ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

On fully practical finite element approximations of degenerate Cahn-Hilliard systems

Barrett, John W.a1, Blowey, James F.a2 and Garcke, Haralda3

a1 Department of Mathematics, Imperial College, London, SW7 2BZ, UK. (

a2 Department of Mathematical Sciences, University of Durham, DH1 3LE, UK.

a3 Institut für Angewandte Mathematik, Wegelerstraße 6, 53115 Bonn, Germany.


We consider a model for phase separation of a multi-component alloy with non-smooth free energy and a degenerate mobility matrix. In addition to showing well-posedness and stability bounds for our approximation, we prove convergence in one space dimension. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. We discuss also how our approximation has to be modified in order to be applicable to a logarithmic free energy. Finally numerical experiments with three components in one and two space dimensions are presented.

(Received April 7 2000)

(Online publication April 15 2002)

Key Words:

  • Phase separation;
  • multi-component systems;
  • degenerate parabolic systems of fourth order;
  • finite element method;
  • convergence analysis.

Mathematics Subject Classification:

  • 35K35;
  • 35K55;
  • 35K65;
  • 65M12;
  • 65M60;
  • 82C26