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.