Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equations with Cordes coefficients - Smears, I (2015)
We propose a discontinuous Galerkin finite element method (DGFEM) for fully nonlinear elliptic Hamilton--Jacobi--Bellman (HJB) partial differential equations (PDE) of second order with Cordes coefficients. Our analysis shows that the method is both consistent and stable, with arbitrarily high-order convergence rates for sufficiently regular solutions. Error bounds for solutions with minimal regularity show that the method is generally convergent under suitable choices of meshes and polynomial degrees. The method allows for a broad range of hp-refinement strategies on unstructured meshes with varying element sizes and orders of approximation, thus permitting up to exponential convergence rates, even for nonsmooth solutions. Numerical experiments on problems with nonsmooth solutions and strongly anisotropic diffusion coefficients demonstrate the significant gains in accuracy and computational efficiency over existing methods. We then extend the DGFEM for elliptic HJB equations to a space-time DGFEM for parabolic HJB equations. The resulting method is consistent and unconditionally stable for varying time-steps, and we obtain error bounds for both rough and regular solutions, which show that the method is arbitrarily high-order with optimal convergence rates with respect to the mesh size, time-step size, and temporal polynomial degree, and possibly suboptimal by an order and a half in the spatial polynomial degree. Exponential convergence rates under combined hp- and τq-refinement are obtained in numerical experiments on problems with strongly anisotropic diffusion coefficients and early-time singularities. Finally, we show that the combination of a semismooth Newton method with nonoverlapping domain decomposition preconditioners leads to efficient solvers for the discrete nonlinear problems. The semismooth Newton method has a superlinear convergence rate, and performs very effectively in computations. We analyse the spectral bounds of nonoverlapping domain decomposition preconditioners for a model problem, where we establish sharp bounds that are explicit in both the mesh sizes and polynomial degrees. We then go beyond the model problem and show computationally that these algorithms lead to efficient and competitive solvers in practical applications to fully nonlinear HJB equations.