Local Discontinuous Galerkin Method For Non-Fickian Difusion In Viscoelastic Polymers

Many practical phenomenons, such as the efect of humidity on thin polymerflms, the penetration and difusion of liquid fow in polymer material, are governedby non-Fickian Difusion in viscoelastic polymers. Numerous experiments have rec-ognized that these phenomenons characterized by a constant velocity spreading ofthe penetrate with a well-defned sharp front, which is generically called non-Fickianbehavior or non-Fickian fow.For industrial applications, the primary interests in the mathematical mod-el of the non-Fickian difusion are the concentration which shows the extent andlocation of the penetrable liquid, the fux which indicates the quantity of the pene-trable liquid through some area of the porous media, and the viscoelastic stress inthe polymer flm. We want to propose a numerical method which can approximatesimultaneously the unknown function(the concentration), the viscoelastic stress inthe polymer flm and the adjoint vector-function (the fux).In this paper, we simulate numerically the following nonlinear non-Fickian D- illusion problem by a local discontinuous Galerkin finite element method. Theoretical analysis in this thesis indicates that this method inherits the advantages of local discontinuous Galerkin method, such as approximating simultaneously the concentration u, the viscoelastic stress a, and the flux function q=D(u)▽u+K(u)σ with high accu-racy. And from the flux q and stress σ we can easily obtain▽u, which describes the well-defined sharp front.We use ε-inequality and Gronwall inequality to analysis the local discontinuous finite element method. And we prove the stability of local discontinuous finite ele-ment scheme under the ratio of grids(κ/h≤l). Further, we introduce L2-projection and use the inductive hypothesis to prove a prior error estimates for the concentra-tion u, the viscoelastic stress σ and the flux q under the the ratio of grids(κ/h≤l). The numerical experiments show that the experimental results and theoretical con-clusions are consistent.We also try to establish an unconditional stability scheme by constructing a backward Euler difference-local discontinuous finite element scheme for the non-Fickian Diffusion problem. We split the error function into two parts, the spatially discrete error and the temporally discrete error. We just prove the temporally dis-crete error and the boundness of the solution of the time-discrete system. From the numerical experiment conducted in this thesis, we see that the backward Euler diference-local discontinuous fnite element scheme should be unconditional stable.