Finite Element Method And Numerical Analysis For The Problem Of Inextensible Immersed Interface In Incompressible Flows

We consider in this paper the immersed interface problem in an incompressible fluid.This kind of interface models has been widely used in a number of fields such as com-posite materials,multi-phase flow,bio-membrane,microbiology,etc.We are particularly-interested in the problems with the inextensibility constraint on the interface,which are commonly used to simulate the red blood cells,drug-carrying capsules,etc.There exist a large body of literatures in numerical methods for the related problems,including finite difference,finite element,and spectral methods.However,there is very limited result about theoretical analysis of these numerical methods.The main purpose of the this pa-per is to design and analyze a finite element,method for the week problem of the immersed interface model in an incompressible fluid,which is indeed a saddle point problem.Pre-cisely,the proposed finite element makes use of the polynomial spaces P2(?)B3,P1,and P1 as approximation spaces of the velocity,pressure,and the surface tension respectively.The well-posedness of the discrete problem is rigorously proved by applying the classical saddle point theory and the macro-element technique.In particular,the inf-sup condition is proved with an inf-sup constant independent of the discrete parameter.Finally,we give the implementation details of the proposed method and present some numerical examples to confirm the theoretical claims.