Preconditioned Iterative Methods Research Articles

A Preconditioned Iterative Method for Saddlepoint Problems

A preconditioned iterative method for indefinite linear systems corresponding to certain saddlepoint problems is suggested. The block structure of the systems is utilized in order to design effective preconditioners, while the governing iterative solver is a standard minimum residual method. The method is applied to systems derived from discretizations of the Stokes problem and mixed formulations of second-order elliptic problems.

A finite element solutionof three‐dimensional mixed convection gas flows in horizontal chnnels using preconditioned iterative metrix methods

AbstractAn algorithm is presented for the finite element solution of three‐dimensional mixed convection gas flows in channels heated from below. The algorithm uses Newton's method and iterative matrix methods. Two iterative solution algorithms, conjugate gradient squared (CGS) and generalized minimal residual (GMERS), are used in conjunction with a preconditioning technique that is simple to implement. The preconditioner is a subset of the full Jacobian matrix centered around the main diagonal but retaining the most fundamental axial coupling of the residual equations. A domain‐renumbering scheme that enhances the overall algorithm performance is proposed on the basis of and analysis of the preconditioner. Comparison with the frontal elimination method demonstrates that the iterative method will be faster when the front width exceeds approximately 500. Techniques for the direct assembly f the problem into a compressed sparse row storage format are demonstrated. Elimination of fixed boundary conditions is shown to decrease the size of the matrix problem by up to 30%. Finally, fluid flow solutions obtained with the numerical technique are presented. These solutions reveal complex three‐dimensional mixed convection fluid flow phenomena at low Reynolds numbers, including the reversal of the direction of longitudinal rolls in the presence of a strong recirculation in the entrance region of the channel.

Constant wavefront iteration methods for nine- and 15-point difference matrices

Classical wavefront preconditioned iteration methods for difference matrices on a rectangular or on a rectangular parallelepipedal domain use wavefronts based on diagonal (line or plane, respectively) orderings of the mesh-point. Since such wavefronts do not have constant widths, they cannot be implemented efficiently on parallel computers. We discuss various methods to get wavefronts with constant width for difference matrices for second order elliptic problems. In particular, we discuss their applications for the nine-point (2D) and 15-point (3D) difference approximations for the Laplacian, which are fourth order accurate for proper choices of the coefficients. It turns out that we can easily get preconditioning methods with wavefronts in the form of vertical or horizontal lines both in 2D and 3D, which have condition number Oh-1 , but for general three space dimensional problems no simple ordering leading to constant plane wavefronts seems to exist in general, for which the corresponding preconditioner has such a small condition number. A crucial property we make use of in the methods is the spectral equivalence between the nine-point and the standard five-point difference matrices and between the 15-point and the standard seven-point difference matrices in two and three space dimensions, respectively. The methods use only nearest neighbor connections and can therefore be implemented efficiently not only on shared memory computers but also on distributed memory computer architectures, such as mesh-connected computer architectures. — Authors' Abstract

A survey of multilevel preconditioned iterative methods

We survey multilevel iterative methods applied for solving large sparse systems with matrices, which depend on a level parameter, such as arise by the discretization of boundary value problems for partial differential equations when successive refinements of an initial discretization mesh is used to construct a sequence of nested difference or finite element meshes.We discuss various two-level (two-grid) preconditioning techniques, including some for nonsymmetric problems. The generalization of these techniques to the multilevel case is a nontrivial task. We emphasize several ways this can be done including classical multigrid methods and a recently proposed algebraic multilevel preconditioning method. Conditions for which the methods have an optimal order of computational complexity are presented.

A Preconditioned Iterative Method for the Solution of the Biharmonic Problem

A Preconditioned Iterative Method for the Solution of the Biharmonic Problem

On sparse and compact preconditioned conjugate gradient methods for partial differential equations

This paper generalises the preconditioning techniques, introduced by Evans [2], and defines sparse and compact preconditioned iterative methods for the numerical solution of the linear system Au = b.The difference between the methods is shown to depend on whether a conditioning matrix R consists of components derived from a splitting or factorization of A. Some theoretical results for the iterative schemes are given when A has particular properties such as consistent ordering, irreducibility, diagonal dominance, positive definiteness, etc., when derived from the finite difference discretisation of a 2nd order self-adjoint elliptic partial differential equation. Finally, the application of both forms of preconditioning to the Conjugate Gradient method is presented and computational results compared.

On efficient time-stepping methods for nonlinear partial differential equations

Efficient procedures for time-stepping Galerkin methods for approximating smooth solutions of quasilinear second-order hyperbolic equations are considered. The techniques presented can be used to analyze approximation procedures for related second-order-in-time quasilinear partial differential equations which have applications including initial-boundary value problems for vibrations (possibly) with inertia, dynamics of rotating fluids, and nonlinear viscoelasticity. The procedure involves the use of a pre-conditioned iterative method for approximately solving the different linear systems of equations arising at each time step in a discrete-time Galerkin method. Optimal order L 2 spatial errors and almost optimal order work estimates are obtained for the second-order hyperbolic case.