Considering saddle-point systems of the Karush–Kuhn–Tucker (KKT) form, we propose approximations of the “ideal” block diagonal preconditioner based on the exact Schur complement proposed by Murphy et al. (SIAM J Sci Comput 21(6):1969–1972, 2000). We focus on the case where the (1,1) block is symmetric and positive definite, but with a few very small eigenvalues that possibly affect the convergence of Krylov subspace methods like Minres. Assuming that these eigenvalues and their associated eigenvectors are available, we first propose a Schur complement preconditioner based on this knowledge and establish lower and upper bounds on the preconditioned Schur complement. We next analyse theoretically the spectral properties of the preconditioned KKT systems using this Schur complement approximation in two spectral preconditioners of block diagonal forms. In addition, we derive a condensed “two in one” formulation of the proposed preconditioners in combination with a preliminary level of preconditioning on the KKT system. Finally, we illustrate on a PDE test case how, in the context of a geometric multigrid framework, it is possible to construct practical block preconditioners that help to improve on the convergence of Minres.

考虑到Karush–Kuhn–Tucker（KKT）形式的鞍点系统，我们根据Murphy等人提出的精确Schur补全提出“理想”块对角预处理器的近似值。（SIAM J Sci Comput 21（6）：1969–1972，2000）。我们关注（1,1）块是对称且为正定的情况，但特征值很小，可能会影响Mrys等Krylov子空间方法的收敛。假设这些特征值及其相关的特征向量可用，我们首先基于此知识提出Schur补码预处理器，并在预处理的Schur补码上建立上下边界。接下来，我们将在块对角线形式的两个光谱预处理器中使用此Schur补码近似理论分析预处理KKT系统的光谱特性。此外，我们结合KKT系统的初步预处理水平，得出了建议的预处理器的浓缩“二合一”公式。最后，我们在一个PDE测试用例中说明了如何在几何多网格框架的背景下构造实用的块预处理器，以帮助改善M inres的收敛性。