  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.
