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.

翻译：本文针对广义达西问题（间断系数的椭圆方程），研究了一种特殊的Galerkin-最小二乘方法（即增广混合有限元方法）及其与标准最小二乘有限元方法的关系。本文提出了两种具有稳健先验和后验误差估计的增广混合有限元方法。作为部分最小二乘方法，增广混合有限元方法比原始的最小二乘有限元方法更具灵活性。增广混合有限元方法的后验误差估计器是对应最小二乘泛函中数值解的评估。作为对比，我们讨论了紧密相关的最小二乘有限元方法的非稳健性。通过数值实验验证了我们的发现。