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Given a domain $\Omega \subset \mathbb{R}^n$, the de Rham complex ofdifferential forms arises naturally in the study of problems inelectromagnetism and fluid mechanics defined on $\Omega$, and itsdiscretization helps build stable numerical methods for such problems. Forconstructing such stable methods, one critical requirement is ensuring that thediscrete subcomplex is cohomologically equivalent to the continuous complex.When $\Omega$ is a hypercube, we thus require that the discrete subcomplex beexact. Focusing on such $\Omega$, we theoretically analyze the discrete de Rhamcomplex built from hierarchical B-spline differential forms, i.e., the discretedifferential forms are smooth splines and support adaptive refinements - theseproperties are key to enabling accurate and efficient numerical simulations. Weprovide locally-verifiable sufficient conditions that ensure that the discretespline complex is exact. Numerical tests are presented to support thetheoretical results, and the examples discussed include complexes that satisfyour prescribed conditions as well as those that violate them.

Locally-verifiable sufficient conditions for exactness of the hierarchical B-spline discrete de Rham complex in $\mathbb{R}^n$