Spectral Collocation Method for Numerical Solution to the Fully Nonlinear Monge-Ampère Equation

Abstract

The Legendre–Gauss–Labatto spectral collocation method is proposed to solve the fully nonlinear Monge-Ampère equation in both two and three dimensional settings with the Dirichlet boundary conditions. The inhomogeneous boundary conditions are effectively handled by converting to homogeneous boundary conditions or modifying the second-order differentiation matrices. We propose a novel approach for approximating the initial value, which significantly reduces the number of iteration steps, thus simplifying the computations compared to existing methods. To overcome the strong nonlinearity of the underlying equation, we employ a fixed point technique which strongly makes use of the stability property of the linearized problem and its spectral collocation approximations. The convergence analysis of the proposed scheme is discussed under \(H^1\)-, \(H^2\)- and \(L^2\)-norms. Numerical examples are presented to validate the theoretical estimates. Several interesting phenomena are observed for the first time and open for mathematical verification.