  Abstract Interpretation approximates the semantics of a program by mimicking
its concrete fixpoint computation on an abstract domain $\mathbb{A}$. The
abstract (post-) fixpoint computation is classically divided into two phases:
the ascending phase, using widenings as extrapolation operators to enforce
termination, is followed by a descending phase, using narrowings as
interpolation operators, so as to mitigate the effect of the precision losses
introduced by widenings. In this paper we propose a simple variation of this
classical approach where, to more effectively recover precision, we decouple
the two phases: in particular, before starting the descending phase, we replace
the domain $\mathbb{A}$ with a more precise abstract domain $\mathbb{D}$. The
correctness of the approach is justified by casting it as an instance of the
A$^2$I framework. After demonstrating the new technique on a simple example, we
summarize the results of a preliminary experimental evaluation, showing that it
is able to obtain significant precision improvements for several choices of the
domains $\mathbb{A}$ and $\mathbb{D}$.
