A simplified HSS preconditioner for generalized saddle point problems (Q291888): Difference between revisions

This paper develops the SHSS preconditioner for generalized saddle point problems \[ \begin{bmatrix} A & B^T\\ -B & C \end{bmatrix}\, \begin{bmatrix} x\\y\end{bmatrix} = \begin{bmatrix} f\\ -g \end{bmatrix} \] that is built on and simplified from the classical HSS preconditioner of \textit{Z.-Z. Bai} et al. [SIAM J. Matrix Anal. Appl. 24, No. 3, 603--626 (2003; Zbl 1036.65032)]. The idea is to leave off parts of the HSS preconditioner \(P_{HSS} = \dfrac{1}{\alpha}\) \[ \begin{bmatrix} \alpha I + A& 0\\ 0 & \alpha I + C \end{bmatrix} \begin{bmatrix} \alpha I & B^T\\ -B & \alpha I\end{bmatrix} \] in the new preconditioner \(P_{SHSS} = \dfrac{1}{\alpha}\) \[ \begin{bmatrix} A& 0\\ 0 & \alpha I \end{bmatrix} \begin{bmatrix} \alpha I & B^T\\ -B & \alpha I\end{bmatrix}. \] The simpler preconditioner \(P_SHSS\) differs from the saddle point matrix \[ \begin{bmatrix} A & B^T\\ -B & C \end{bmatrix} \] only in the upper triangular block. This allows the thus preconditioned algorithm to solve only two linear systems where the more complex HSS methods need to solve three. Moreover, the spectral distribution of the preconditioned matrix improves significantly, leading to much shorter iteration sequences. A detailed spectral analysis of SHSS is included. Numerical experiments deal with preconditioned GMRES for the Stokes equation in 2D. Extensive numerical tests show the advantages (fewer iterations, CPU speedup by factors above \(5\), same accuracy) of SHSS over HSS as well as choices for a near optimal parameter \(\alpha\).