Time-space Finite Element Methods For The Incompressible Fluid Flows And Their Coupling Problems

The viscous incompressible Navier-Stokes(NS)equation is a typical nonlinear equation,which describes the motion law of fluid.Its efficient,stable,and high-precision numerical simulation is of great significance in both scientific computing theory and engineering applications.The coupled model of(Navier-)Stokes equation can usually describe multiple physical phenomena in multiple regions,it is widely used in many fields such as hydraulics,environmental science,biological fluid mechanics and so on.In this thesis,we propose and develop several high efficient finite element methods for incompressible fluid flows and its coupling problems.Firstly,for the incompressible NS equation,by combining the penalty method and the modular Grad-Div technique,we develop the first-order and second-order temporal discretization penalty-Grad-Div algorithms.The algorithms uncoupled the fluid veloc-ity and pressure variables based on the penalty method,then update velocity using the modular Grad-Div technique.We extend the proposed algorithms to the variable time step ?t and variable penalty parameter.Finally,numerical experiments validate the efficiency of the proposed algorithms.Secondly,for the non-stationary natural-convection equation,based on first-order backward Euler time discretization and time filters,we propose and analysis the second-order decoupled algorithms.The accuracy of numerical solutions obtained by first-order backward Euler scheme can be improved to second-order by time filters.Moreover,the time step is adjusted by estimating the numerical error of each step through the time filters,we extend the algorithms to adaptive time step algorithms.Numerical experi-ments validate the theoretical analysis and demonstrate the reliability of the proposed algorithms.Thirdly,for the non-stationary Stokes-Darcy model,noticing that the fluid in dif-ferent physical regions has different velocities,we propose a second-order partitioned method with multiple-time-step technique,which allows different time steps in the dif-ferent regions,and the time filter is used to improve the accuracy of the solution.Due to the difference of velocity between the free flow region and the porous media flow region,the error estimation of fluid in different regions will be very different at each time level,by combining the multiple-time-step technique and the time filter,we design new error estimate method and time step adjustment strategy,and the single adaptive algorithm and parallel adaptive algorithm are obtained.Numerical experiments verify the theoretical analysis and illustrate that the efficiency of the adaptive algorithms.Fourthly,for the non-stationary Dual-Porosity-Stokes model,we develop first-order algorithm with multiple-time-step technique.In the porous media(i.e.,the Darcy domain),the Dual-Porosity-Stokes model is modeled by the coupling system of two kinds of fluid flows in the microfracture and matrix.The proposed algorithm allows us-ing different time steps to advance the flow in the Stokes region,the fracture region,and the matrix region.We analyze the unconditional stability and error estimates of the par-titioned scheme.Numerical experiments verify the efficiency of the multiple-time-step technique.Fifthly,for the non-stationary Stokes-Darcy-Transport model,we propose stabi-lized finite element method and coupled and decoupled algorithms.The decoupled algorithm is a non-iterative partitioned scheme which splits the coupled problem into four subproblems(Stokes subproblem,Darcy subproblem,two transport subproblem).The stability of the proposed algorithms can be ensured by an interface stabilized term which depending on mesh size.The numerical experiments are performed to illustrate the theoretical analysis and demonstrate the applicability of the proposed algorithms.