A Study Of Unconditionally Stable Numerical Scheme For Cahn-Hilliard-Hele-Shaw System With Flory-Huggins Potential

The Cahn-Hilliard-Hele-Shaw(CH-HS)diffusion interface model describes the phase separation process of viscous binary fluid.The motion of a binary fluid mixture between two parallel plates is separated by a narrow gap,commonly known as the hele-shaw flow The CH-HS system has important application background in mathematical physics.chemical engineering and material science,for instance,tumor growth,cell separation.and displacement of binary fluids in two-phase flow in porous media.In recent years.there are many researches on CH-HS system with polynomial potential,but there are few studies on the finite element method of CH-HS system with logarithmic potential.we hope to get some new conclusions from this aspect.In this thesis,the CH-HS system with logarithmic potential is studied as followsIn the first part,the coupled CH-HS system with variable coefficients is studied and analyzed.Due to the domain of the Flory-Huggins potential function is restricted,some constraint conditions are needed to prove the energy stability.First of all,in order to maintain the unconditional stability of energy,the domain of Flory-Huggins potential function is extended from(-1,1)to(-?,?)by using the regularization procedure Secondly,a fully discrete scheme based on the energy functional convex splitting method is constructed,and the optimal error estimation of ? and p is obtained by applying the mean value theorem to the term with variable flow coefficient.Finally,according to the results of numerical experiments,the influence of different parameters on the error and convergence order is obtainedIn the second part,the decoupled finite element method for solving CH-HS system is analyzed.In order to solve the problem of the coupling system,a fully discrete scheme for solving the decoupling finite element scheme of CH-HS system is presented.In the first place,the explicit treatment of the pressure gradient term in the Darcy equation makes it unnecessary to update the pressure gradient when solving the equation with the nonlinear term of Cahn-Hilliard(CH)equation.Next,to update the pressure,we just need to solve a Poisson equation at each time step by using an incremental pressure-correction technique for the pressure gradient in Darcy equation,and the unconditional stability and convergence of the scheme are proved.Finally,some numerical experiments are implemented to illustrate the theoretical analysis.