New modified shift-splitting preconditioners for non-symmetric saddle point problems

Abstract

Zhou et al. and Huang et al. have proposed the modified shift-splitting (MSS) preconditioner and the generalized modified shift-splitting (GMSS) for non-symmetric saddle point problems, respectively. They have used symmetric positive definite and skew-symmetric splitting of the (1, 1)-block in a saddle point problem. In this paper, we use positive definite and skew-symmetric splitting instead and present new modified shiftsplitting (NMSS) method for solving large sparse linear systems in saddle point form with a dominant positive definite part in (1, 1)-block. We investigate the convergence and semi-convergence properties of this method for nonsingular and singular saddle point problems. We also use the NMSS method as a preconditioner for GMRES method. The numerical results show that if the (1, 1)-block has a positive definite dominant part, the NMSS-preconditioned GMRES method can cause better performance results compared to other preconditioned GMRES methods such as GMSS, MSS, Uzawa-HSS and PU-STS. Meanwhile, the NMSS preconditioner is made for non-symmetric saddle point problems with symmetric and non-symmetric (1, 1)-blocks.

Fractional Boundary Element Solution for Nonlinear Nonlocal Thermoelastic Problems of Anisotropic Fibrous Polymer Nanomaterials

“…1. illustrates the CPU time and iterations number for regularized [43], generalized modified shift-splitting (GMSS) [44] and new modified shift-splitting (NMSS) [45] iterative methods at every discretization level, where the number of the equation is represented between parentheses. This table demonstrates the superiority of the NMSS over the regularized and GMSS approaches.…”

“…( 7), ( 35) and (62), we obtain general solution 𝑢 (𝐺𝑆) of the considered problem as follows 𝑢 (GS) = 𝑢 PTES + 𝑢 𝐹𝑆𝑇𝐷𝑆 + 𝑢 NNES (63) where the general solution 𝑢 (GS) of our considered problem is constructed as the sum of the following three solutions: polymer thermoelastic solution 𝑢 PTES , fractional size-and temperaturedependent solution 𝑢 𝐹𝑆𝑇𝐷𝑆 , and nonlinear nonlocal elasticity solution 𝑢 NNES . The new modified shiftsplitting (NMSS) [45] has been used to solve the resulting linear Eqs. ( 7), ( 35) and (62) arising from BEM.…”

Fractional Boundary Element Solution for Nonlinear Nonlocal Thermoelastic Problems of Anisotropic Fibrous Polymer Nanomaterials

“…The integral equation corresponding to (32) can be defined as [13] (P)θ(P, To solve the domain integrals in Equation (35), we used the same process as Fahmy [14] and the techniques from [24,25] to obtain the following system:…”

“…Hence, from Equations ( 7), (35), and (62), we obtain general solution u (GS) of the considered problem as follows:…”

A Local Positive (Semi)Definite Shift-Splitting Preconditioner for Saddle Point Problems with Applications to Time-Harmonic Eddy Current Models

“…Recently, a lot of efforts have been spent on iteration methods for the problem (1.1). The list of the methods studied includes classical Uzawa iteration method [2] and its generalisations [13,20,28,31], Hermitian and skew-Hermitian splitting (HSS) iteration methods [11] and its variants [9,10,26,33,45], shift-splitting iteration methods [1,14,[22][23][24]29,41,42], residual reduction algorithms [3] and Krylov subspace iteration methods [40]. If is a non-Hermitian and/or ill-conditioned matrix, the preconditioning is often used to accelerate the convergence.…”