A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier–Stokes equation

On the Generalized Deteriorated Positive Semi-Definite and Skew-Hermitian Splitting Preconditioner

“…So, it seems that the GVDPSS preconditioner with proper parameters α and β is more closer to the coefficient matrix A than the VDPSS preconditioner due to the independence of the parameters and, as a result, the corresponding preconditioned matrix will have a well-clustered spectrum. It can be seen that by choosing different values for the parameters α and β, the GVDPSS preconditioner coincides with some existing preconditioners such as the RHSS preconditioner [11], the REHSS preconditioner [16], the RDPSS preconditioner [9] and the VDPSS preconditioner [18].…”

Inexact modified positive-definite and skew-Hermitian splitting preconditioners for generalized saddle point problems

“…The linear independence of these eigenvectors can be derived by the method studied in Cao et al 10 [Theorem 3.2] with only technical modifications. Hence, we omit the rest of the proof here.…”

“…LetŜ 2 R m 3 m be a nonsingular matrix and a be a positive constant. Let the IMPSS preconditioner P IMPSS and the optimal IMPSS preconditioner P OIMPSS be defined as in equations (7) and (10), respectively. Then, both the degree of the minimal polynomial of the preconditioned matrix P À1…”

“…Much effort has gone into improving the DPSS preconditioner. For example, based on the relaxed techniques studied in previous works, [11][12][13] Cao et al 10,14 proposed a relaxed deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioner for equation (1) when C = 0 and a simplified Hermitian and skew-Hermitian splitting (SHSS) preconditioner when the submatrices A and C are Hermitian, the resulting preconditioners are closer to the generalized saddle point matrix than the DPSS-based preconditioner. Zhou et al 15 proposed a relaxed block triangular splitting preconditioner for generalized saddle point matrix by considering the coefficient matrix as a threeby-three block matrix.…”

Preconditioning optimal control of the unsteady Burgers equations withH1regularized term

“…However, it is never used without the application of appropriate preconditioning, which is chosen to enhance the convergence behavior for the iterative method. In the last years, many efficient preconditioners have been introduced, e.g., see [3][4][5][6]8,9,13,21,25].…”