Xiaobing Feng, University of Tennessee

Fully Nonlinear Second Order PDEs: Theory, Numerical Methods, and Applications

Abstract: In the past thirty years tremendous progresses have been made on the development of the viscosity solution theory for fully nonlinear 2rd order PDEs. However, in contrast with the success of the PDE theory, until very recently there has been essentially no progress on how to reliably compute these viscosity solutions. This lack of progress is due to the facts that (a) viscosity solutions often are only conditionally unique; (b) the notion of viscosity solutions is nonvariational and nonconstructive, hence, it is extremely difficult (if it is possible) to mimic it at the discrete level. In this talk, I shall first review some recent advances (and attempts) in numerical methods for fully nonlinear 2rd order PDEs, in particular, the Monge-Ampere type PDEs. I shall then focus on discussing a newly developed methodology (called the vanishing moment method) and the induced new notion of weak solutions (called moment solutions) as well the relationship between viscosity solutions and moment solutions. Recent developments in Galerkin type methods such as finite element methods, spectral methods, mixed methods for fully nonlinear 2rd order PDEs based on the vanishing moment methodology will also be reviewed. Finally, I shall present some numerical results for the Monge-Ampere equation and the prescribed Gauss curvature equation, and also discuss a few applications such as the semigeostrophic flow and the Monge-Kantorovich optimal mass transport which all give rise interesting fully nonlinear PDE problems.

THURSDAY, March 19, 2009
Time: 1:30 P.M.
Location: PPB 300