Optimized Ventcel-Schwarz methods for the Cahn-Hilliard equation discretized by the stabilized linear Crank-Nicolson scheme

Abstract: The stabilized linear Crank-Nicolson (SL-CN) scheme is a very important time discretization for the Cahn-Hilliard (CH) equation since it is an unconditionally energy stable method of second order, and allows to use time steps as large as possible to reduce the total calculation. Though, it still requires a very large amount of calculations for simulating the CH equation because of the essential nature of the CH equation. To accelerate the simulation process, we propose in this paper 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, we derive using Fourier analysis the convergence factor, which takes on two different forms according to the size of the time steps. By solving the hard min-max problem of convergence factors using asymptotic analysis, we rigorously optimized the convergence factors for each case and obtained the optimized transmission parameters in explicit form, and the estimate for the corresponding convergence rates. The theoretical results are illustrated by several numerical examples.
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