In this paper, a new block preconditioner is proposed for the saddle point problem arising from the Neumann boundary control problem. In order to deal with the singularity of the stiffness matrix, the saddle point problem is first extended to a new one by a regularization of the pure Neumann problem. Then after row permutations of the extended saddle point problem, a new block triangular preconditioner is constructed based on an approximation of the Schur complement. We analyze the eigenvalue properties of the preconditioned matrix and provide eigenvalue bounds. Numerical results illustrate the efficiency of the proposed preconditioning method.

翻译：本文针对诺伊曼边界控制问题产生的鞍点问题，提出了一种新的块预处理方法。为处理刚度矩阵的奇异性，首先通过对纯诺伊曼问题进行正则化，将原鞍点问题扩展为新形式。随后对扩展鞍点问题进行行置换，基于Schur补的近似构造了新的块三角预处理子。我们分析了预处理矩阵的特征值性质并给出了特征值界。数值结果验证了所提预处理方法的有效性。