Convergence to equilibrium for a second-order time semi-discretization of the Cahn-Hilliard equation

Paola Francesca Antonietti (1) , Benoît Merlet (2) , Morgan Pierre (3) , Marco Verani (1)

1 MOX - Modeling and Scientific Computing [Milano]
2 RAPSODI - Reliable numerical approximations of dissipative systems
3 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]

We consider a second-order two-step time semi-discretization of the Cahn-Hilliard equation with an analytic nonlinearity. The time-step is chosen small enough so that the pseudo-energy associated with the discretization is nonin-creasing at every time iteration. We prove that the sequence generated by the scheme converges to a steady state as time tends to infinity. We also obtain convergence rates in the energy norm. The proof is based on the Lojasiewicz-Simon inequality.

Mots clés: backward differentiation formula; gradient flow; Cahn-Hilliard; Lojasiewicz-Simon inequality

Domaines: Analyse numérique [math.NA]

Paola Francesca Antonietti, Benoît Merlet, Morgan Pierre, Marco Verani. Convergence to equilibrium for a second-order time semi-discretization of the Cahn-Hilliard equation. AIMS Mathematics, 2016, Nonlinear Evolution PDEs, Interfaces and Applications, 1 (3), pp.178-194. ⟨10.3934/Math.2016.3.178⟩. ⟨hal-01355956⟩