Tree-cotree gauging for two-dimensional hierarchical splines

TL;DR

This work tackles non-uniqueness in magnetostatics and eddy-current problems formulated with the magnetic vector potential by extending tree-cotree gauging to two-dimensional hierarchical B-splines on the Greville grid, with a dedicated spanning tree built per refinement level and a multi-level gauge $\mathcal{M}_{L} = \bigcup_{\ell=0}^{L} \mathcal{T}_{\ell}^{\mathcal{A}}$. Each level undergoes a minimum spanning tree construction (e.g., Kruskal) to select boundary, active, and deactivated edges, yielding a gauge that combines into a multi-level cotree while excluding deactivated basis functions from the hierarchical space. Numerical tests solve the Maxwell eigenproblem in $V_h \subset \mathbf{H}_0(\mathbf{curl};\Omega)$ with $\mathbf{E}_h \in V_h$ and unit material properties, showing that the gauged formulation removes zero eigenvalues and matches the nonzero spectrum of the ungauged problem for both $p=1$ and $p=3$, on domains with simple refinement and with a hole. Results validate the approach, demonstrating correct gauge behavior for hierarchical splines on 2D domains and indicating potential extension to the 3D case, including topology-sensitive gauges.

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.

Figures (1)

• Figure 1.3: Illustration of the trees built on each level for degree $p=1$, for a three-level mesh. The first three figures show the tree of each single level, where the whole gray region corresponds to $G_{\ell,\ell}$, and the dark gray region corresponds to $G_{\ell,\ell+1}$. At each level, we first build the tree on the boundary of $G_{\ell,\ell}$ (blue edges), then on the active edges (red edges), and finally on the deactivated edges (cyan edges). The right figure (d) shows the multi-level tree, combining active edges of the three levels.

Theorems & Definitions (2)

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