An approach for solving interface problems is the diffuse interface theory, which was originally

developed as methodology for modeling and approximating solid-liquid phase transitions in which the effects

of surface tension and non-equilibrium thermodynamic behavior may be important at the surface. The

diffuse interface model describes the interface by a mixing energy represented as a layer of small thick-

ness. This idea can be traced to van der Waals, and is the foundation for the phase-field theory for phase

transition and critical phenomena. Thus, the structure of the interface is determined by molecular forces;

the tendencies for mixing and de-mixing are balanced through the non-local mixing energy. The method

uses an auxiliary function (so-called phase-field function) to localize the phases, assuming distinct values

in the bulk phases (for instance 1 in a phase and -1 in the other one) away from the interfacial regions

over which the phase function varies smoothly.

During the talk I will present the main ideas to approximate the Cahn-Hilliard model, a classical

Phase field model, introducing different numerical schemes and showing the advantage and disadvantages

of each scheme. The key point is to try to preserve the properties of the original models while the

numerical schemes are efficient in time.

Finally, I will show how these ideas for designing numerical schemes to approximate phase-fields

models can be extended to other applications.