  In this paper, for the generalized Darcy problem (an elliptic equation with
discontinuous coefficients), we study a special partial Least-Squares
(Galerkin-least-squares) method, known as the augmented mixed finite element
method, and its relationship to the standard least-squares finite element
method (LSFEM). Two versions of augmented mixed finite element methods are
proposed in the paper with robust a priori and a posteriori error estimates.
Augmented mixed finite element methods and the standard LSFEM uses the same a
posteriori error estimator: the evaluations of numerical solutions at the
corresponding least-squares functionals. As partial least-squares methods, the
augmented mixed finite element methods are more flexible than the original
LSFEMs. As comparisons, we discuss the mild non-robustness of a priori and a
posteriori error estimates of the original LSFEMs. A special case that the
$L^2$-based LSFEM is robust is also presented for the first time. Extensive
numerical experiments are presented to verify our findings.
