Hierarchical B-spline complexes of discrete differential forms
This work provides a theoretical and practical framework for adaptive isogeometric analysis in electromagnetics and fluid mechanics, but the results are incremental as they extend existing B-spline complexes to hierarchical settings.
The paper introduces hierarchical B-spline complexes of discrete differential forms for arbitrary spatial dimension, enabling adaptive isogeometric solutions for electromagnetics and fluid mechanics. It provides exactness conditions and demonstrates stable approximations for Maxwell eigenproblems and Stokes problems, with numerical results showing promise.
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.