A 2D finite element model is presented for the prediction of the advancing front in the thickness of a part during mold filling in the polymer injection molding process. The full Navier-Stokes equations are considered for the determination of the velocity and the pressure. The computation of the flow front uses the Arbitrary Lagrangian-Eulerian approach and is coupled with an automatic remeshing technique triggered only when the mesh becomes distorted. Three series of numerical tests with a Newtonian fluid are presented. The first one deals with the fountain-flow in a flat gap and a tube and allows for validation of the code. The second example in a 90-degrees bend illustrates the capability of the proposed method to deal with multiple fronts, and the third one deals with jetting.