Construction of exact refinements for the 2-dimensional HB/THB-spline de Rham complex
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When working with a hierarchical B-spline complex, it is not true in general that this complex will generate an exact subcomplex of its continuous counterpart for an arbitrary refinement, as shown in Evans et al. (2018), Shepherd and Toshniwal (2024). In this paper, we focus on providing an overview of the inexact refinements described above, as well as a way to add degrees of freedom to the hierarchical B-spline de Rham complex that transform them into exact refinements, thus avoiding spurious solutions. The proofs either work in general dimensions, or can easily be adapted to do so, but we mostly restrict ourselves to the 2-dimensional case, due to the significant simplifications in terms of the computational implementation.