Fully nonlinear PDEs are referred to the class of nonlinear PDEs which are nonlinear in the highest order derivatives of the unknown functions appearing in the equations,  they arise from many fields in science and engineering such as astrophysics, antenna design, differential geometry, geostrophic fluid dynamics, materials science, mathematical finance, meteorology, optimal transport, and stochastic control. This class of PDEs are known to be very difficult to study analytically and to approximate numerically. In this talk I shall review and discuss some latest advances in developing efficient numerical methods for fully nonlinear second (and first) order PDEs such as the Monge-Ampere type equations and Hamilton-Jacobi-Bellman equations. Background materials on the viscosity solution theory for fully nonlinear PDEs will be briefly reviewed. The focus of the talk will be on discussing various numerical approaches/methods/ideas and their pros and cons for constructing numerical methods which can reliably approximate viscosity solutions of fully nonlinear second order PDEs. Numerical experiments and application problems as well as open problems in numerical fully nonlinear PDEs will also be presented.

Dr. Xiaobing Feng is a professor in the Department of Mathematics at the University of Tennessee and served as an Associate Department Head and the Director of Graduate Studies. He obtained his Ph.D. degree from Purdue University in Computational and Applied Mathematics under the direction of the late Professor Jim Douglas, Jr. His primary research interest is numerical solutions of deterministic and stochastic nonlinear PDEs which arise from various applications including fluid and solid mechanics, subsurface flow and poroelasticity, phase transition, forward and inverse scattering, image processing, optimal control, systems biology, and data assimilation.  He has done some seminal work on numerical analysis of phase field models and their sharp interface limits and discontinuous Galerkin (DG) methods for high frequency waves. He has also pioneered the vanishing moment methodology for approximating viscosity solutions of fully nonlinear PDEs and lately become a leading expert on numerical analysis of stochastic Stokes/Navier-Stokes equations.