The first-order system LL* (FOSLL*) approach for general second-order
elliptic partial differential equations was proposed and analyzed in [10], in
order to retain the full efficiency of the L2 norm first-order system
least-squares (FOSLS) ap- proach while exhibiting the generality of the
inverse-norm FOSLS approach. The FOSLL* approach in [10] was applied to the
div-curl system with added slack vari- ables, and hence it is quite
complicated. In this paper, we apply the FOSLL* approach to the div system and
establish its well-posedness. For the corresponding finite ele- ment
approximation, we obtain a quasi-optimal a priori error bound under the same
regularity assumption as the standard Galerkin method, but without the
restriction to sufficiently small mesh size. Unlike the FOSLS approach, the
FOSLL* approach does not have a free a posteriori error estimator, we then
propose an explicit residual error estimator and establish its reliability and
efficiency bound