Improved accuracy of monotone finite difference schemes on point clouds and regular grids

TL;DR

This paper develops higher-accuracy finite difference schemes for solving nonlinear degenerate elliptic PDEs on point clouds and regular grids, improving convergence rates by optimizing spatial and angular resolutions.

Contribution

The authors introduce geometrically motivated schemes achieving second-order accuracy in angular resolution on point clouds and grids, surpassing previous methods.

Findings

Achieved $ ext{O}(R + d heta^2)$ accuracy on point clouds.

Achieved $ ext{O}(R^2 + d heta^2)$ accuracy on uniform grids.

Enhanced convergence rates for anisotropic PDEs like Monge-Ampere.

Abstract

Finite difference schemes are the method of choice for solving nonlinear, degenerate elliptic PDEs, because the Barles-Sougandis convergence framework [Barles and Sougandidis, Asymptotic Analysis, 4(3):271-283, 1991] provides sufficient conditions for convergence to the unique viscosity solution [Crandall, Ishii and Lions, Bull. Amer. Math Soc., 27(1):1-67, 1992]. For anisotropic operators, such as the Monge-Ampere equation, wide stencil schemes are needed [Oberman, SIAM J. Numer. Anal., 44(2):879-895]. The accuracy of these schemes depends on both the distances to neighbors, $R$, and the angular resolution, $d\theta$. On uniform grids, the accuracy is $\mathcal O(R^2 + d\theta)$. On point clouds, the most accurate schemes are of $\mathcal O(R + d\theta)$, by Froese [Numerische Mathematik, 138(1):75-99, 2018]. In this work, we construct geometrically motivated schemes of higher accuracy in both cases: order $\mathcal O(R + d\theta^2)$ on point clouds, and $\mathcal O(R^2 + d\theta^2)$ on uniform grids.