Tree-cotree gauging for two-dimensional hierarchical splines

TL;DR

This paper extends tree-cotree gauging techniques to hierarchical splines, enabling gauge fixing in adaptive isogeometric analysis, with numerical validation on Maxwell eigenvalue problems.

Contribution

It introduces a method to construct spanning trees for hierarchical splines, generalizing existing techniques to multi-level spline spaces for improved gauge fixing.

Findings

The method is valid for degree p=1 splines and finite element meshes with hanging nodes.

Numerical results confirm the correctness of the proposed gauging technique.

The approach enhances adaptive isogeometric analysis in electromagnetics.

Abstract

In magnetostatics and eddy current problems, formulated in terms of the magnetic vector potential, the solution is not unique, because the addition of an irrotational function to the solution remains a valid solution. The tree-cotree decomposition is a gauging technique to recover uniqueness when using finite elements, which consists in considering the mesh as a graph, and building a spanning tree on that graph. The idea has been recently extended to isogeometric analysis, applying the construction of the spanning tree on the control mesh, or equivalently, on the Greville grid. In the present paper we extend the construction to hierarchical splines, a set of splines with multi-level structure for adaptive refinement, by constructing a spanning tree for each single level. Since for degree $p=1$ the spaces of finite elements and hierarchical splines coincide, the presented construction is also valid for quadrilateral finite element meshes with hanging nodes. To assess the correctness of the method, we present numerical results for Maxwell eigenvalue problem.