A discrete fracture model for the flow of compressible, multicomponent fluids in homogeneous, heterogeneous, and fractured media is presented in single phase. In the numerical model we combine the mixed finite element (MFE) and the discontinuous Galerkin (DG) methods. We use the cross-flow equilibrium concept to approximate the fractured matrix mass transfer. The discrete fracture model is numerically superior to the single-porosity model and overcomes limitations of the dual-porosity models including the use of a shape factor. The MFE method provides a direct and accurate approximation for the velocity field, which is crucial for the convective terms in the flow equations. The DG method associated with a slope limiter is used to approximate the species balance equations. This method can capture the sharp moving fronts. The calculation of the fracture-fracture flux across three and higher intersecting fracture branches is a challenge. In this work, we provide an accurate approximation of these fluxes by using the MFE formulation. Numerical examples in unfractured and fractured media illustrate the efficiency and robustness of the proposed numerical model.