High-order Temporal Discrete Finite Element Methods For Two Classes Of Coupled Models Of Navier–stokes Equations

Fluid mechanics and phase field models play an important role in studying the laws of motion of fluids and have been successfully used to simulate dynamic processes in many fields.The analysis of finite element methods(FEM)for such problems is always a hot issue that attract lots of concerns.In this thesis,to study the application of the time-accurate FEM to fluid mechanics problems(such as the Navier–Stokes equations and the Cahn–Hilliard–Navier–Stokes equations),a second-order backward difference formula finite element method(BDF-FEM)numerical scheme is designed for solving the fluid mechanics and phase field problems.Moreover,this thesis analyzes the unconditional en-ergy stability and convergence of the proposed numerical scheme,and meanwhile gives the numerical experiments to verify the theoretical analysis.Firstly,the physical background and significance of some problems in this thesis,such as the incompressible fluid phase field model and its numerical methods,are briefly introduced.Meanwhile,several preliminaries,including basics of Sobolev spaces,some important inequalities and projection estimation in finite element,are reviewed.Secondly,a second-order decoupled BDF-FEM scheme for solving the incompress-ible Navier–Stokes(NS)equations is constructed.The“decoupled”technique removes the incompressible constricts from the flow equations by introducing an intermediate vari-able and treating the pressure terms explicitly.Then the unconditional energy stability and unique solvability of the numerical scheme could be obtained.Furthermore,by intro-ducing an intermediate function,this thesis proves the optimal convergence rates for the discrete scheme,which is O(h^r+1+?~2)for the velocity variable in?^?([0,T],L~2)-norm.Here,r represents the degree of the polynomial function space,and h and?are the spatial and temporal size,respectively.Finally,a second-order decoupled scheme for solving the Cahn–Hilliard–Navier–Stokes(CHNS)system is discussed,by combining the Invariant Energy Quadratization(IEQ)method for the Cahn–Hilliard equations and the decoupled method for the Navier–Stokes equations.A combination of the FEM and the second-order BDF has been adopted in space and time,respectively.In particular,a second-order Adams–Bashforth extrap-olation is used to linearize the nonlinear terms.Meanwhile,it will remain the uncondi-tional energy stability for the proposed scheme.Moreover,the optimal convergence rates O(h^r+?~2)are derived,for the phase variable bounded in the?^?([0,T];H~1)-norm,the chemical potential in the?~2([0,T];H~1)-norm,and the velocity variable in the?^?([0,T];L~2)-norm.In addition,some numerical experiments are presented for the last two parts above to verify the stability and convergence of the discrete scheme.