2010/07/05 by Brittany D. Froese, Adam M. Oberman, Froese, Brittany D. +1 · 3 citations
Mathematics · #Nonlinear Partial Differential Equations #Meromorphic and Entire Functions #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1007.0765
The elliptic Monge-Amp `ere equation is a fully nonlinear Partial\nDifferential Equation that originated in geometric surface theory and has been\napplied in dynamic meteorology, elasticity, geometric optics, image processing\nand image registration. Solutions can be singular, in which case standard\nnumerical approaches fail. Novel solution methods are required for stability\nand convergence to the weak (viscosity) solution.\n In this article we build a wide stencil finite difference discretization for\nthe MA equation. The scheme is monotone, so the Barles-Souganidis theory\nallows us to prove that the solution of the scheme converges to the unique\nviscosity solution of the equation.\n Solutions of the scheme are found using a damped Newton's method. We prove\nconvergence of Newton's method and provide a systematic method to determine a\nstarting point for the Newton iteration.\n Computational results are presented in two and three dimensions, which\ndemonstrates the speed and accuracy of the method on a number of exact\nsolutions, which range in regularity from smooth to non-differentiable.\n