Optimized Schwarz Methods For Cahn-Hilliard Equation Discretized With Small Time Steps

Cahn-Hilliard was originally proposed to describe the phase separation phenomenon for binary alloy in the quenching process and has now been widely applied in many other scientific fields,such as image processing and polymer materials.Because of its non-linearity,numerical simulation is a main tool for investigating the solution behavior of Cahn-Hilliard equation.While because the phase separation procedure evolves very fast in time,it requires very small time step size to describe the corresponding phenomenon In this paper we investigate a coupled second order elliptic system of constant coefficients which is resulted from many different unconditionally energy stable time discretization scheme.By using asymptotic analysis to a min-max problem of convergence factor that is obtained by Fourier analysis in the case of two-subdomain domain decomposition,we obtain the optimized transmission parameters in explicit form applied in Robin-like and two-sided Robin-like transmission conditions for both overlapping and non-overlapping Schwarz domain decomposition algorithms,and we obtain as well the asymptoitc esti-mates of corresponding convergence factors.While for the Ventcel transmission condi-tions,we determine reasonable choices of the optimized transmission paramters by equi-oscillation.In the end,we use numerical examples to illustrate our theoretical findings.