  Fully nonlinear numerical simulation of two dimensional Faraday waves between
two incompressible and immiscible fluids is performed by adopting the
phase-field method with the Cahn-Hilliard equation due to Jacqmin (1999) [J.
Comput. Phys., v.155, 96]. Its validation is checked against the linear theory.
In a nonlinear regime, qualitative comparison is made with an earlier
vortex-sheet simulation of two dimensional Faraday waves by Wright et al.
(2000) [J. Fluid Mech., v.400, 1]. The vorticity outside the interface region
is studied in this comparison. The period tripling state, which is observed in
the quasi-two dimensional experiment by Jiang et al. (1998) [J. Fluid Mech.,
v.369, 273], is successfully simulated with the present phase-field method.
