Jacobi Relaxation Research Articles

Development of semi-coarsening techniques

Departing from Mulder's semi-coarsening technique for first order PDEs, the notion of a grid of grids is introduced and a multi-level finite-volume technique for second order elliptic PDEs is developed. Various grid transfer operators are investigated, in combination with damped Jacobi relaxation. Convergence rates as they are predicted by Fourier local mode analysis are compared with practical measurements. The wide variety of grids at our disposal leads to the notion of coherent representations of a function on different grids. A sawtooth multi-level algorithm is proposed for the case of multiple semi-coarsening. A hierarchical set of basis functions for finite volumes on sparse grids is briefly discussed.

Multi-staging of Jacobi relaxation in multigrid methods for steady Euler equations, II

Multi-stage versions of Jacobi relaxation are studied for use in multigrid methods for steady Euler equations. In particular, versions adapted to the nonlinear TVD discretization are analysed. It is shown that multi-stage relaxation versions lead to more efficient multigrid methods than possible with classic relaxation methods.

Random-walk interpretations of classical iteration methods

We give a simple framework for computing relative convergence rates for relaxation methods with discrete Laplace operators (five point or nine point). This gives relations between the convergence rate for Jacobi, point Gauss Seidel, and various block relaxation strategies, essentially by inspection. The framework is a random walk interpretation of Jacobi relaxation that extends to these other relaxation methods.

MULTI-STAGE JACOBI RELAXATION IN MULTIGRID METHODS FOR THE STEADY EULER EQUATIONS

SUMMARY Mulli-stage versions of the Jacobi relaxation methods are developed for use in multigrid methods for the steady Euler equations. The non-linear TVD-discretization, both on structured and unstructured grids, is employed. Different multigrid formulations are compared : the defect correction procedure based on the first order discretization; the mixed discretization formulation with the TVD-operator on the finest grid and a linear operator on the coarser grids; the full second order formulation employing the TVD-operator on all grids. The full second order requires the use of implicit residual weighting. Optimized coefficient sets in the multi-staging are constructed for every formulation. It is shown that the mixed discretization is, in general, the most efficient. The speed of convergence on unstructured grids is much greater than on structured grids.

Multi-staging of Jacobi relaxation to improved smoothing properties of multigrid methods for steady Euler equations

Multi-stage versions of Jacobi relaxation are studied for use in multigrid methods for steady Euler equations. It is shown that these multi-stage versions allow much more general and much more efficient multigrid methods than possible with classic relaxation methods.

Efficient modified Jacobi relaxation for minimizing the energy functional.

We present an efficient scheme of diagonalizing large Hamiltonian matrices in a self-consistent manner. In the framework of the preconditioned conjugate gradient minimization of the energy functional, we replace the modified Jacobi relaxation for preconditioning and use for band-by-band minimization the restricted-block Davidson algorithm, in which only the previous wave functions and the relaxation vectors are included additionally for subspace diagonalization. Our scheme is found to be comparable with the preconditioned conjugate gradient method for both large ordered and disordered Si systems, while it is more rapidly converged for systems with transition-metal elements.

Nonlinear relaxation/quasi-Newton algorithm for the compressible Navier-Stokes equations

A highly efficient implicit method for the computation of steady, two-dimensional compressible Navier-Stokes flowfields is presented. The discretization of the governing equations is hybrid in nature, with flux-vector splitting utilized in the streamwise direction and central differences with flux-limited artificial dissipation used for the transverse fluxes. Line Jacobi relaxation is used to provide a suitable initial guess for a new nonlinear iteration strategy based on line Gauss-Seidel sweeps. The applicability of quasi-Newton methods as convergence accelerators for this and other line relaxation algorithms is discussed, and efficient implementations of such techniques are presented. Convergence histories and comparisons with experimental data are presented for supersonic flow over a flat plate and for several high-speed compression corner interactions. Results indicate a marked improvement in computational efficiency over more conventional upwind relaxation strategies, particularly for flowfields containing large pockets of streamwise subsonic flow.

