On standard finite difference discretizations of the elliptic MongeAmpere equation
Abstract
Given an orthogonal lattice with mesh length h on a bounded convex domain, we propose to approximate the Aleksandrov solution of the MongeAmpere equation by regularizing the data and discretizing the equation in a subdomain using the standard finite difference method. The Dirichlet data is used to approximate the solution in the remaining part of the domain. We prove the uniform convergence on compact subsets of the solution of the discrete problems to an approximate problem on the subdomain. The result explains the behavior of methods based on the standard finite difference method and designed to numerically converge to nonsmooth solutions. We propose an algorithm which for smooth solutions appears faster than the popular Newton's method with a high accuracy for non smooth solutions. The convergence of the algorithm is independent of how close to the numerical solution the initial guess is, upon rescaling the equation and given a user's measure of the closeness of an initial guess.
 Publication:

arXiv eprints
 Pub Date:
 November 2013
 arXiv:
 arXiv:1311.2812
 Bibcode:
 2013arXiv1311.2812A
 Keywords:

 Mathematics  Numerical Analysis