Expanded Mixed Finite Element Simulation For Soboley Equations

Many pratical phenomenons, such as the effect of humidity on thin polymer films, the penetration and diffusion of liquid flow in polymer material, are governed by Sobolev equations. Numerous experiments have recognized 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-Pickian flow. For industrial applications, the primary interests in the mathematical model of the nonlinear Sobolev equation are the concentration u which shows the extent and location of the penetrable liquid, the fluxσwhich indicates the quantity of the penetrable liquid through some area of the porous media, and the gradient Vu which predicts where and when the non-Fickian behavior takes place. In this thesis, we propose an H1-Galerkin expanded mixed finite element method to simulate linear and nonlinear Sobolev equations. The theoretical analysis and numerical experiments show that the proposed method in this thesis can approximate the unknown function(the concentration), its gradient and the adjoint vector-function(the flux), and so, is a high-performence numerical method for Sobolev equations.As a more simple case we consider the following linear Sobolev probolem An H1-Galerkin expanded mixed finite element method is proposed, the equivalence between the boundary value problem and the variational formulation are proved and then the existence and uniqueness of the method is given. Theoretical analysis indicate that this mehtod inherits the advantages of H1-Galerkin method and expanded mixed finite element method, such as approximating the concentration u, the gradient of the concentration p=(?)u and the fluxσ= a(x,t)pt+b(x,t)p simultaneously with high accuracy, being not subject to LBB consistent conditions in the choice of the finite element spaces and possessing optimal L2 error estimates without introducing curl operators.In the elastic limit case, we derive a mathematical model- for non-Fickian flow as a kind of nonlinear Sobolev equation In order to apply H1-Galerkin expanded mixed finite element method to the nonlin-ear Sobolev equation, we derive a variational formulation and proved the equivalence, then propose an semi-discrete H1-Galerkin expandad mixed finite element method and prove the existence and uniqueness of the discrete scheme. An optimal L2 error esti-mates for u, p=(?)u andσ=E(u)pt +D(u)p are derived with an inductive hypothesis. Numerical experiments confirm the theoretical finding. This shows that H1-Galerkin expanded mixed fnite element method is a powerful numerical method for non-Fickian flow, which can overcome the difficulties arising from calculating the inverse of a small diffusive coefficient E(u) within a low permeability zone and avoid the troubles resulted from representation of the time derivatives.