Stable discretizations of the Cahn-Hilliard-Gurtin equations

We study space and time discretizations of the Cahn-Hilliard-Gurtin equations with a polynomial nonlinearity. We first consider a space semi-discrete version of the equations, and we prove in particular that any solution converges to a steady state (as in the continuous case). Then, we study the numerical stability of the fully discrete scheme obtained by applying the Euler backward scheme to the space semi-discrete problem. In particular, we show that this fully discrete problem is unconditionally stable. Numerical simulations in one space dimension conclude the paper.

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