We propose local multigrid solvers for adaptively refined isogeometric discretizations using (truncated) hierarchical B-splines ((T)HB-splines). Smoothing is only performed in or near the refinement areas on each level, leading to a computationally efficient solving strategy. We prove robust convergence of the proposed solvers with respect to the number of levels and the mesh sizes of the hierarchical discretization space under the assumption that the hierarchical mesh satisfies an admissibility condition, i.e., the number of interacting mesh levels is uniformly bounded. We also provide several numerical experiments. The main analytical tools are quasi-interpolators for THB-splines and the abstract convergence theory of subspace correction methods.

中文翻译：

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用于分层样条空间中自适应等几何分析的局部多重网格求解器

我们提出了局部多重网格求解器，用于使用（截断的）分层 B 样条（（T）HB 样条）进行自适应细化等几何离散化。平滑仅在每个级别的细化区域中或附近执行，从而产生计算高效的求解策略。在分层网格满足可接受性条件的假设下，我们证明了所提出的求解器在分层离散空间的层数和网格大小方面的稳健收敛性，即相互作用的网格层数是一致有界的。我们还提供了几个数值实验。主要的分析工具是THB样条的准插值器和子空间校正方法的抽象收敛理论。