An a priori analysis for a generalized local projection stabilized finite element solution of the Darcy equations is presented in this paper. A first-order nonconforming \(\mathbb {P}^{nc}_{1}\) finite element space is used to approximate the velocity, whereas the pressure is approximated using two different finite elements, namely piecewise constant \(\mathbb {P}_{0}\) and piecewise linear nonconforming \(\mathbb {P}^{nc}_{1}\) elements. The considered finite element pairs, \(\mathbb {P}^{nc}_{1}/\mathbb {P}_{0}\) and \(\mathbb {P}^{nc}_{1}/\mathbb {P}^{nc}_{1} \), are inconsistent and incompatibility, respectively, for the Darcy problem. The stabilized discrete bilinear form satisfies an inf-sup condition with a generalized local projection norm. Moreover, a priori error estimates are established for both finite element pairs. Finally, the validation of the proposed stabilization scheme is demonstrated with appropriate numerical examples.

本文对Darcy方程的广义局部投影稳定有限元解进行了先验分析。一阶不合格\（\ mathbb {P} ^ {nc} _ {1} \）有限元空间用于近似速度，而压力使用两个不同的有限元近似，即分段常数\（\ mathbb {P} _ {0} \）和分段线性不合格\（\ mathbb {P} ^ {nc} _ {1} \）元素。所考虑的有限元对\（\ mathbb {P} ^ {nc} _ {1} / \ mathbb {P} _ {0} \）和\（\ mathbb {P} ^ {nc} _ {1} / \ mathbb {P} ^ {nc} _ {1} \）对于Darcy问题，分别是不一致和不兼容的。稳定的离散双线性形式满足具有广义局部投影范数的insup条件。此外，针对两个有限元对建立先验误差估计。最后，通过适当的数值示例证明了所提出的稳定方案的有效性。