The Velocity Correction Projection Finite Element Methods For Kelvin-Voigt Viscoelastic Fluid Equations

This thesis focuses on the velocity-correction projection finite element method and the modified characteristic projection finite element method for the Kelvin-Voigt viscoelastic fluid model.We consider the Kelvin-Voigt viscoelastic fluid equation in the following form:(?) where Ω (?)Rd,d=2,3 is a convex polygonal domain and I=(0,T],u stands for the vector velocity,p denotes the scalar pressure.κ,Re are retardation time and the Reynolds coefficient,respectively,and f is the body force,u0 the initial velocity.The first part introduces the background of Kelvin-Voigt viscoelastic fluid model,the related research of modified characteristic projection finite element method and the structure of the paper.The second part mainly introduces some inequalities and spaces often used in this paper,and gives some basic lemma and operator definitions.In part 3,the rotational velocity-correction projection finite element method for Kelvin-Voigt viscoelastic fluid is presented.In this part,the basic regularity hypothesis for theoretical analysis is given,and the velocity-correction projection finite element algorithm in time discrete form is provided.Then,select appropriate test functions to analyze the theory.The stability and convergence of the algorithm are obtained by the theoretical analysis.The numerical results show that the velocity correction projection finite element method is stable and has the optimal convergence order.In part 4,the modified characteristics finite element method is applied to the Kelvin-Voigt viscoelastic fluid equation.The time derivative term and convective term are converted into linear terms by the characteristic line method,and the time discrete and fully discrete formal algorithms are obtained by velocity-correction.And then the stability and convergence of the two algorithms are obtained through mathematical induction.The final numerical examples show that our algorithm is valid for differentκ,Re,and the convergence order of the algorithm is optimal.