Georgia Institute of Technology College of Sciences

In this talk I shall present some latest advances on developing numerical methods (such as finite difference methods, Galerkin methods, discontinuous Galerkin methods) for fully nonlinear second order PDEs including Monge-Ampere type equations and Hamilton-Jacobi-Bellman equations. The focus of this talk is to present a new framework for constructing finite difference methods which can reliably approximate viscosity solutions of these fully nonlinear PDEs. The connection between this new framework with the well-known finite difference theory for first order fully nonlinear Hamilton-Jacobi equations will be explained. Extensions of these finite difference techniques to discontinuous Galerkin settings will also be discussed.