Improved accuracy of monotone finite difference schemes on point clouds and regular grids
This work provides higher-order accurate schemes for solving anisotropic PDEs like Monge-Ampere, benefiting computational scientists who rely on monotone finite difference methods.
The authors improved the accuracy of monotone finite difference schemes for solving degenerate elliptic PDEs, achieving O(R + dθ^2) on point clouds and O(R^2 + dθ^2) on uniform grids, compared to previous O(R + dθ) and O(R^2 + dθ) respectively.
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, $dθ$. On uniform grids, the accuracy is $\mathcal O(R^2 + dθ)$. On point clouds, the most accurate schemes are of $\mathcal O(R + dθ)$, by Froese [Numerische Mathematik, 138(1):75-99, 2018]. In this work, we construct geometrically motivated schemes of higher accuracy in both cases: order $\mathcal O(R + dθ^2)$ on point clouds, and $\mathcal O(R^2 + dθ^2)$ on uniform grids.