This paper extends the analysis of the self-excitated vibrations of a drilling structure presented in an earlier paper [T. Richard, C. Germay, E. Detournay, A simplified model to explore the root cause of stick-slip vibrations in drilling systems with drag bits, Journal of Sound and Vibration 305 (3) (2007) 432-456] by basing the formulation of the model on a continuum representation of the drillstring rather than on a characterization of the drilling structure by a 2 degree of freedom system. The particular boundary conditions at the bit-rock interface, which according to this model are responsible for the self-excited vibrations, account for both cutting and frictional contact processes. The cutting process combined with the quasi-helical motion of the bit leads to a regenerative effect that introduces a coupling between the axial and torsional modes of vibrations and a state-dependent delay in the governing equations, while the frictional contact process is associated with discontinuities in the boundary conditions when the bit sticks in its axial and angular motion. The dynamic response of the drilling structure is computed using the finite element method. While the general tendencies of the system response predicted by the discrete model are confirmed by this computational model (for example that the occurrence of stick-slip vibrations as well as the risk of bit bouncing are enhanced with an increase of the weight-on-bit or a decrease of the rotational speed), new features in the self-excited response of the drillstring can be detected. In particular, stick-slip vibrations are predicted to occur at natural frequencies of the drillstring different from the fundamental one (as sometimes observed in field operations), depending on the operating parameters.