Mixed finite element approximation of the Hamilton-Jacobi-Bellman equation with Cordes coefficients

A mixed finite element approximation of $H^2$ solutions to the fully nonlinear Hamilton--Jacobi--Bellman equation, with coefficients that satisfy the Cordes condition, is proposed and analyzed. A priori and a posteriori bounds on the approximation error are proved. The contributions from the a posteriori error estimator can be used as refinement indicators in an adaptive mesh-refinement algorithm. The convergence of this procedure is proved and empirically studied in numerical experiments.