Improved Accuracy of Monotone Finite Difference Schemes on Point Clouds and Regular Grids

Chris Finlay, Adam M. Oberman

Abstract: Finite difference schemes are the method of choice for solving nonlinear, degenerate elliptic PDEs, because the Barles--Souganidis convergence framework [G. Barles and P. E. Souganidis, Asymptotic Anal., 4 (1991), pp. 271--283] provides sufficient conditions for convergence to the unique viscosity solution [M. G. Crandall, H. Ishii, and P.-L. Lions, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1--67]. For anisotropic operators, such as the Monge--Ampère equation, wide stencil schemes are needed [A. M. Oberman, SIAM J. Numer. Anal., 44 (2006), pp. 879--895]. The accuracy of these schemes depends on both the distances to neighbors, $R$, and the angular resolution, $d\theta$. On regular 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 [Numer. Math., 138 (2018), pp. 75--99]. 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 regular grids.