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

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, $d\hat{I}¸$. On uniform grids, the accuracy is $\mathcal{O}(R^2+d\hat{I}¸)$. On point clouds, the most accurate schemes are of $\mathcal{O}(R+d\hat{I}¸)$, by Froese [Numerische Mathematik, 138(1):75-99, 2018]. In this work, we construct geometrically motivated schemes of higher accuracy in both cases: order $\mathcal{O}(R+d\hat{I}{¸}^2)$ on point clouds, and $\mathcal{O}(R^2+d\hat{I}{¸}^2)$ on uniform grids.