Line Iterative Methods for Cyclically Reduced Discrete Convection-Diffusion Problems

An analytic and empirical study of line iterative methods for solving the discrete convection-diffusion equation is performed. The methodology consists of performing one step of the cyclic reduction method, followed by iteration on the resulting reduced system using line orderings of the reduced grid. Two classes of iterative methods are considered: block stationary methods, such as the block Gauss–Seidel and SOR methods, and preconditioned generalized minimum residual methods with incomplete LU preconditioners. New analysis extends convergence bounds for constant coefficient problems to problems with separable variable coefficients. In addition, analytic results show that iterative methods based on incomplete LU preconditioners have faster convergence rates than block Jacobi relaxation methods. Numerical experiments examine additional properties of the two classes of methods, including the effects of direction of flow, discretization, and grid ordering on performance.

A multi-grid algorithm for mixed problems with penalty

In this paper we describe a multi-grid algorithm for the finite element approximation of mixed problems with penalty by the MINI-element. It is proved that the convergence rate of the algorithm is bounded away from 1 independently of the meshsize and of the penalty parameter. For convenience, we only discuss Jacobi relaxation as smoothing operator in detail.

A Note on the Use of Symmetric Line Gauss–Seidel for the Steady Upwind Differenced Euler Equations

Symmetric Line Gauss–Seidel (SLGS) relaxation, when used to compute steady solutions to the upwind differenced Euler equations of gas dynamics, is shown to be unstable. The instability occurs for the long waves. If SLGS is used in a multigrid scheme, stability is restored. However, the use of an unstable relaxation scheme will not provide a robust multigrid code. Damped Symmetric Point Gauss–Seidel relaxation is stable and provides similar multigrid convergence rates at much lower cost. However, it fails if the flow is aligned with the grid over a substantial part of the computational domain. Damped Alternating Direction Line Jacobi relaxation can overcome this problem.

Multigrid method for stability problems

The problem of calculating the stability of steady state solutions of differential equations is treated. Leading eigenvalues (i.e., having maximal real part) of large matrices that arise from discretization are to be calculated. An efficient multigrid method for solving these problems is presented. The method begins by obtaining an initial approximation for the dominant subspace on a coarse level using a damped Jacobi relaxation. This proceeds until enough accuracy for the dominant subspace has been obtained. The resulting grid functions are then used as an initial approximation for appropriate eigenvalue problems. These problems are solved first on coarse levels, followed by refinement until a desired accuracy for the eigenvalues has been achieved. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a nonstandard way in which the right-hand side of the coarse grid equations involves unknown parameters to be solved for on the coarse grid. This in particular leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem that are presented demonstrate the effectiveness of the method proposed. Using an FMG algorithm a solution to the level of discretization errors is obtained in just a few work units (less than 10), where a work unit is the work involved in one Jacobi relaxation on the finest level.

Fourier acceleration of iterative processes in disordered systems

Technical details are given on how to use Fourier acceleration with iterative processes such as relaxation and conjugate gradient methods. These methods are often used to solve large linear systems of equations, but become hopelessly slow very rapidly as the size of the set of equations to be solved increases. Fourier acceleration is a method designed to alleviate these problems and result in a very fast algorithm. The method is explained for the Jacobi relaxation and conjugate gradient methods and is applied to two models: the random resistor network and the random central-force network. In the first model, acceleration works very well; in the second, little is gained. We discuss reasons for this. We also include a discussion of stopping criteria.

A multigrid method for the membrane problem

A multigrid method for the membrane problem is studied. It turns out that some extra devices are needed in problems of elasticity if locking is present on the auxiliary grids. Symmetric successive relaxation is recommended for the smoothing steps whenever the matrices can be kept in the fast storage. Otherwise one may choose Jacobi relaxation. Finally some comments on preconditioned conjugate gradient methods are given.

A Multilevel Algorithm for Mixed Problems

We describe a multilevel algorithm for the numerical solution of symmetric indefinite problems which arise, e.g. from mixed finite element approximations of the Stokes equation. The main difficulty, besides the indefiniteness, is the lack of regularity of the solution of the corresponding continuous problem. This is overcome by introducing a scale of mesh-dependent norms. The convergence rate of the described algorithm is bounded independently of the meshsize. For convenience we only discuss Jacobi relaxation as smoothing operator in detail. In the last section we comment on Lanczos-type smoothing procedures.