We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh size h whereas the coarse scale ε is dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical solution to the viscosity solution as ε, h → 0, and ε ≳ (h|log h|)1/2. In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution.
Accepté le :
DOI : 10.1051/m2an/2018067
Mots-clés : Fully nonlinear equations, discrete maximum principle, finite elements
@article{M2AN_2019__53_2_351_0,
author = {Salgado, Abner J. and Zhang, Wujun},
title = {Finite element approximation of the {Isaacs} equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {351--374},
publisher = {EDP-Sciences},
volume = {53},
number = {2},
year = {2019},
doi = {10.1051/m2an/2018067},
zbl = {1433.65311},
mrnumber = {3939307},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018067/}
}
TY - JOUR AU - Salgado, Abner J. AU - Zhang, Wujun TI - Finite element approximation of the Isaacs equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 351 EP - 374 VL - 53 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018067/ DO - 10.1051/m2an/2018067 LA - en ID - M2AN_2019__53_2_351_0 ER -
%0 Journal Article %A Salgado, Abner J. %A Zhang, Wujun %T Finite element approximation of the Isaacs equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 351-374 %V 53 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018067/ %R 10.1051/m2an/2018067 %G en %F M2AN_2019__53_2_351_0
Salgado, Abner J.; Zhang, Wujun. Finite element approximation of the Isaacs equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 2, pp. 351-374. doi : 10.1051/m2an/2018067. http://www.numdam.org/articles/10.1051/m2an/2018067/
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