Optimized Schwarz Methods for the Cahn–Hilliard Equation

Abstract

The Cahn–Hilliard equation was originally proposed to describe the phase separation phenomenon for a binary alloy in the quenching process and now has been widely applied in many other scientific fields. To investigate its solution behavior, one mainly depends on the numerical simulations because of the nonlinearity. However, it would require very small time steps to describe the phase separation procedure since it evolves very quickly in time, which implies that an efficient solver for the spatial problem at each time level is very important. In this paper, we investigate a coupled second order elliptic system of constant coefficients, which occurs in many different unconditionally energy stable time discretization schemes for the Cahn–Hilliard equation. For a min-max problem of convergence factor that is obtained via Fourier analysis in the case of two-subdomain domain decomposition, by using an asymptotic analysis we obtain the optimized transmission parameters in explicit form applied in the Robin and the two-sided Robin transmission conditions for both the overlapping and the nonoverlapping Schwarz domain decomposition algorithms, and we obtain as well the corresponding asymptotic convergence rates. Particularly, we also explore and analyze the many-subdomain cases directly and find that the algorithms are scalable until a certain number of subdomains is reached for small time steps, even if there is no coarse grid correction. We finally use numerical examples to illustrate our theoretical findings.