论文标题
基于符号的鞍形方法中的基于符号的收敛分析
Symbol based convergence analysis in multigrid methods for saddle point problems
论文作者
论文摘要
在求解stokes方程时，在各种应用中出现了鞍点问题。可以将它们配制为使系统矩阵是对称的，但不确定，因此通常用于证明无法应用多机收敛的变分收敛理论。在Numerische Mathematik Notay的2016年论文中，提出了一种不同的代数方法，该方法分析了正确预处的鞍点问题，证明了两网格方法的收敛性。在本文中，我们分析了鞍点问题，其中块是在此框架内循环的。我们能够得出足够的条件以使收敛，并为鞍点问题的预处理和使用的点更顺畅地提供了最佳参数。该分析基于循环块的生成符号。此外，我们表明该结构可以保持在粗糙的水平上，从而使该方法在W-或V-Cycle中递归应用，并证明“级别独立性”属性。数值结果证明了所提出的方法在循环体和Toeplitz情况下的效率。
Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the Two-Grid method. In the present paper we analyze saddle point problems where the blocks are circulant within this framework. We are able to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, we show that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W- or V-cycle and proving the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.
