Finite element approximation of an Allen-Cahn/Cahn-Hilliard system

We consider an Allen–Cahn/Cahn–Hilliard system witha non-degenerate mobility and (i) a logarithmic free energyand (ii) a non-smooth free energy (the deep quench limit). Thissystem arises in the modelling of phase separation and orderingin binary alloys. In particular we prove in each case that thereexists a unique solution for sufficiently smooth initial data.Further, we prove an error bound for a fully practical piecewiselinear finite element approximation of (i) and (ii) in one andtwo space dimensions (and three space dimensions for constantmobility). The error bound being optimal in the deep quenchlimit. In addition an iterative scheme for solving the resultingnonlinear discrete system is analysed. Finally some numericalexperiments are presented.