  In the first part of this article, we study feedback stabilization of a
parabolic coupled system by using localized interior controls. The system is
feedback stabilizable with exponential decay $-\omega<0$ for any $\omega>0$. A
stabilizing control is found in feedback form by solving a suitable algebraic
Riccati equation. In the second part, a conforming finite element method is
employed to approximate the continuous system by a finite dimensional discrete
system. The approximated system is also feedback stabilizable (uniformly) with
exponential decay $-\omega+\epsilon$, for any $\epsilon>0$ and the feedback
control is obtained by solving a discrete algebraic Riccati equation. The error
estimate of stabilized solutions as well as stabilizing feedback controls are
obtained. We validate the theoretical results by numerical implementations.
