| In this paper,we study the viscous Cahn-Hilliard equation and the Cahn-HilliardHele-Shaw system.On the one hand,a mixed finite element method for the CahnHilliard equation with logarithmic potential energy is studied,in order to obtain secondorder scheme in time,we propose a backward difference(BDF)numerical scheme.On the other hand,we present the Cahn-Hilliard-Hele-Shaw system with double potential,by introducing Lagrange multiplier,we get second-order numerical scheme in time.The details are as follows:In the first part,the background and method of the Cahn-Hilliard equation,viscous Cahn-Hilliard equation and Cahn-Hilliard-Hele-Shaw system are introduced.In the second part,the mixed finite element method is used to solve the viscous Cahn-Hilliard equation with logarithmic potential in space,in time we adopt the backward difference numerical scheme.For the potential energy function,we use convex splitting method(i.e.the energy functional is divided into convex part and concave part,this framework treats the concave part explicitly and treated the convex part implicitly),we construct a second-order linear scheme.We add artificial stability terms to ensure that the proposed scheme is unconditionally stable,the optimal error estimation of u is obtained by rigorous theoretical analysis.The theory is verified by numerical examples and we obtained the influence of different parameters on the error and convergence order.In the third part,the mixed finite element method is used to solve the CahnHilliard-Hele-Shaw system in space,in time we use Crank-Nicolson numerical scheme.In order to obtain a second-order linear numerical scheme,we introduce a Lagrange multiplier r.With auxiliary variable r we construct a fully discrete,second-order in time linear scheme,and we verify that the scheme is unconditionally stable and the optimal error estimation of ? is obtained.Finally,the theory is verified by numerical examples.Finally,we give a summary and some expectations. |
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