Discontinuous Galerkin Finite Element Approximation of Hamilton--Jacobi--Bellman Equations with Cordes Coefficients

We propose an $hp$-version discontinuous Galerkin finite element method for fully nonlinear second-order elliptic Hamilton--Jacobi--Bellman equations with Cordes coefficients. The method is proved to be consistent and stable, with convergence rates that are optimal with respect to mesh size, and suboptimal in the polynomial degree by only half an order. Numerical experiments on problems with nonsmooth solutions and strongly anisotropic diffusion coefficients illustrate the accuracy and computational efficiency of the scheme. An existence and uniqueness result for strong solutions of the fully nonlinear problem and a semismoothness result for the nonlinear operator are also provided.