Research On The Fully Discrete Finite Element Method For The Fluid-fluid Interaction Equation

Based on the finite element method,this paper constructs the viscosity-splitting method and the grad-div stabilized finite element schemes for the time-dependent fluid-fluid interaction model.Firstly,a fully discrete viscosity-splitting finite element method is developed and studied for the fluid-fluid interaction model.This method applies decomposition technique of viscosity in time and mixed finite element method in space,where the temporal term includes two steps.In the first step,a backward Euler scheme is utilized for the temporal discretization,semi-implicit scheme is applied for the nonlinearity term and the geometric averaging method is used to deal with the fluid interface.Then,in the second step,we only solve a linear Stokes problem without spatial iteration per time step for each individual domain.The viscosity-splitting finite element method splits nonlinearity and incompressibility.Thereby,a more complicated solution system is decomposed into two simple systems for calculation,which reduce the amount of required calculation.Moreover,the stability and convergence of the method are established by rigorous analysis.Finally,numerical experiments are presented to show the performance of the proposed method.Secondly,we construct two fully discrete grad-div stabilized finite element schemes for the fluid-fluid interaction model.The first scheme is standard graddiv stabilized scheme.This method is to add two gradient-divergence stabilization terms to the classic backward Euler format,and its purpose is to compensate for the lack of mass conservation in the post-Euler method,and to achieve a better approximation.And the other one is modular grad-div stabilized scheme which adds to Euler backward scheme an update step.This method overcomes the phenomenon of processor collapse caused by large stabilization parameters in the standard method,and the calculation time does not increase with the increase of the stabilization coefficient.Finally,both of the above grad-div stabilization methods are unconditionally stable and can calculate the problem with small viscosity coefficient.Moreover,stability and error estimates of these schemes are given.At the end,the correctness and validity of the proposed method are verified by numerical examples.