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：我们提出了一种用于Stokes问题的新型稳定等几何公式，其中目标几何通过重叠NURBS（非均匀有理B样条）补丁获得，即一个补丁以任意但预定义的层次顺序叠加在另一个补丁之上。所有可见区域构成计算域，而独立补丁通过Nitsche公式在可见界面处耦合。这种几何表示不可避免地涉及裁剪，可能产生极小尺寸的裁剪单元（称为坏单元），从而导致不稳定问题。受严格保证裁剪几何稳定性的最小稳定化方法[1]启发，本文将该方法推广到重叠补丁上的Stokes问题。我们方法的核心是对压力和速度空间进行不同处理：对速度的稳定化针对界面上的通量项进行，而压力则在所有坏单元中进行稳定化。我们通过全面的理论研究提供了先验误差估计。通过一系列数值测试，我们首先表明实现了最优收敛速度，这与我们的理论结果一致。其次，我们表明与无稳定化的结果相比，使用所提出的稳定化方法后，压力的精度提高了几个数量级。最后，我们还展示了所提方法在捕捉解场局部特征方面的灵活性和效率。