On the iterative methods for solving linear systems of equations

Dutto, Laura C.

doi:10.1080/12506559.1993.10511091

Cited by 6 publications

(3 citation statements)

References 54 publications

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“…A compressed sparse row storage scheme is used, where only the nonzero entries of the matrix are stored row by row. The matrix A is time dependent, due to the time-varying normal components at the boundary, and an iterative method is appropriate for solving (52): see Dutto (1993) for a survey of such solution methods for linear systems. Since the global matrix A is nonsymmetric due to the Coriolis terms, a generalized minimal residual (GMRES) iterative method has been adopted.…”

Section: K Equationmentioning

confidence: 99%

Finite Elements for Shallow-Water Equation Ocean Models

Roux¹,

Staniforth²,

Lin³

1998

Mon. Wea. Rev.

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The finite-element spatial discretization of the linear shallow-water equations on unstructured triangular meshes is examined in the context of a semi-implicit temporal discretization. Triangular finite elements are attractive for ocean modeling because of their flexibility for representing irregular boundaries and for local mesh refinement. The semi-implicit scheme is beneficial because it slows the propagation of the high-frequency small-amplitude surface gravity waves, thereby circumventing a severe time step restriction. High-order computationally expensive finite elements are, however, of little benefit for the discretization of the terms responsible for rapidly propagating gravity waves in a semi-implicit formulation. Low-order velocity/surface-elevation finite-element combinations are therefore examined here. Ideally, the finite-element basis-function pair should adequately represent approximate geostrophic balance, avoid generating spurious computational modes, and give a consistent discretization of the governing equations. Existing finite-element combinations fail to simultaneously satisfy all of these requirements and consequently suffer to a greater or lesser extent from noise problems. An unconventional and largely unknown finite-element pair, based on a modified combination of linear and constant basis functions, is shown to be a good compromise and to give good results for gravity-wave propagation.

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Section: K Equationmentioning

confidence: 99%

Finite Elements for Shallow-Water Equation Ocean Models

Roux¹,

Staniforth²,

Lin³

1998

Mon. Wea. Rev.

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show abstract

“…However, the use of direct methods is limited by both storage requirements and computing time, which prevent their use in large problems. This drawback has been countered by recent developments in non-stationary iterative solution algorithms [5,4] such as Krylov subspace methods. Besides requiring fewer computer resources, using an iterative solver provides exibility in controlling the extent to which the linear system is solved, contrary to direct methods, which yield the exact solutions.…”

Section: Introductionmentioning

confidence: 98%

“…This aspect becomes dominant as the number of grid points increases. Rapid advances in computer speed and available memory combined with recent developments in non-stationary iterative solvers and preconditioners [3][4][5] for non-symmetric matrices have enabled the development of fully coupled algorithms for the solution of the Navier-Stokes equations. Although research in the ÿnite volume community has been ambivalent on the eciency of fully coupled methods compared to the more common segregated solution procedures [6,7], there is common agreement on the added robustness of fully coupled methods.…”

Section: Introductionmentioning

confidence: 99%

Development of a fully coupled control‐volume finite element method for the incompressible Navier–Stokes equations

Ammara

Masson

2004

Numerical Methods in Fluids

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SUMMARYThis paper proposes and investigates fully coupled control-volume ÿnite element method (CVFEM) for solving the two-dimensional incompressible Navier-Stokes equations. The proposed method borrows many of its features from the segregated CVFEM described by Baliga et al. Thus ÿnite-volume discretization is employed on a colocated grid using either the MAW or the FLO schemes and an element-by-element assembling procedure is applied for the construction of the discretizations equations. In this paper, and unlike the case for most fully coupled formulations available in the literature, the Poisson pressure equation has been retained from the segregated approach. The use of a pressure equation leads to an unfavourable size increase of the fully coupled linear system, but signiÿcantly improves the system's conditioning. The fully coupled system obtained is solved using an ILUT preconditioned GMRES algorithm. The other important element in this paper is the proposal of a Newton linearization of the convection terms in lieu of the common Picard iteration procedure. A systematic comparison between two segregated and four fully coupled fomulations has been presented which has allowed for an evaluation of the individual beneÿts and strengths of the coupling and linearization procedure by studying lid-driven cavity problems and ows past a circular cylinder. All coupled formulations have proven to be signiÿcantly superior both in robustness and e ciency, as compared with the segregated formulation. In some circumstances, the coupled methods yield a converged solution of the system of discretized equations constructed using the FLO scheme, while the segregated formulations diverge. Compared to Picard's linearization, Newton's linearization is more e cient at reducing the number of iterations needed to converge, but requires more computational e ort per iteration from the linear equation solver. Furthermore, the Jacobian matrix should include contributions from the nonlinearity appearing at both the governing-equation level and the interpolation-scheme level to ensure Newton's method convergence. The key element in guaranteeing successful, fully coupled solutions lies in the use of an e cient linear equation solver and preconditioner.

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An iterative algorithm for finite element analysis

Laouafa

Royis

2004

Journal of Computational and Applied Mathematics

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International audienceIn this paper, we state in a new form the algebraic problem arising from the one-field displacement finite element method (FEM). The displacement approach, in this discrete form, can be considered as the dual approach (force or equilibrium) with subsidiary constraints. This approach dissociates the nonlinear operator to the linear ones and their sizes are linear functions of integration rule which is of interest in the case of reduced integration. This new form of the problem leads to an inexpensive improvement of FEM computations, which acts at local, elementary and global levels. We demonstrate the numerical performances of this approach which is independent of the mesh structure. Using the GMRES algorithm we build, for nonsymmetric problems, a new algorithm based upon the discretized field of strain. The new algorithms proposed are more closer to the mechanical problem than the classical ones because all fields appear during the resolution process. The sizes of the different operators arising in these new forms are linear functions of integration rule, which is of great interest in the case of reduced integration

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On the iterative methods for solving linear systems of equations

Cited by 6 publications

References 54 publications

Finite Elements for Shallow-Water Equation Ocean Models

Finite Elements for Shallow-Water Equation Ocean Models

Development of a fully coupled control‐volume finite element method for the incompressible Navier–Stokes equations

An iterative algorithm for finite element analysis

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