Augmented mixed finite element methods for elliptic equations with discontinuous coefficients: Robust A Priori and A Posteriori Error Estimates

In this paper, for the generalized Darcy problem (an elliptic equation with discontinuous coefficients), we study a special 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. As partial least-squares methods, the augmented mixed finite element methods are more flexible than the original least-squares finite element methods. The a posteriori error estimators of the augmented mixed finite element methods are the evaluations of the numerical solutions at the corresponding least-squares functionals. As comparisons, we discuss the non-robustness of the closely related least-squares finite element methods. Numerical experiments are presented to verify our findings.