  We investigate torsional Alfv\'en oscillations of relativistic stars with a
global dipole magnetic field, via two-dimensional numerical simulations. We
find that a) there exist two families of quasi-periodic oscillations (QPOs)
with harmonics at integer multiples of the fundamental frequency, b) the
lower-frequency QPO is related to the region of closed field lines, near the
equator, while the higher-frequency QPO is generated near the magnetic axis, c)
the QPOs are long-lived, d) for the chosen form of dipolar magnetic field, the
frequency ratio of the lower to upper fundamental QPOs is ~0.6, independent of
the equilibrium model or of the strength of the magnetic field, and e) within a
representative sample of equations of state and of various magnetar masses, the
Alfv\'en QPO frequencies are given by accurate empirical relations that depend
only on the compactness of the star and on the magnetic field strength. The
lower and upper QPOs can be interpreted as corresponding to the edges or
turning points of an Alfv\'en continuum, according to the model proposed by
Levin (2007). Several of the low-frequency QPOs observed in the X-ray tail of
SGR 1806-20 can readily be identified with the Alfv\'en QPOs we compute. In
particular, one could identify the 18Hz and 30Hz observed frequencies with the
fundamental lower and upper QPOs, correspondingly, while the observed
frequencies of 92Hz and 150Hz are then integer multiples of the fundamental
upper QPO frequency (three times and five times, correspondingly). With this
identification, we obtain an upper limit on the strength of magnetic field of
SGR 1806-20 (if is dominated by a dipolar component) between ~3 and $7\times
10^{15}$G. Furthermore, we discuss the implications for the high-density EOS of
compact stars. (Abridged)
