An Expanded Mixed Finite Element Method For Non-Fickian Diffusion In Viscoelastic Polymers

Many practical phenomenons, such as the effect of humidity on thin polymer films, the penetration and diffusion of liquid flow in polymer material, are governed by non-Fickian Diffusion in viscoelastic pofymers. Numerous experiments have rec-ognized that these phenomenons characterized by a constant velocity spreading of the penetrate with a well-defined sharp front, which is generically called non-Fickian behavior or non-Fickian flow.For industrial applications, the primary interests in the mathematical model of the non-Fickian diffusion are the concentration which shows the extent and location of the penetrable liquid, the flux which indicates the quantity of the penetrable liquid through some area of the porous media, the gradient which predicts where and when the non-Fickian behavior takes place, and the viscoelastic stress in the polymer film. We want to propose a numerical method which can approximate the unknown function(the concentration), its gradient, the viscoelastic stress in the polymer film and the adjoint vector-function (the flux).Since the expanded mixed finite element method can approximate the unknown function, its gradient and the adjoint vector-function simultaneously, so it becomes a powerful numerical method for partial differential equations. Further it can simulate well the flow within a low permeability zone for without calculating the invese of a small diffusion coefficient.In this thesis, according to classical continuum models for diffusion in materials, a flux containing a contribution from viscoelastic stress and viscoelastic relaxation equation, we derive the mathematical model for non-Fickian diffusion and propose an expanded mixed finite element method. Moreover, we prove the equivalence between the boundary value problem and the variational formulation and give the existence and uniqueness of the discretization solution.For theoretical analysis, we introduce L2-projection, Raviart-Thomas projec-tion, elliptical projection,ε-inequality and Gronwall inequality. Theoretical analy-sis indicates that this method inherits the advantage of the expanded mixed finite element method, such as approximating the concentration u, the gradient of the concentration p=-D▽u. the viscoelastic stress in the polymer filmσand the flux J=-D(u)▽u-K(u)σsimultaneously with high accuracy and possessing optimal L2 error estimates.We derive a mathematical model for nonlinear non-Fickian. In order to apply an expanded mixed finite element method to the nonlinear equation, we derive a variational formulation and propose an semi-discrete expanded mixed finite element method. For theoretical analysis, we introduce L2-projection, Raviart-Thomas pro-jection, elliptical projection,ε-inequality and Gronwall inequality. The theoretical analysis shows that this method has the advantage of the expanded mixed finite ele- ment method,namel y,we derive the optimal L2 error estimates for u,σ,p=-D▽u and the flux J=-D(u)▽u-K(u)σ.