Precise evaluation of the dynamics of granular material during hopper discharge, particularly the velocity field coupled with the stress field, has been an important area of research for many years. In this paper, a finite element method (FEM) based on the Eulerian formulation is described and validated to meet this need. It is demonstrated that this method outperforms the ordinary Lagrangian-formulation method in resolving the problem of mesh distortion, thereby is capable of simulating the complete emptying process of a hopper. On this basis, various discharge behaviours of a conical hopper are studied, including the mass and funnel flow modes, the mass flow rate and the wall pressure. The results are in general agreement with those from experiments and recognized correlations in all the examined aspects, which validates the applicability of the Eulerian FEM. This continuum method ought to be of practical significance because it is computationally tractable, and viable in dealing with complex silo geometries and variable flow patterns of granular materials.