Abstract Over the years, Isogeometric Analysis has shown to be a successful alternative to the Finite Element Method (FEM). However, solving the resulting linear systems of equations efficiently remains a challenging task. In this paper, we consider a p -multigrid method, in which coarsening is applied in the spline degree p instead of the mesh width h , and compare it to h -multigrid methods. Since the use of classical smoothers (e.g. Gauss–Seidel) results in a p -multigrid/ h -multigrid method with deteriorating performance for higher values of p , the use of an ILUT smoother is investigated as well. Numerical results and a spectral analysis indicate that the use of this smoother exhibits convergence rates essentially independent of h and p for both p -multigrid and h -multigrid methods. In particular, we compare both coarsening strategies (e.g. coarsening in h or p ) adopting both smoothers for a variety of two and three dimensional benchmarks. Furthermore, the ILUT smoother is compared to a state-of-the-art smoother (Hofreither and Takacs 2017) using both coarsening strategies. Finally, the proposed p -multigrid method is used to solve linear systems resulting from THB-spline discretizations.

中文翻译：

---

p-多重网格方法及其与等几何分析中的 h-多重网格方法的比较

摘要 多年来，等几何分析已被证明是有限元方法 (FEM) 的成功替代方法。然而，有效地求解由此产生的线性方程组仍然是一项具有挑战性的任务。在本文中，我们考虑 ap -multigrid 方法，其中粗化应用于样条度 p 而不是网格宽度 h ，并将其与 h -multigrid 方法进行比较。由于使用经典平滑器（例如 Gauss-Seidel）会导致 ap -multigrid/h -multigrid 方法在更高的 p 值下性能恶化，因此也研究了 ILUT 平滑器的使用。数值结果和频谱分析表明，对于 p 多重网格和 h 多重网格方法，使用这种平滑器表现出基本上独立于 h 和 p 的收敛速度。特别是，我们比较了两种粗化策略（例如 在 h 或 p 中粗化）为各种二维和三维基准采用平滑器。此外，ILUT 平滑器与使用两种粗化策略的最先进的平滑器（Hofreither 和 Takacs 2017）进行了比较。最后，所提出的 p 多重网格方法用于求解由 THB 样条离散化产生的线性系统。