The Research Of Finite Element Discretization And Numerical Solution For The Cahn-hilliard Equation

Cahn Hilliard equation is a fourth order nonlinear diffusion equations, this paper focuses on finite element discrete format as well as the numerical solution of the equation The continuous and local discontinuous finite element methods are used to solve the Cahn Hilliard equation in one - dimensional and two - dimen-sional, In the one dimensional case, we use both continuous finite element and local discontinuities are approximated to the space in the first, Two semi-discrete formats are derived, and we have introduce the stability of the energy function to prove the format of them, and then the time with the forward Euler difference derived full discrete format, In the two-dimensional case, We also do both contin-uous finite element and local discontinuous finite element approximation to space,And then the time discretion using Crank-Nicolson discrete method, third-order TVD Runge-Kutta discrete method, it is concluded that the full discrete format.In the last part of the article, two forms of one dimensional case to do the numer-ical calculation, to calculate the nonlinear term and given boundary conditions,different numerical solution of initial value, It is verified that the two numerical schemes can guarantee the conservation of mass and the energy attenuation, At the same time, we find that the explicit form of the local discontinuous finite element is not as stable as the continuous element in solving the given equation,The continuous element more strict with time step, easy to burst.