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.

翻译：我们提出了一种协调有限元方法来逼近具有Dirichlet边界且系数满足Cordes条件的二阶Hamilton-Jacobi-Bellman方程强解。我们证明了完全非线性Hamilton-Jacobi-Bellman方程的连续半光滑牛顿法的收敛性。对该方程应用此线性化，会产生一系列非散度形式的线性椭圆边值问题的递归序列。我们通过Lakkis & Mousavi [2021, arxiv:1909.00491]的最小二乘梯度恢复方法数值处理此类边值问题。我们为近似解提供了最优阶的先验和后验误差界。后验误差被用于驱动自适应加密过程。最后，我们通过均匀网格和自适应网格上的计算机实验来验证理论结果。