We consider the SIMPLE preconditioning for block two-by-two generalized saddle point problems; this is the general nonsymmetric, nonsingular case where the (1,2) block needs not to equal the transposed (2,1) block, and the (2,2) block may not be zero. The eigenvalue analysis of the SIMPLE preconditioned matrix is presented. The relationship between the two different formulations spectrum of the SIMPLE preconditioned matrix is established by using the theory of matrix eigenvalue, and some corresponding results in recent article by Li and Vuik (2004) are extended.
Shi-Liang Wu. Feng Chen. Xiao-Qi Niu. "A Note on the Eigenvalue Analysis of the SIMPLE Preconditioning for Incompressible Flow." J. Appl. Math. 2012 (SI12) 1 - 7, 2012. https://doi.org/10.1155/2012/564132