Mixed finite element approximation of periodic Hamilton--Jacobi--Bellman problems with application to numerical homogenization

TL;DR

This paper develops and analyzes a mixed finite element method for solving periodic Hamilton--Jacobi--Bellman equations, with applications to numerical homogenization of effective Hamiltonians, supported by numerical experiments.

Contribution

It introduces a new mixed finite element approach for nonlinear HJB equations and applies it to numerical homogenization, providing rigorous analysis and practical validation.

Findings

Effective approximation of the Hamilton--Jacobi--Bellman solutions

Successful numerical homogenization of the effective Hamiltonian

Numerical experiments confirm the scheme's accuracy

Abstract

In the first part of the paper, we propose and rigorously analyze a mixed finite element method for the approximation of the periodic strong solution to the fully nonlinear second-order Hamilton--Jacobi--Bellman equation with coefficients satisfying the Cordes condition. These problems arise as the corrector problems in the homogenization of Hamilton--Jacobi--Bellman equations. The second part of the paper focuses on the numerical homogenization of such equations, more precisely on the numerical approximation of the effective Hamiltonian. Numerical experiments demonstrate the approximation scheme for the effective Hamiltonian and the numerical solution of the homogenized problem.