A Proof of the Higher Order Accuracy of the Patchy Method for Solving the Hamilton-Jacobi-Bellmamn Equation

We describe a finite element method for solving the Hamilton-Jacobi-Bellman equation of an infinite horizon nonlinear optimal control problem. We prove that in regions where the true solution is smooth and a strict Lyapunov function, the absolute error of the computed solution is higher order; the analytical error bound grows as the product of a fixed power of the step size times a term that grows exponentially in the distance from the origin of the state space. We provide examples that illustrate the higher order accuracy of the method.