A second order splitting method for the Cahn-Hilliard equation. (English) Zbl 0668.65097

The Cahn-Hilliard equation models the phase separation which occurs upon the quenching of a binary mixture into the spinodal region: \[ (1)\quad \partial u/\partial t=\Delta \phi (u)-\gamma \Delta^ 2u\quad x\in \Omega,\quad t>0, \] plus initial and boundary conditions and with \(\phi (u)=\psi '(u);\quad \psi (u)=\gamma_ 2(u^ 4/4)+\gamma_ 1(u^ 4/4)+\gamma_ 1(u^ 3/3)+(\gamma_ 0/2)u^ 2;\) \(\gamma_ 2>0\). Here u(x,t) means the concentration of one component of such a mixture. By introducing the chemical potential \(w=\phi (u)-\gamma \Delta u,\) the author substitutes to (1) a second order system of partial differential equations. Then, he considers a semi-discrete finite element method (in space) which requires only continuous elements.
The optimal order error bounds are derived in various norms for an implementation which uses mass lumping. The continuous problem has an energy based Lyapunov functional and it proved that this property still holds for the discrete problem.
Reviewer: M.Bernadou


65Z05 Applications to the sciences
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76T99 Multiphase and multicomponent flows
76Z05 Physiological flows
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