# Research on the Monge–Ampère equation

**The Monge–Ampère equation.** Joint work with R. H. Nochetto (UMD, USA).

Link to our paper: Optimal pointwise error estimates for two-scale methods for the Monge–Ampère equation.

Consider the Monge–Ampère equation with Dirichlet boundary conditions: find a convex function $u$ such that

\[\begin{equation} \label{E:MA} \det{D^2u(x)} = f(x), \; x \in \Omega \subset \mathbb{R}^d; \quad u(x) = g(x), \; x \in \partial \Omega, \end{equation}\]where the dimension $d\ge2$, $\Omega$ is a strictly convex domain, and functions $f \ge 0$ and $g$ are given.

We perform error analysis for the two-scale method proposed and studied in (R. H. Nochetto, D. Ntogkas, W. Zhang 2019 Math. Comp.) and (R. H. Nochetto, D. Ntogkas, W. Zhang 2019 IMA J. Numer. Anal.). In this method, the fine scale $h$ is the mesh size of the piecewise linear FE space and the coarse scale $\delta > h$ is used to compute centered finite differences in the discrete operator $T_{\delta,h}$. Our technique hinges on the discrete comparison principle, which follows from the monotonicity of the two-scale method, and the construction of suitable discrete barrier functions. We obtain optimal errro estimate for solution $u \in C^{2+k,\alpha}(\overline{\Omega})$ for $k = 0,1$ and $0 < \alpha \le 1$. When $k = \alpha = 1$, we obtain linear rate in $h$. Furthermore, we also derive error estimates for uniformly convex $u$ with Sobolev regularity and obtain linear rate when $u \in W^4_p(\Omega)$ with $p > d$. It is worth to mention that our framework of error analysis is applicable for a wide class of monotone methods for Hamilton-Jacobi-Bellman equations.