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We propose a conforming finite element method to approximate the strongsolution of the second order Hamilton-Jacobi-Bellman equation with Dirichletboundary and coefficients satisfying Cordes condition. We show the convergenceof the continuum semismooth Newton method for the fully nonlinearHamilton-Jacobi-Bellman equation. Applying this linearization for the equationyields a recursive sequence of linear elliptic boundary value problems innondivergence form. We deal numerically with such BVPs via the least-squaresgradient recovery of Lakkis & Mousavi [2021, arxiv:1909.00491]. We provide anoptimal-rate apriori and aposteriori error bounds for the approximation. Theaposteriori error are used to drive an adaptive refinement procedure. We closewith computer experiments on uniform and adaptive meshes to reconcile thetheoretical findings.

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