Abstract: In this talk I shall first present a brief overview about recent advances in numerical fully nonlinear PDEs. I shall then discuss in details a newly developed narrow-stencil finite difference framework for approximating viscosity solutions of fully nonlinear second order PDEs (such as Hamilton-Jacobi-Bellman and Monge-Ampere equations). The focus of the talk will be on discussing how to compensate the loss of monotonicity of the schemes (due to the use of narrow stencils) in order to ensure the convergence of the schemes, and to explain some key new concepts such as generalized monotonicity, consistency and numerical moment. The connection between the proposed methods and some well-known finite difference methods for first order Hamilton-Jacobi equations will be explained. Finally I shall briefly explain how to extend these finite difference techniques to a (high order) discontinuous Galerkin setting.

报告人简介：凤小兵教授（http://www.math.utk.edu/~xfeng/）是数值偏微分方程领域的知名专家，他在有限元方法，多重网格方法和区域分解法的理论分析和应用方面做了很多优秀的工作。最近他的研究兴趣包括带随机效应的微分方程的数值分析，和完全非线性方程的数值方法。本次邀请他介绍一下最近在完全非线性方程（例如Monge-Ampere, HJB) 方面的工作，这个方向是近几年的数值偏微分方程的一个热门领域。