In this talk, 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 present a sufficient and necessary condition guaranteeing exactness of the hierarchical B-spline complex for arbitrary spatial dimension, as well as a set of local, easy-to-compute, and sufficient exactness conditions for the two-dimensional setting. Abstract results from cohomology theory are employed to prove these conditions. We examine the stability properties of the hierarchical B-spline complex through numerical example, and we demonstrate that the complex 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.
Co-Authors: Michael A. Scott, Kendrick Shepherd, Derek Thomas, Rafael Vazquez
Thackeray Hall 427