  In this work, we analyze porous elastic system with microtemperature from
second spectrum viewpoint. Indeed, by using the classical Faedo-Galerkin method
combined with the a priori estimates, we prove the existence and uniqueness of
a global solution of this problem. Then we prove that this solution is
exponentially stable without assuming the condition of equal wave speeds. Then,
we introduce a finite element approximation and we prove that the associated
discrete energy decays. Finally, we obtain some a priori error estimates
assuming additional regularity on the solution and we present some numerical
results which demonstrate the accuracy of the approximation and the behaviour
of the solution
