Construction of exact refinements for the two-dimensional hierarchical B-spline de Rham complex

TL;DR

This paper develops a constructive algorithm to refine hierarchical B-spline spaces for the 2D de Rham complex, ensuring structure preservation, admissibility, and improved numerical stability in electromagnetism and fluid mechanics simulations.

Contribution

It introduces a new algorithm for exact refinement of hierarchical B-spline complexes that maintains structure and admissibility, enhancing numerical stability and convergence.

Findings

The algorithm guarantees structure preservation during refinement.

Admissibility is maintained across all complex spaces under common restrictions.

Numerical results show improved stability in Maxwell and vector Laplace problems.

Abstract

The de Rham complex arises naturally when studying problems in electromagnetism and fluid mechanics. Stable numerical methods to solve these problems can be obtained by using a discrete de Rham complex that preserves the structure of the continuous one. This property is not necessarily guaranteed when the discrete function spaces are hierarchical B-splines, and research shows that an arbitrary choice of refinement domains may give rise to spurious harmonic fields that ruin the accuracy of the solution. We will focus on the two-dimensional de Rham complex over the unit square $\Omega \subseteq \mathbb{R}^2$, and provide theoretical results and a constructive algorithm to ensure that the structure of the complex is preserved: when a pair of functions are in conflict some additional functions, forming an L-chain between the pair, are also refined. Another crucial aspect to consider in the hierarchical setting is the notion of admissibility, as it is possible to obtain optimal convergence rates of numerical solutions and improved stability by limiting the multi-level interaction of basis functions. We show that, under a common restriction, the admissibility class of the first space of the discrete complex persists throughout the remaining spaces. As such, admissible refinement can be combined with our new algorithm to obtain admissible meshes that also respect the structure of the de Rham complex. Moreover, we detail how our algorithm can be easily included in standard adaptive mesh refinement schemes. Finally, we include numerical results that motivate the importance of the previous concerns for the vector Laplace and Maxwell eigenvalue problems.