Unconditionally Stable Schemes In Energy For The Viscous Cahn-Hilliard Equation And The Penalty Ericksen-Leslie Equation

In this paper,unconditionally stable finite element numerical schemes in energy for the viscous Cahn-Hilliard equation and the Ericksen-Leslie equation are studied.For the viscous Cahn-Hilliard equation,a finite element numerical method is proposed,which adopts the second-order scheme(the modified Crank-Nicolson scheme)in time and the hybrid finite element method in space.For the Ericksen-Leslie equation,a first-order convex splitting numerical scheme for the penalized Ericksen-Leslie equation and a second-order scheme(it combines the Crank-Nicolson scheme and the modified Crank-Nicolson scheme)with Lagrange multiplier in the penalized saddle point structure arc presented.The details arc as follows.In Chapter 1,the research background,numerical methods and research progress of viscous Cahn-Hilliard equation and Ericksen-Leslie equation are introduced.In Chapter 2,the finite element numerical method for the viscous Cahn-Hilliard equation with logarithmic potential is studied.Firstly,the logarithmic potential function F(u)is regularized to expand its domain from(-1,1)to(-∞,+∞).Based on it,a new numerical scheme is proposed,which adopts the second-order scheme in time(the modified Crank-Nicolson scheme)and the mixed finite element method in space.Through theoretical calculation and analysis,it is proved that the numerical scheme is unconditionally stable in energy,and the error estimation is carried out.Finally,several numerical examples are given to verify the effectiveness and accuracy of the proposed numerical scheme.In Chapter 3,two linear unconditionally stable finite element numerical schemes in energy for the Ericksen-Leslie equation are presented.Firstly,based on the modified convex splitting method(for the Ginzburg-Landau function,the linear term is treated implicitly and the nonlinear term is treated explicitly),a first-order scheme for the penalty Ericksen-Leslie equation is proposed.Secondly,for the penalty Ericksen-Leslie equation in saddle point structure,a new second-order scheme is proposed,which combines the Lagrange multiplier method and the modified Crank-Nicolson method.Through a series of theoretical calculations and analysis,the unconditional stability of numerical schemes in energy are proved respectively.Finally,the stability,accuracy and effectiveness of the proposed numerical schemes are verified by numerical simulation.In Chapter 4,The results of this paper are summarized and the research content is prospected.