Hierarchical B-spline complexes of discrete differential forms

TL;DR

This paper introduces a hierarchical B-spline complex for discrete differential forms applicable in adaptive isogeometric analysis, ensuring exactness and stability in electromagnetics and fluid mechanics problems across arbitrary dimensions.

Contribution

It develops a new hierarchical B-spline complex framework with necessary and sufficient conditions for exactness in any dimension, and analyzes its stability and applicability.

Findings

Ensures exactness of the hierarchical B-spline complex under specific conditions.

Provides stable approximations for Maxwell and Stokes problems.

Demonstrates promising numerical results in adaptive isogeometric solutions.

Abstract

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.