  We consider the Fast Diffusion Equation $u_t=\Delta u^m$ posed in a bounded
smooth domain $\Omega\subset \RR^d$ with homogeneous Dirichlet conditions; the
exponent range is $m_s=(d-2)_+/(d+2)<m<1$. It is known that bounded positive
solutions $u(t,x)$ of such problem extinguish in a finite time $T$, and also
that such solutions approach a separate variable solution $u(t,x)\sim
(T-t)^{1/(1-m)}S(x)$, as $t\to T^-$. Here we are interested in describing the
behaviour of the solutions near the extinction time. We first show that the
convergence $u(t,x)\,(T-t)^{-1/(1-m)}$ to $S(x)$ takes place uniformly in the
relative error norm. Then, we study the question of rates of convergence of the
rescaled flow. For $m$ close to 1 we get such rates by means of entropy methods
and weighted Poincar\'e inequalities. The analysis of the latter point makes an
essential use of fine properties of the associated stationary elliptic problem
$-\Delta S^m= {\bf c} S$ in the limit $m\to 1$, and such a study has an
independent interest.
