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.

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