Second-Order In Time Decoupled Stabilized Finite Element Methods For Fluid-Fluid Interaction Problem

The flow-flow coupling model is widely used in the global climate system,regional weather simulation,air and ocean coupling,and the interaction between the atmosphere and ocean,cardiovascular modeling.In this paper,for the flow-flow coupling problem with dynamic interface condi-tions,Nitsche's type boundary conditions are imposed on the interface,and a spatial semi-discrete coupling scheme based on Nitsche's interface method is obtained,and analyze and prove of this coupling scheme regarding the stability of the energy norm and consistency with the original problem.On the basis of semi-discrete space,we use the second-order backward Euler discrete format for time to obtain the fully discrete coupled format of the original problem.By decomposing the whole region into two sub-regions and decoupling the full-coupling problem into two sub-regional problems with time,a staggered format of coupled sub-problems with time advancement is ob-tained.The staggered format produces only two sub-problem solutions per time step.This method combines the explicit Robin-Robin interface coupling conditions of time and the weakly consistent interface pressure stability term.By introducing the rela-tionship between full pressure and static pressure and velocity,we redefine the original flow-flow coupling problem.For the full pressure formula of the coupling problem,we obtain an a priori energy estimate that guarantees decoupling stability.At the end of the article,three numerical experiments to verify the effectiveness of the proposed algorithm.the Robin-Robin type static pressure formula(4.1)staggered decoupling algorithm and the Robin-Robin type full pressure formula(5.1)staggered decoupling algorithm are used to triangulate the above true solution flow coupling prob-lem,select different finite element spaces for the above real solution examples for calcu-lation and simulation.It is verified that the relative errors of the L~2-norm of the velocity in the two regions,the H~1-norm of the velocity in the two regions and the L~2-norm of the pressure have grid optimal order.Then for the second-order format of BDF2standard time,let h=O(?t)to get the L~2-norm of the velocity of the two regions with respect to time has approximately O(h~2)relative error order,the two-region velocity H~1-norm and the pressure L~2-norm have approximately O(h)relative error order with respect to time,It is further verified that the Robin-Robin type static pressure formula staggered decoupling algorithm(4.1)and the Robin-Robin type full pressure formula staggered decoupling algorithm(5.1)also have the optimal error order for time.The nu-merical simulation of the driving flow model of the opposite cavity is carried out,and the results show that it is in line with the actual situation of the driving flow,and the effectiveness of the proposed algorithm is verified.