This paper introduces a novel adaptive refinement strategy for Isogeometric Analysis (IGA) using Truncated Hierarchical B-splines (THB-splines). The strategy is motivated by the fact that certain applications may benefit from adaptive refinement schemes, which lead to a higher degree of structure in the locally-refined mesh than usual, and building this structure a priori can simplify the implementation in those contexts. Specifically, we look at two applications: formulation of an L²-stable local projector for THB-splines a la Bézier projection [Dijkstra and Toshniwal (2023)], and adaptive structure-preserving discretizations using THB-splines [Evans et al. (2020), Shepherd and Toshniwal (2024)]. Previously proposed approaches for these applications require mesh modifications to preserve critical properties of the spline spaces, such as local linear independence or the exactness of the discrete de Rham complexes. Instead, we propose a macro-element-based refinement approach based on refining q=q₁×⋯×q_n blocks of elements, termed q-boxes, where the block size q is chosen based on the spline degree p and the specific application. • For the Bézier projection for THB-splines, we refine p-boxes (i.e., q=p). We show that THB-splines are locally linearly independent on p-boxes, which allows for a simple extension of the Bézier projection algorithm to THB-splines. This new formulation significantly improves upon the approach previously proposed by Dijkstra and Toshniwal (2023). • For structure-preserving discretizations, we refine (p+1)-boxes (i.e., q=p+1). We prove that this choice of q ensures that the mesh satisfies the sufficient conditions presented in Shepherd and Toshniwal (2024) for guaranteeing the exactness of the THB-spline de Rham complex a priori and in an arbitrary number of dimensions. This is crucial for structure-preserving discretizations, as it eliminates the need for additional mesh modifications to maintain the exactness of the complex during adaptive simulations. The effectiveness of the proposed framework is demonstrated through theoretical proofs and numerical experiments, including optimal convergence for adaptive approximation and the simulation of the incompressible Navier-Stokes equations.

This paper introduces a Variational Multiscale Stabilization (VMS) formulation of the incompressible Navier–Stokes equations that utilizes the Finite Element Exterior Calculus (FEEC) framework. The FEEC framework preserves the geometric and topological structure of continuous spaces and PDEs in the discrete spaces and model, and helps build stable and convergent discretizations. For the Navier–Stokes equations, this structure is encoded in the de Rham complex. In this work, we consider the vorticity–velocity–pressure formulation discretized within the FEEC framework. We model the effect of the unresolved scales on the finite-dimensional solution by introducing appropriate fine-scale governing equations, which we also discretize using the FEEC approach. This preserves the structure of the continuous problem in both the coarse- and fine-scale solutions; for instance, both the coarse- and fine-scale velocities are pointwise incompressible. We demonstrate that the resulting formulation is residual-based, energetically stable, and optimally convergent. Moreover, our fine-scale model allows for an efficient computational approach — by decoupling fine-scale problems on one element from another, it is possible to solve them in parallel. In fact, the fine-scale equations can be entirely eliminated during matrix assembly, thereby leading to a VMS formulation where the problem size is governed by the coarse-scale discretization only. Finally, the proposed formulation applies to both the lowest regularity discretizations of the de Rham complex and high-regularity isogeometric discretizations. We validate our theoretical results through numerical experiments, simulating steady-, unsteady-, viscous-, and inviscid-flow problems, and empirically demonstrating Reynolds robustness of the formulation. These tests show that the stabilized solutions are qualitatively better than the unstabilized ones, converge at optimal rates, and, as the mesh is refined, the stabilization is asymptotically turned off.

In this paper we propose a local projector for truncated hierarchical B-splines (THB-splines). The local THB-spline projector is an adaptation of the Bézier projector proposed by Thomas et al. (Comput Methods Appl Mech Eng 284, 2015) for B-splines and analysis-suitable T-splines (AS T-splines). For THB-splines, there are elements on which the restrictions of THB-splines are linearly dependent, contrary to B-splines and AS T-splines. Therefore, we cluster certain local mesh elements together such that the THB-splines with support over these clusters are linearly independent, and the Bézier projector is adapted to use these clusters. We introduce general extensions for which optimal convergence is shown theoretically and numerically. In addition, a simple adaptive refinement scheme is introduced and compared to Giust et al. (Comput. Aided Geom. Des. 80, 2020), where we find that our simple approach shows promise.