

AB -
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.

LA - eng

KW - Finite difference schemes; partial differential equations; viscosity solutions; Monge-Ampère equation; finite difference schemes

UR - http://eudml.org/doc/250759

ER -

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- G. Loeper and F. Rapetti, Numerical solution of the Monge-Ampère equation by a Newton's algorithm. C. R. Math. Acad. Sci. Paris340 (2005) 319–324. Zbl1067.65119
- 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. Zbl1124.65103
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