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：本文研究了在单位正方形/立方体域上，由正则$B$-样条有限元离散$\bm{curl}-\bm{curl}$和$\bm{grad}-div$椭圆问题所产生的大型线性系统。我们考虑了同时满足齐次本质边界条件和自然边界条件的系统。我们的目标是基于Hiptmair等人提出的辅助空间预条件方法\cite{hiptmair2007nodal}，开发一种最优且鲁棒的预条件策略。所提出的方法被证明对网格尺寸具有鲁棒性，同时我们还展示了如何将其与文献\cite{mazza2019isogeometric}中的广义局部Toeplitz（GLT）序列分析相结合，推导出一种对样条阶数最优且稳定的算法。通过数值测试验证了该方法的有效性。