Differential complexes provide a natural framework for encoding the key algebraic structures involved in the analysis and computation of PDE systems and can be utilised to verify the satisfaction of conservation laws governing the physical phenomena described by these systems. In this talk, we present a framework for developing discrete numerical methods that preserve the key structural properties of the underlying continuous problems in arbitrary dimensions. Such a framework is essential as it provides a unified, dimension-independent approach and reveals structural insights that may not be apparent in lower-dimensional settings. We construct structure-preserving finite element methods on n-dimensional cubical meshes using the Stokes complex derived from the de Rham complex. We develop low-order, conforming finite element spaces that form an exact subcomplex of the Stokes complex at the discrete level and yield pointwise divergence-free velocity fields. Thus, we ensure the exact satisfaction of the conservation of mass for incompressible fluid flow at the discrete level. These elements exhibit enhanced stability and pressure-robustness and can be used in discretizing problems involving incompressible fluid flow in arbitrary dimensions. We then extend this framework via a tensor-product Bernstein–Gelfand–Gelfand (BGG) diagram to derive additional complexes, such as the elasticity and div–div complexes, involving polynomial spaces of arbitrary degree and regularity, as well as second-order differential operators. These complexes can be used to obtain finite element and spline discretizations of various problems in geometry and continuum mechanics.