Through the seminal works of Buffa et al., the fruitful integration of the two discretization paradigms of Isogeometric Analysis (IGA) and Finite Element Exterior Calculus (FEEC) was demonstrated already in 2011. The latter concept explores and provides design steps for the approximation of so-called abstract Hodge–Laplace problems, which, in turn, represent PDEs derived from Hilbert complexes. The introduction of isogeometric discrete differential forms in [1] laid the foundation for discretizing classical de Rham complexes in a structure-preserving manner using B-splines, meaning the discretization of Hodge–Laplace problems derived from de Rham sequences. Although the FEEC theory was developed for general closed Hilbert complexes, and while Hilbert sequences play a role in various physical applications, connecting IGA and FEEC often proves challenging or is sometimes not directly clear. This is especially true for Hilbert complexes that also encompass differential operators of higher orders. We present two approaches to obtain well-posed discretizations of a whole class of Hodge–Laplace problems using IGA. We focus on mixed weak formulations of saddle-point structure and second-order Hilbert complexes. In particular, we go beyond the standard de Rham case and demonstrate that ideas from FEEC and IGA are useful for non-de Rham chains as well. A central tool for describing the underlying settings and for choosing the finite element spaces is the Bernstein–Gelfand–Gelfand (BGG) construction considered by Arnold and Hu in [2]. Our approach allows us to incorporate geometries with curved boundaries, which is not directly possible with classical FEEC approaches, and also provides suitable discretizations in arbitrary dimensions. We show error estimates for both approximation methods and explain their applicability in the field of linear elasticity theory. The theoretical discussions and estimates are further illustrated with various numerical examples performed utilizing the GeoPDEs package [3]. REFERENCES [1] A. Buffa, J. Rivas, G. Sangalli and R. Vázquez. Isogeometric discrete differential forms in three dimensions. SIAM J. Numer. Anal., 49:818-844, 2011. [2] D. N. Arnold and K. Hu. Complexes from complexes. Foundations of Computational Mathematics, 21:1739–1774, 2021. [3] C. de Falco, A. Reali and R. Vázquez. GeoPDEs: A research tool for isogeometric analysis of PDEs. Advances in Engineering Software, 42:1020-1034, 2011.