In this paper, we apply streamline-diffusion and Galerkin-least-squares finite element methods for 2D steady-state two-phase model in the cathode of polymer electrolyte fuel cell (PEFC) that contains a gas channel and a gas diffusion layer (GDL). This two-phase PEFC model is typically modeled by a modified Navier-Stokes equation for the mass and momentum, with Darcy's drag as an additional source term in momentum for flows through GDL, and a discontinuous and degenerate convection-diffusion equation for water concentration. Based on the mixed finite element method for the modified Navier-Stokes equation and standard finite element method for water equation, we design streamline-diffusion and Galerkin-least-squares to overcome the dominant convection arising from the gas channel. Meanwhile, we employ Kirchhoff transformation to deal with the discontinuous and degenerate diffusivity in water concentration. Numerical experiments demonstrate that our finite element methods, together with these numerical techniques, are able to get accurate physical solutions with fast convergence.