Numerical solution of Hamilton—Jacobi—Bellman equations

Hamilton—Jacobi—Bellman (HJB) equations are a class of fully nonlinear second-order partial differential equations (PDE) of elliptic or parabolic type that originate from Stochastic Optimal Control Theory. These PDE are fully nonlinear in the sense that the nonlinear terms include the second partial derivatives of the unknown solution; this strong nonlinearity severely restricts the range of numerical methods that are known to be convergent. These problems have traditionally been solved with low order monotone schemes, often of finite difference type, which feature certain limitations in terms of efficiency and practicability.

In this summary talk of my DPhil studies, we will be interested in the development of hp-version discontinuous Galerkin finite element methods (DGFEM) for the class of HJB equations that satisfy a Cordès condition. First, we will show the novel techniques of analysis used to find a stable and convergent scheme in the elliptic setting, and then we will present recent work on their extension to parabolic problems. The resulting method is very nonstandard, provably of high order, and it even allows for exponential convergence under hp-refinement. We present numerical experiments showing the accuracy, computational efficiency and flexibility of the scheme