In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo [7]. Numerical results for a variety of finite element discretizations of both steady and unsteady incompre...

In this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo in [8]. Numerical results for a variety of finite element discretizations of both steady and unsteady incom...

We consider the iterative solution of linear systems arising from four convection–diffusion model problems: scalar convection–diffusion problem, Stokes problem, Oseen problem and Navier–Stokes problem. We design preconditioners for these model problems that are based on Kronecker product approximations (KPAs). For this we first identify explicit Kronecker product structure of the coefficient ma...

This paper proposes a preconditioner for the conjugate gradient method (CG) that is designed for solving systems of equations Ax = bi with different right-hand-side vectors or for solving a sequence of slowly varying systems Akx = bk. The preconditioner has the form of a limited memory quasi-Newton matrix and is generated using information from the CG iteration. The automatic preconditioner doe...

We present an algebraic structured preconditioner for the iterative solution of large sparse linear systems. The preconditioner is based on a multifrontal variant of sparse LU factorization used with nested dissection ordering. Multifrontal factorization amounts to a partial factorization of a sequence of logically dense frontal matrices, and the preconditioner is obtained if structured factori...

Recently, the authors introduced a preconditioner for the linear systems that arise from fully implicit Runge-Kutta time stepping schemes applied to parabolic PDEs [9]. The preconditioner was a block Jacobi preconditioner, where each of the blocks were based on standard preconditioners for low-order time discretizations like implicit Euler or Crank-Nicolson. It was proven that the preconditione...

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year’s Report, St. Hugh’s College, ...

Block preconditioner with circulant blocks (BPCB) has been used for solving linear systems with block Toeplitz structure since 1992 [R. Chan, X. Jin, A family of block preconditioners for block systems, SIAM J. Sci. Statist. Comput. (13) (1992) 1218–1235]. In this new paper, we use BPCBs to general linear systems (with no block structure usually). The BPCBs are constructed by partitioning a gen...

1 Ghent University, Department of Agricultural Economics, Coupure links 653, B-9000 Gent, Belgium 2 University of Surrey, School of Psychology, Guildford Surrey, GU2 7XH, United Kingdom 3 International Rice Research Institute (IRRI), Los Baños, Laguna, Philippines 4 CIS-IUL, Instituto Universitario de Lisboa, 1649-026, Lisbon, Portugal 5 University of Évora, Department of Pyschology, 7005-345 É...