Optimized Ventcel-Schwarz methods for the Cahn-Hilliard equation discretized by the stabilized linear Crank-Nicolson scheme
One of the most important methods for the time discretization of the Cahn-Hilliard (CH) equation is the stabilized linear Crank-Nicolson (SL-CN) scheme. While this method is an unconditionally energy stable method of second order (allowing to use time steps as large as possible to reduce the total calculation), it is well known that it requires a very large amount of calculations for simulating the CH equation. To accelerate the simulation process, it is proposed to solve the differential system resulting from the time discretization, by an optimized Schwarz method using a newly proposed Ventcel transmission condition. For a setting of two-subdomain domain decomposition with or without overlap, the authors obtain the convergence factor. This is derived using Fourier analysis and takes on two different forms according to the size of the time steps. The convergence factors are optimized for each case, optimized transmission parameters are obtained and so are the estimates for the convergence rates. This is achieved by solving the min-max problem of convergence factors using an asymptotic analysis. Numerical examples are included to illustrate the theoretical findings.
scientific article; zbMATH DE number 7608425
scientific article