  In this paper we discuss the error estimations for the div least-squares
finite element method on elliptic problems. Compared with the previous work, we
present a complete error analysis, which improves the current
\emph{state-of-the-art} results. The error estimations for both the scalar and
the flux variables are established by dual arguments, and in most cases, only
an $H^{1+\varepsilon}$ regularity is used. Numerical experiments strongly
confirm our analysis.
