Rates of convergence in WP 2-norm for the Monge-Ampère equation

Michael Neilan, Wujun Zhang

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)3099-3120
Number of pages22
JournalSIAM Journal on Numerical Analysis
Volume56
Issue number5
DOIs
StatePublished - Jan 1 2018

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Discrete Alexandroff maximum principle
  • Monge-Ampère equation
  • W error estimate

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