A least-squares Galerkin approach to gradient recovery for Hamilton-Jacobi-Bellman equation with Cordes coefficients

TL;DR

This paper introduces a finite element method combined with least-squares gradient recovery to solve the Hamilton-Jacobi-Bellman equation with Cordes coefficients, providing convergence analysis and adaptive refinement strategies.

Contribution

It develops a novel least-squares Galerkin approach for gradient recovery in solving fully nonlinear Hamilton-Jacobi-Bellman equations with proven error bounds.

Findings

Optimal-rate a priori error bounds established

Effective adaptive mesh refinement demonstrated

Numerical experiments confirm theoretical convergence

Abstract

We propose a conforming finite element method to approximate the strong solution of the second order Hamilton-Jacobi-Bellman equation with Dirichlet boundary and coefficients satisfying Cordes condition. We show the convergence of the continuum semismooth Newton method for the fully nonlinear Hamilton-Jacobi-Bellman equation. Applying this linearization for the equation yields a recursive sequence of linear elliptic boundary value problems in nondivergence form. We deal numerically with such BVPs via the least-squares gradient recovery of Lakkis & Mousavi [2021, arxiv:1909.00491]. We provide an optimal-rate apriori and aposteriori error bounds for the approximation. The aposteriori error are used to drive an adaptive refinement procedure. We close with computer experiments on uniform and adaptive meshes to reconcile the theoretical findings.