In the first part of the paper, we study the discontinuous Galerkin (DG) and C0 interior penalty (C0-IP) finite element approximation of the periodic strong solution to the fully nonlinear second-order Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation with coefficients satisfying the Cordes condition. We prove well-posedness and perform abstract a posteriori and a priori analyses which apply to a wide family of numerical schemes. These periodic problems arise as the corrector problems in the homogenization of HJBI equations. The second part of the paper focuses on the numerical approximation to the effective Hamiltonian of ergodic HJBI operators via DG/C0-IP finite element approximations to approximate corrector problems. Finally, we provide numerical experiments demonstrating the performance of the numerical schemes.
Keywords: Hamilton–Jacobi–Bellman and HJB–Isaacs equations, nondivergence-form elliptic PDE, Cordes condition, nonconforming finite element methods, homogenization
@article{M2AN_2022__56_2_679_0,
author = {Kawecki, Ellya L. and Sprekeler, Timo},
title = {Discontinuous {Galerkin} and $C^0${-IP} finite element approximation of periodic {Hamilton{\textendash}Jacobi{\textendash}Bellman{\textendash}Isaacs} problems with application to numerical homogenization},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {679--704},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {2},
doi = {10.1051/m2an/2022017},
mrnumber = {4393617},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022017/}
}
TY - JOUR AU - Kawecki, Ellya L. AU - Sprekeler, Timo TI - Discontinuous Galerkin and $C^0$-IP finite element approximation of periodic Hamilton–Jacobi–Bellman–Isaacs problems with application to numerical homogenization JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 679 EP - 704 VL - 56 IS - 2 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022017/ DO - 10.1051/m2an/2022017 LA - en ID - M2AN_2022__56_2_679_0 ER -
%0 Journal Article %A Kawecki, Ellya L. %A Sprekeler, Timo %T Discontinuous Galerkin and $C^0$-IP finite element approximation of periodic Hamilton–Jacobi–Bellman–Isaacs problems with application to numerical homogenization %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 679-704 %V 56 %N 2 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022017/ %R 10.1051/m2an/2022017 %G en %F M2AN_2022__56_2_679_0
Kawecki, Ellya L.; Sprekeler, Timo. Discontinuous Galerkin and $C^0$-IP finite element approximation of periodic Hamilton–Jacobi–Bellman–Isaacs problems with application to numerical homogenization. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 2, pp. 679-704. doi: 10.1051/m2an/2022017
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