We consider isogeometric discretizations of the Poisson model problem, focusing on high polynomial degrees and strong hierarchical refinements. We derive a posteriori error estimates by equilibrated fluxes, i.e., vector-valued mapped piecewise polynomials lying in the $\boldsymbol{H}({\rm div})$ space which appropriately approximate the desired divergence constraint. Our estimates are constant-free in the leading term, locally efficient, and robust with respect to the polynomial degree. They are also robust with respect to the number of hanging nodes arising in adaptive mesh refinement employing hierarchical B-splines. Two partitions of unity are designed, one with larger supports corresponding to the mapped splines, and one with small supports corresponding to mapped piecewise multilinear finite element hat basis functions. The equilibration is only performed on the small supports, avoiding the higher computational price of equilibration on the large supports or even the solution of a global system. Thus, the derived estimates are also as inexpensive as possible. An abstract framework for such a setting is developed, whose application to a specific situation only requests a verification of a few clearly identified assumptions. Numerical experiments illustrate the theoretical developments.

翻译：我们考虑泊松模型问题的等几何离散化，重点研究高多项式次数和强层次加密。我们通过均衡流推导后验误差估计，即位于$\boldsymbol{H}({\rm div})$空间中的向量值映射分片多项式，该多项式适当逼近所需的散度约束。我们的估计在主项中不含常数、局部高效且对多项式次数具有鲁棒性。此外，该估计对采用层次B样条的自适应网格加密中产生的悬挂节点数量也具有鲁棒性。我们设计了两类单位分解：一类具有对应于映射样条的较大支撑，另一类具有对应于映射分片多重线性有限元帽基函数的小支撑。均衡仅在较小支撑上执行，避免了在大支撑上进行均衡甚至求解全局系统所需的高计算代价。因此，所推导的估计也尽可能低代价。我们为此类设定开发了一个抽象框架，该框架在特定情形下的应用仅需验证少数明确假设。数值实验验证了理论发展。