2017/12/07 by Michael Neilan, Neilan, Michael, Wujun Zhang +1
Mathematics · #35B50 #35D30 #35J96 #65N12 #65N15 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1712.02492
openalex publication_date 2017/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop discrete W2p-norm error estimates for the Oliker-Prussner method applied to the Monge-Ampère equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second order difference. In addition, we show that the size of the complement of the contact set is controlled by the consistency of the method. Combining both observations, we show that the error estimate ‖u - uh‖W2p ≤ C h1/p if p > d and ‖u - uh‖W2p ≤ C h1/d (ln(\frac 1 h ))1/d if p ≤ d. Here the constant C depends on ‖u‖C3,1(Ω), the dimension d, and the constant p. Numerical examples are given in two space dimensions and confirm that the estimate is sharp in several cases.