A Relaxed HSS Preconditioner for Saddle Point Problems from Meshfree Discretization

Abstract

In this paper, a relaxed Hermitian and skew-Hermitian splitting (RHSS) preconditioner is proposed for saddle point problems from the element-free Galerkin (EFG) discretization method. The EFG method is one of the most widely used meshfree methods for solving partial differential equations. The RHSS preconditioner is constructed much closer to the coefficient matrix than the well-known HSS preconditioner, resulting in a RHSS fixed-point iteration. Convergence of the RHSS iteration is analyzed and an optimal parameter, which minimizes the spectral radius of the iteration matrix is described. Using the RHSS preconditioner to accelerate the convergence of some Krylov subspace methods (like GMRES) is also studied. Theoretical analyses show that the eigenvalues of the RHSS preconditioned matrix are real and located in a positive interval. Eigenvector distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are obtained. A practical parameter is suggested in implementing the RHSS preconditioner. Finally, some numerical experiments are illustrated to show the effectiveness of the new preconditioner.Mathematics subject classification: 65F10.

On semi-convergence of a class of relaxation methods for singular saddle point problems

“…This kind of system of linear equations arises in a variety of scientific and engineering applications, such as computational fluid dynamics, constrained optimization, optimal control, weighted least-squares problems, electronic networks, computer graphic, the constrained least squares problems and generalized least squares problems etc; see [2,9,18,23,28,29] and the references therein. In addition, we can also obtain saddle point linear systems from the meshfree discretization of some partial differential equations [12,20] or the mixed hybrid finite element discretization of second order elliptic problems [11]. A comprehensive summary about various applications leading to saddle point matrices and a general framework of preconditioning methods and their theoretical analyses were given in [8].…”

On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

“…This kind of system of linear equations arises in a variety of scientific and engineering applications, such as computational fluid dynamics, constrained optimization, optimal control, weighted least-squares problems, electronic networks, computer graphic etc; see [1][2][3][4] and the references therein. In addition, we can also obtain saddle-point linear systems from the mixed or hybrid finite element discretization of secondorder elliptic problems [5] or the meshfree discretization of some partial differential equations [6,7]. When matrix B is column rank-deficient, i.e., rankðBÞ < m 6 n, matrix M is singular.…”

Scaled norm minimization method for computing the parameters of the HSS and the two‐parameter HSS preconditioners

“…Due to its promising performance and elegant mathematical properties, the HSS scheme immediately attracted wide attention and was used to solve different kinds of problems, such as saddle‐point problems, complex linear systems, certain singular problems, and nonlinear problems . As it is used as a solver, the HSS method was also used as a preconditioner to accelerate the convergence speed of the Krylov subspace methods . This preconditioner has the form of …”

“…23,24 As it is used as a solver, the HSS method was also used as a preconditioner to accelerate the convergence speed of the Krylov subspace methods. [25][26][27][28] This preconditioner has the form of…”