Abstract
In this paper, we present a new block preconditioner for solving the saddle point linear systems. The proposed method is developed from an augmented reformulation of the saddle point problem into a new linear system with an almost block triangular coefficient matrix. Theoretical results are derived on the eigenvalue distribution of the preconditioned matrix, and an efficient algorithmic implementation is developed and presented. Several numerical examples are reported to support the theoretical findings and to illustrate the favourable convergence properties of the proposed preconditioner, also compared to other popular solvers for saddle point problems.