Among the multiple stages of the resin transfer molding (RTM) processes, flow and mold filling of injected resin correspond to the most complex and crucial stage. During the latter, air bubble agglomeration must be avoided and complete wetting of fibers must be achieved in order to ensure the maximum quality of the parts at the lowest possible manufacturing time. Focusing on these manufacturing issues, a mathematical model and a numerical resolution are presented to predict the resin flow throughout the fiber reinforcement inside the mold cavity. The methodology employs conventional finite element techniques for solving the flow problem through a porous medium governed by Darcy's law and mass conservation. Simultaneously, a state of the art numerical scheme known as the discontinuous Galerkin method is implemented to determine the location and shape of the advancing flow fronts ruled by a hyperbolic transport equation. These two schemes are implemented to work with a two-dimensional domain, handling diverse geometries with multiple injection and ventilation ports. The results for key process parameters, such as filling time and position of the advancing flow fronts, show a good agreement with results from analytical solutions for particular cases and from empirical data. When several simulated results are taken into account in the design process of RTM cavities, the overall process could be enhanced.