This paper presents a study of large linear systems resulting from the regular $B$-splines finite element discretization of the $\bm{curl}-\bm{curl}$ and $\bm{grad}-div$ elliptic problems on unit square/cube domains. We consider systems subject to both homogeneous essential and natural boundary conditions. Our objective is to develop a preconditioning strategy that is optimal and robust, based on the Auxiliary Space Preconditioning method proposed by Hiptmair et al. \cite{hiptmair2007nodal}. Our approach is demonstrated to be robust with respect to mesh size, and we also show how it can be combined with the Generalized Locally Toeplitz (GLT) sequences analysis presented in \cite{mazza2019isogeometric} to derive an algorithm that is optimal and stable with respect to spline degree. Numerical tests are conducted to illustrate the effectiveness of our approach.

翻译：本文介绍了对单位平方/立方体域中由$\bm{curl}-\bm{curl}-\bm{curl}}美元和$\bm{grad}-div美元等离散元素生成的大型线性系统的研究。 我们考虑的系统既具有同质的基本条件,也具有自然边界条件。 我们的目标是根据Hiptmair et al.\cite{hittmair2007nodal} 提出的辅助空间先质调节方法,制定一项最佳和稳健的前提条件战略。 我们的方法在网状大小方面表现得非常有力, 我们还展示了如何与在\cite{mazzazzal2019logizology}中介绍的通用局部托普利茨(GLT)序列分析相结合,以得出与样度最佳和稳定的算法。 进行了数值测试,以说明我们的方法的有效性。</s>