In this paper, we describe and analyze the spectral properties of a number of exact block preconditioners for a class of double saddle point problems. Among all these, we consider an inexact version of a block triangular preconditioner providing extremely fast convergence of the FGMRES method. We develop a spectral analysis of the preconditioned matrix showing that the complex eigenvalues lie in a circle of center (1,0) and radius 1, while the real eigenvalues are described in terms of the roots of a third order polynomial with real coefficients. Numerical examples are reported to illustrate the efficiency of inexact versions of the proposed preconditioners, and to verify the theoretical bounds.

翻译：本文描述并分析了一系列精确块预处理子在一类双鞍点问题中的谱性质。其中，我们考虑了一种块三角预处理子的非精确版本，该版本能够实现FGMRES方法极其快速的收敛。我们对预处理后的矩阵进行了谱分析，结果表明：复特征值位于以(1,0)为圆心、半径为1的圆内，而实特征值则可用一个实系数三次多项式的根来描述。文中还给出了数值算例，以说明所提预处理子非精确版本的有效性，并验证理论界。