Title: Phase field method for geometric moving interface problems and their numerical approximations

Abstract:  In this talk I shall first give a brief introduction to the phase field method for general geometric moving interface problems. The focus will be on presenting its idea, formulation,and relationship to other methods for moving interface problems such as the level set method.The second part of the talk will devote to discussing some latest advances in developing efficient numerical methods for solving phase field models. Two best known such models (namely, the Allen-Cahn and Cahn-Hilliard equations) will be used as specific examples to illustrate the main mathematical and numerical issues associated with those phase field models. The focuses will be on discussing the recent developments in establishing the convergence of the numerical interfaces to the sharp interface limits of those two phase field models (namely, the mean curvature flow and the Hele-Shaw flow), as both the numerical mesh parameters and the phase field parameter tend to zero, and to present the main ideas for establishing those results, as well as to discuss generalizations of those ideas and results to other related phase field models.