2018/07/13 by Christopher C. Finlay, Adam M. Oberman, Finlay, Chris +1
Economics, Econometrics and Finance · Engineering · Mathematics · #35J15 #35J25 #65N06 #65N12 #65N22 #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1807.05150
openalex publication_date 2018/07/13 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
Finite difference schemes are the method of choice for solving nonlinear,\ndegenerate elliptic PDEs, because the Barles-Sougandis convergence framework\n[Barles and Sougandidis, Asymptotic Analysis, 4(3):271-283, 1991] provides\nsufficient conditions for convergence to the unique viscosity solution\n[Crandall, Ishii and Lions, Bull. Amer. Math Soc., 27(1):1-67, 1992]. For\nanisotropic operators, such as the Monge-Ampere equation, wide stencil schemes\nare needed [Oberman, SIAM J. Numer. Anal., 44(2):879-895]. The accuracy of\nthese schemes depends on both the distances to neighbors, R, and the angular\nresolution, d\θ. On uniform grids, the accuracy is mathcal O(R2 +\nd\θ). On point clouds, the most accurate schemes are of mathcal O(R +\nd\θ), by Froese [Numerische Mathematik, 138(1):75-99, 2018]. In this work,\nwe construct geometrically motivated schemes of higher accuracy in both cases:\norder mathcal O(R + d\θ2) on point clouds, and mathcal O(R2 +\nd\θ2) on uniform grids.\n