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

Auxiliary Splines Space Preconditioning for B-Splines Finite Elements:  The case of $\bm{H}(\bm{curl},\Omega)$ and $\bm{H}(div,\Omega)$ elliptic  problems