In this talk, I will introduce different modeling and numerical methods for fluid-structure interaction (FSI) problems based upon different numerical strategies and solution interests, where, the arbitrary Lagrangian-Eulerian(ALE) method will be introduced first for different motions of structure,then the full Eulerian-phase field method and the fictitious domain method will be addressed for the case ALE method cannot handle. More beyond, I will extend my talk from FSI problems to other multiphysics problems and interface problems, introduce my recent work on the Stokes-Darcy-Brinkman coupling problem, the parabolic interface problem and the Stokes interface problem. Relevant numerical analyses (well-posedness, stability and convergence properties) and realistic applications of FSI problems will be illustrated as well in this talk.

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