  Using the third-order WKB approximation and monodromy methods, we investigate
the influence of Lorentz violating coefficient $b$ (associated with a special
axial-vector $b_{\mu}$ field) on Dirac quasinormal modes in the Schwarzschild
black hole spacetime. At fundamental overtone, the real part decreases linearly
as the parameter $b$ increases. But the variation of the imaginary part with
$b$ becomes more complex. For the larger multiple moment $k$, the magnitude of
imaginary part increases with the increase of $b$, which means that presence of
Lorentz violation makes Dirac field damps more rapidly. At high overtones, it
is found that the real part of high-damped quasinormal frequency does not tend
to zero, which is quite a different from the symptotic Dirac quasinormal modes
without Lorentz violation.
