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，重点关注高多项式次数与强层次化细分。通过平衡通量（即位于$\boldsymbol{H}({\rm div})$空间中、恰当逼近预期散度约束的向量值映射分片多项式）推导后验误差估计。我们的估计在主导项中无常数依赖、具有局部有效性，并对多项式次数保持鲁棒性。该估计对采用层次化B样条的自适应网格细化中出现的悬挂节点数量同样具有鲁棒性。我们设计了两种单位分解：一种采用更大支撑（对应映射样条），另一种采用小支撑（对应映射分片多重线性有限元帽基函数）。平衡过程仅在小支撑上执行，避免了在大支撑上平衡甚至求解全局系统所需的高昂计算代价。因此，所推导的估计同样保持了最低经济成本。本文建立了适用于此类场景的抽象框架，针对具体情境的应用仅需验证若干明确设定的假设条件。数值实验验证了理论成果。