  Piezoelectric materials exhibit electric responses to mechanical stress, and
mechanical responses to electric stress. The PDE model, describing the
longitudinal oscillations on the beam, with two boundary feedback controllers
is known to have exponentially stable solutions. However, the reduced model by
the semi-discretized Finite Elements is shown to lack of exponential stability
uniformly as the discretization parameter tends to zero. This is due to the
loss of uniform gap among the high-frequency eigenvalues. In this paper, an
alternate Finite-Difference based model reduction is investigated by cleverly
reducing the order of the model together with the consideration of equidistant
grid points and averaging operators. This new model reduction successfully
retains the exponential stability uniformly as the discretization parameter
tends to zero. Moreover, it does not need a further numerical Fourier
filtering. Our results are based on a careful construction of a Lyapunov
function. The numerical simulations are provided to compare reduced models and
to show the strength of introduced results.
