Diegel, Amanda E.; Wang, Cheng; Wise, Steven M. Stability and convergence of a second-order mixed finite element method for the Cahn-Hilliard equation. (English) Zbl 1433.80005 IMA J. Numer. Anal. 36, No. 4, 1867-1897 (2016). Summary: In this paper, we devise and analyse an unconditionally stable, second-order-in-time numerical scheme for the Cahn-Hilliard equation in two and three space dimensions. We prove that our two-step scheme is unconditionally energy stable and unconditionally uniquely solvable. Furthermore, we show that the discrete phase variable is bounded in \(L^\infty (0,T; L^\infty)\) and the discrete chemical potential is bounded in \(L^\infty (0,T; L^2)\), for any time and space step sizes, in two and three dimensions, and for any finite final time \(T\). We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions. We include in this work a detailed analysis of the initialization of the two-step scheme. Cited in 96 Documents MSC: 80M10 Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 80A22 Stefan problems, phase changes, etc. 82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics 82M10 Finite element, Galerkin and related methods applied to problems in statistical mechanics Keywords:Cahn-Hilliard equation; spinodal decomposition; mixed finite element methods; energy stability; error estimates; second-order accuracy PDFBibTeX XMLCite \textit{A. E. Diegel} et al., IMA J. Numer. Anal. 36, No. 4, 1867--1897 (2016; Zbl 1433.80005) Full Text: DOI arXiv