We investigate the homogeneous Dirichlet problem for the Fast Diffusion Equation u_t=\Delta u^m, posed in a smooth bounded domain \Omega\subset \mathbb{R}^N, in the exponent range m_s=(N-2)_+/(N+2)<m<1. It is known that bounded positive solutions extinguish in a finite time T>0, and also that they approach a separate variable solution u(t,x)\sim (T-t)^{1/(1-m)}S(x), as t\to T^-. It has been shown recently that v(x,t)=u(t,x)\,(T-t)^{-1/(1-m)} tends to S(x) as t\to T^-, uniformly in the relative error norm. Starting from this result, we investigate the fine asymptotic behaviour and prove sharp rates of convergence for the relative error. The proof is based on an entropy method relying on a (improved) weighted Poincaré inequality, that we show to be true on generic bounded domains. Another essential aspect of the method is the new concept of "almost orthogonality", which can be thought as a nonlinear analogous of the classical orthogonality condition needed to obtain improved Poincaré inequalities and sharp convergence rates for linear flows.
