Convergence Rate Estimates for Aleksandrov's Solution to the Monge--Ampère Equation

In this paper, we establish convergence rate estimates for convex solutions to the Dirichlet problem of the Monge--Ampere equation $det D^2u=f$ in $Omega$, where $f$ is a positive and continuous function and $Omega$ is a bounded convex domain in the Euclidean space $mathbb{R}^n$. We approximate the solution $u$ by a sequence of convex polyhedra $v_h$, which are generalized solutions to the Monge--Ampere equation in the sense of Aleksandrov, and the associated Monge--Ampere measures $nu_h$ are supported on a properly chosen grid in $Omega$. We will derive the convergence rate estimates for the cases when $f$ is smooth, Holder continuous, and merely continuous.