On large time-stepping methods for the Cahn--Hilliard equation
Applied Numerical Mathematics
Journal of Scientific Computing
A class of stable spectral methods for the Cahn-Hilliard equation
Journal of Computational Physics
Error Estimation of a Class of Stable Spectral Approximation to the Cahn-Hilliard Equation
Journal of Scientific Computing
A nonconforming finite element method for the Cahn-Hilliard equation
Journal of Computational Physics
SIAM Journal on Numerical Analysis
An Adaptive Time-Stepping Strategy for the Molecular Beam Epitaxy Models
SIAM Journal on Scientific Computing
On linear schemes for a Cahn-Hilliard diffuse interface model
Journal of Computational Physics
Numerical and computational efficiency of solvers for two-phase problems
Computers & Mathematics with Applications
High accuracy solutions to energy gradient flows from material science models
Journal of Computational Physics
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We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation ut + Δ(ɛΔu−ɛ−1f(u)) = 0, where ɛ 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on ** only in some lower polynomial order for small ɛ. The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as ɛ → 0 in [29].