Date(s) - 11/04/2017
11 h 00 min - 12 h 00 min
Elliptic and parabolic Hamilton—Jacobi—Bellman equations are an important class of second-order fully nonlinear PDEs, with applications to stochastic optimal control problems in engineering and finance. It is known that existing finite difference and finite element methods based on discrete maximum principles can often be guaranteed to converge to the viscosity solution in the small mesh limit. However, the requirement for a discrete maximum principle imposes severe restrictions on the choice of mesh, the order of convergence and the size of the stencil for strongly anisotropic problems, which can limit the computational efficiency on practical mesh sizes. This motivates the search for more flexible high-order methods that achieve the key properties of consistency, stability and convergence without discrete maximum principles. In this talk, we will present how these challenges are overcome in the context of fully nonlinear second-order elliptic and parabolic Hamilton–Jacobi–Bellman equations that satisfy a structural property named the Cordes condition. We construct an hp-version discontinuous Galerkin finite element method which is motivated by the PDE theory of the problem. Both the continuous and discrete analyses are based on a variational strong monotonicity argument which establishes well-posedness of the fully nonlinear HJB PDE in the class of strong solutions, and of the discrete numerical scheme. We show that the numerical method is consistent and stable, with error bounds that are optimal in the mesh size, and suboptimal in the polynomial degrees, as standard for hp-version DGFEM. For parabolic problems, the discretisation is extended by a high-order DG time-stepping method, permitting high-order approximation in both time and space. Numerical experiments demonstrate the accuracy and efficiency of the numerical scheme on problems featuring strongly anisotropic diffusion coefficients and singular solutions, including exponential convergence rates under hp-refinement. This is a joint work with Prof. Endre Süli, University of Oxford.