TY - JOUR
T1 - Rates of convergence in WP 2-norm for the Monge-Ampère equation
AU - Neilan, Michael
AU - Zhang, Wujun
N1 - Funding Information: ∗Received by the editors December 7, 2017; accepted for publication (in revised form) August 6, 2018; published electronically October 16, 2018. http://www.siam.org/journals/sinum/56-5/M116040.html Funding: The work of the first author was partially supported by NSF grants DMS-1541585 and DMS-1719829. The work of the second author was supported by the start up funding at Rutgers University and NSF grant DMS-1818861. †Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260 (neilan@pitt.edu). ‡Department of Mathematics, Rutgers University, Piscataway, NJ 08854 (wujun@math.rutgers. edu). Funding Information: The work of the first author was partially supported by NSF grants DMS-1541585 and DMS-1719829. The work of the second author was supported by the start up funding at Rutgers University and NSF grant DMS-1818861. Publisher Copyright: © 2018 Society for Industrial and Applied Mathematics.
PY - 2018
Y1 - 2018
N2 - We develop discrete Wp 2-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 ku−uhkWf,p 2 (Nh I) converges in order O(h1/p) if p > d and converges in order O(h1/d ln(h 1 )1/d) if p ≤ d, where k·kWf,p 2 (NhI ) is a weighted Wp 2-type norm, and the constant C > 0 depends on kukC3,1(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.
AB - We develop discrete Wp 2-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 ku−uhkWf,p 2 (Nh I) converges in order O(h1/p) if p > d and converges in order O(h1/d ln(h 1 )1/d) if p ≤ d, where k·kWf,p 2 (NhI ) is a weighted Wp 2-type norm, and the constant C > 0 depends on kukC3,1(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.
KW - Discrete Alexandroff maximum principle
KW - Monge-Ampère equation
KW - W error estimate
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U2 - https://doi.org/10.1137/17M1160409
DO - https://doi.org/10.1137/17M1160409
M3 - Article
VL - 56
SP - 3099
EP - 3120
JO - SIAM Journal on Numerical Analysis
JF - SIAM Journal on Numerical Analysis
SN - 0036-1429
IS - 5
ER -