Optimized Schwarz Methods For Cahn-Hilliard Equation

Cahn-Hilliard equation is an important forth order nonlinear diffusion equation.In this work,we will solve the Cahn-Hilliard equation by Schwarz domain decomposition method on the basis of time-marching method.We first show that the classical Schwarz method of Cahn-Hilliard equation is convergent,but with poor performence.Then we derive the optimal transmission condition to accelerate the Schwarz subdomain iterations,but it cannot be used because of the non-local pseudo-differential operators,which would not be numerically tractable.We thus want to approximate them by local operators(OO0,OO2,O2S)which result in the corresponding optimized transmission conditions after optimization.We gave the explicit formulas for the optimized transmission conditions.Finally,numerical simulations are conducted in the setting of two subdomains to validate the theoretical findings.