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

Least-Squares versus Partial Least-Squares Finite Element Methods:  Robust A Priori and A Posteriori Error Estimates of Augmented Mixed Finite  Element Methods