| In this paper,the finite element algorithms of the Cahn-Hilliard equation with logarithmic potential and the Cahn-Hilliard-Hele-Shaw system are presented.On the one hand,a second-order finite element numerical scheme is proposed for the CahnHilliard equation with logarithmic potential and concentration mobility.On the other hand,based on the leap-frog time-marching scheme,a second-order linear numerical method is given for the Cahn-Hilliard-Hele-Shaw system.The details are as follows:In the first chapter,we introduce the research background and status of the Cahn-Hilliard equation,the Cahn-Hilliard equation with concentration mobility and the Cahn-Hilliard-Hele-Shaw system.In the second chapter,a second-order numerical algorithm of the Cahn-Hilliard equation with logarithmic potential and concentration mobility is presented.Firstly,the domain of the logarithmic Flory-Huggins potential F(u)is extended from(-1,1)to(-∞,∞)by the regularization method.A finite element scheme is proposed by using the second-order Crank-Nicolson scheme in time and the mixed finite element method in space.Then,we add two artificial stability terms(AτΔ(uhn+1-uhn)and B(uhn+1-2uhn+uhn-1))to ensure the energy stability of the scheme and give the error estimation by the rigorous theoretical analysis.The validity of the proposed scheme is conducted via numerical examples in the end.In the third chapter,we propose a second-order linear finite element numerical scheme of the Cahn-Hilliard-Hele-Shaw system.First of all,a finite element scheme is proposed by using the second-order Crank-Nicolson-Leapfrog scheme in time and the mixed finite element method in space.Then,the proposed scheme is proved to satisfy the energy principle,which shows the scheme is unconditionally stable in energy.In particular,error estimates is given by important inequalities and projection theorems.Finally,numerical examples are presented to verify the stability of the scheme and accuracy of theoretical analyses.The coarsening dynamics of the Cahn-Hilliard-HeleShaw system are also given.Finally,we give a summary and expectations. |
