Hierarchical B-spline complexes of discrete differential forms


B-spline, discrete differentials, isogeometric analysis


In this paper we introduce the hierarchical B-spline complex of discrete differential forms for arbitrary spatial dimension. This complex may be applied to the adaptive isogeometric solution of problems arising in electromagnetics and fluid mechanics. We derive a sufficient and necessary condition guaranteeing exactness of the hierarchical B-spline complex for arbitrary spatial dimension, and we derive a set of local, easy-to-compute and sufficient exactness conditions for the two-dimensional setting. We examine the stability properties of the hierarchical B-spline complex, and we find that it yields stable approximations of both the Maxwell eigenproblem and Stokes problem provided that the local exactness conditions are satisfied. We conclude by providing numerical results showing the promise of the hierarchical B-spline complex in an adaptive isogeometric solution framework.

Original Publication Citation

J. A. Evans, M. A. Scott, K. M. Shepherd, D. C. Thomas, and R. V. Hern´andez. “Hierarchical B-spline complexes of discrete differential forms,” IMA Journal of Numerical Analysis, vol. 40, no. 1, p. 422–473, 2020.

Document Type

Peer-Reviewed Article

Publication Date



Oxford Academic




Ira A. Fulton College of Engineering


Civil and Environmental Engineering

University Standing at Time of Publication

Graduate Student