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Stability and convergence of a second-order mixed finite element method for the Cahn-Hilliard equation. (English) Zbl 1433.80005

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.

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
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