We present a novel stabilized isogeometric formulation for the Stokes problem, where the geometry of interest is obtained via overlapping NURBS (non-uniform rational B-spline) patches, i.e., one patch on top of another in an arbitrary but predefined hierarchical order. All the visible regions constitute the computational domain, whereas independent patches are coupled through visible interfaces using Nitsche's formulation. Such a geometric representation inevitably involves trimming, which may yield trimmed elements of extremely small measures (referred to as bad elements) and thus lead to the instability issue. Motivated by the minimal stabilization method that rigorously guarantees stability for trimmed geometries [1], in this work we generalize it to the Stokes problem on overlapping patches. Central to our method is the distinct treatments for the pressure and velocity spaces: Stabilization for velocity is carried out for the flux terms on interfaces, whereas pressure is stabilized in all the bad elements. We provide a priori error estimates with a comprehensive theoretical study. Through a suite of numerical tests, we first show that optimal convergence rates are achieved, which consistently agrees with our theoretical findings. Second, we show that the accuracy of pressure is significantly improved by several orders using the proposed stabilization method, compared to the results without stabilization. Finally, we also demonstrate the flexibility and efficiency of the proposed method in capturing local features in the solution field.

翻译：我们针对Stokes问题提出了一种新型的稳定等几何公式，其中目标几何通过重叠NURBS（非均匀有理B样条）补片获得，即一个补片以任意但预定义的层次顺序放置在另一个补片之上。所有可见区域构成计算域，而独立补片通过可见界面上的Nitsche公式耦合。这种几何表示不可避免地涉及裁剪，这可能导致产生极小尺寸的裁剪单元（称为坏单元），从而引发稳定性问题。受针对裁剪几何严格保证稳定性的最小稳定化方法[1]启发，在本工作中我们将其推广至重叠补片上的Stokes问题。我们方法的核心在于对压力和速度空间采取不同的处理方式：对界面通量项进行速度稳定化，而对所有坏单元中的压力进行稳定化。我们提供了先验误差估计及全面的理论研究。通过一系列数值测试，我们首先表明达到了最优收敛阶，这与我们的理论结果一致。其次，与未采用稳定化的结果相比，我们提出的稳定化方法将压力精度显著提升了数个数量级。最后，我们还展示了所提出方法在捕获解场局部特征方面的灵活性和高效性。