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Abstract
Abstract Given a domain $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on $$\Omega $$ Ω , and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When $$\Omega $$ Ω is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such $$\Omega $$ Ω , we theoretically analyze the discrete de Rham complex built from hierarchical B-spline differential forms, i.e., the discrete differential forms are smooth splines and support adaptive refinements—these properties are key to enabling accurate and efficient numerical simulations. We provide locally-verifiable sufficient conditions that ensure that the discrete spline complex is exact. Numerical tests are presented to support the theoretical results, and the examples discussed include complexes that satisfy our prescribed conditions as well as those that violate them.
Highlights
While many partial differential equations (PDEs) may be couched as minimization problems, a large swath of them are more naturally described as saddle-point problems
We have presented a theoretical analysis of a discrete de Rham complex built using hierarchical B-splines on a hypercube ⊂ Rn
We have presented locally-verifiable conditions that are sufficient for ensuring exactness of this discrete de Rham complex
Summary
While many partial differential equations (PDEs) may be couched as minimization problems, a large swath of them are more naturally described as saddle-point problems. A generalization of the classical finite element method [17], the isogeometric analysis philosophy relies on the use of spline functions [9, 23] for describing both the domain on which the problem is posed as well as the discrete solution. One objective of this approach is to greatly simplify the application of numerical methods to geometries of engineering interest, which are themselves designed within computer-aided design software using spline functions [13]. Note that this assumption is locally-verifiable, i.e., it can be checked in a local manner for any given hierarchical B-spline mesh We start this manuscript by recalling the basics of the de Rham complex of differential forms
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