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

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.

25 pages

References in corpus (4)

• Unified analysis of discontinuous Galerkin and $C^0$-interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations
• Convergence of adaptive discontinuous Galerkin and $C^0$-interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations
• Mixed finite element approximation of periodic Hamilton--Jacobi--Bellman problems with application to numerical homogenization
• Optimal convergence rates for elliptic homogenization problems in nondivergence-form: analysis and numerical illustrations

Cited by in corpus (5)

• Optimal rate of convergence in periodic homogenization of viscous Hamilton-Jacobi equations
• Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form
• Homogenization of nondivergence-form elliptic equations with discontinuous coefficients and finite element approximation of the homogenized problem
• Computational multiscale methods for nondivergence-form elliptic partial differential equations
• Finite element approximation of stationary Fokker--Planck--Kolmogorov equations with application to periodic numerical homogenization