Discontinuous Galerkin and $C^0$-IP finite element approximation of periodic Hamilton--Jacobi--Bellman--Isaacs problems with application to numerical homogenization

TL;DR

This paper develops and analyzes discontinuous Galerkin and $C^0$-IP finite element methods for solving periodic Hamilton--Jacobi--Bellman--Isaacs equations, with applications to numerical homogenization and effective Hamiltonian approximation.

Contribution

It introduces a unified analysis framework for DG and $C^0$-IP schemes applied to nonlinear second-order PDEs with periodic coefficients, including homogenization applications.

Findings

Proved well-posedness of the schemes.

Established a posteriori and a priori error estimates.

Numerical experiments confirmed the schemes' effectiveness.

Abstract

In the first part of the paper, we study the discontinuous Galerkin (DG) and $C^0$ interior penalty ($C^0$-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/$C^0$-IP finite element approximations to approximate corrector problems. Finally, we provide numerical experiments demonstrating the performance of the numerical schemes.