Stabilized Finite Element And Space-Time Finite Element Method For Kelvin-Voigt Viscoelastic Fluid Flow Model

As an important part of partial differential equations,the viscoelastic fluid flow model of non-newtonian fluids has been widely used in food,molten plastics,biological fluids and so on.In recent years,many important studies have been made on the theoretical analysis and numerical simulation for the viscoelastic fluid models,among which finding an efficient and stable fixed finite element method is a key component of the numerical method for solving the viscoelastic fluid models.In this paper,we mainly research finite element method for the Kelvin-Voigt viscoelastic fluid flow model.Firstly,consider using the stabilized method for the Kelvin-Voigt viscoelastic fluid flow model with the lowest equal order mixed finite element pair.This method is a pressure projection method based on the difference of two local Gauss integration,which can not only overcome the restriction of the so-called inf-sup condition,but also do not need to introduce the edge-based date structures.Compared with other stabilized methods for the Kelvin-Voigt viscoelastic fluid flow model,we introduce some new techniques in this paper,such as the L'hopital law,negative norm technique,etc.Under some reasonable assumptions of velocity and pressure,we obtain the optimal error estimates of velocity and pressure.Secondly,a fully discrete stabilized finite element method is introduced based on the backward Euler scheme treats the time derivative terms.The linear terms are treated with implicit scheme,while the nonlinear term is treated by semi-implicit way.This method is suitable for linear discrete systems with large time steps.By establishing corresponding dual linearized Kelvin-Voigt model,unconditional stability results of numerical solutions in various norms are established.The effectiveness of the method is verified by numerical experiments.Finally,we used the space-time finite element method to solve the Kelvin-Voigt model.Firstly,the stability and convergence of the one level space-time finite element method are studied,secondly,the stability and convergence of the multilevel space-time finite element method are considered.Finally,some numerical results are presented to verify the established theoretical analysis and show the performances of the developed numerical method.