We develop a multigrid solver for the second biharmonic problem in the context of Isogeometric Analysis (IgA), where we also allow a zero-order term. In a previous paper, the authors have developed an analysis for the first biharmonic problem based on Hackbusch’s framework. This analysis can only be extended to the second biharmonic problem if one assumes uniform grids. In this paper, we prove a multigrid convergence estimate using Bramble’s framework for multigrid analysis without regularity assumptions. We show that the bound for the convergence rate is independent of the scaling of the zero-order term and the spline degree. It only depends linearly on the number of levels, thus logarithmically on the grid size. Numerical experiments are provided which illustrate the convergence theory and the efficiency of the proposed multigrid approaches.

我们在等几何分析 (IgA) 的背景下为第二个双调和问题开发了一个多重网格求解器，其中我们还允许使用零阶项。在之前的一篇论文中，作者基于 Hackbusch 的框架对第一个双调和问题进行了分析。如果假设网格均匀，这一分析只能扩展到二次双调和问题。在本文中，我们使用 Bramble 的多重网格分析框架证明了多重网格收敛估计，无需进行规律性假设。我们证明收敛速度的界限与零阶项和样条度的缩放无关。它仅与级别数成线性关系，因此与网格大小成对数关系。提供的数值实验说明了所提出的多重网格方法的收敛理论和效率。