Two numerical methods for the elliptic Monge-Ampère equation

摘要

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.