Convergence Of Multigrid Method Research Articles

Compact Fourier Analysis for Designing Multigrid Methods

The convergence of multigrid methods can be analyzed based on a Fourier analysis of the method or by proving certain inequalities that have to be fulfilled by the smoother and by the coarse grid correction separately. Here, we analyze the multigrid method for the constant coefficient Poisson equation with a compact Fourier analysis using the formalism of multilevel Toeplitz matrices and their generating functions or symbols. The Fourier analysis is applied for determining the smoothing factor and the overall error of the combined smoothing and coarse grid correction error reduction of a twogrid step (TGS) by representing the twogrid step explicitly by a symbol. If the effects of the smoothing correction and the coarse grid correction are orthogonal to each other, then, in a twogrid step, the error is removed in one step, and the twogrid method can be considered as a direct solver. If the coarse linear system is identical to the original matrix, then the same projection and smoother again make the twogrid step a direct solver. Hence, the multigrid cycle has to be applied only once with one smoothing step on each level, and therefore the whole multigrid method can be considered as a direct solver. In this paper we want to identify multigrid as a direct solver in one-dimension with a coarse system derived by linear and constant interpolation, and in two-dimensions for a modified projection related to [U. Trottenberg, C.W. Osterlee, and A. Schüller, Multigrid, Academic Press, San Diego, 2001, paragraph A.2.3]. By studying not only smoothing and coarse grid correction separately but the symbol of the full twogrid step as well, we are furthermore able to derive better convergence estimates and information on how to find efficient combinations of projection and smoother. As smoothers we consider also approximate inverse smoothers, colored smoothers that take into account the different character of grid points, and rank-reducing smoothers that coincide with the given matrix on a large number of entries. Numerical examples show the effectiveness of the new approach.

An improved and accelerated non-linear multigrid method for total-variation denoising

Fast solution of the non-linear partial differential equations (PDEs) arising from image restoration is of practical importance. The standard multigrid methods do not work well, because of the highly discontinuous coefficients of the underlying non-linear PDEs. We present two related global but linear smoothers that help the convergence of multigrid methods. Furthermore, the Krylov acceleration technique is combined with the proposed multigrid method to improve performance. Numerical experiments are shown.

Convergence of multigrid method for edge-based finite-element method

This paper discusses robustness of the multigrid (MG) method against distortion of finite elements. The convergence of MG method becomes considerably worse as the finite elements become flat. It is shown that the smoother used in the MG method cannot effectively eliminate the high-frequency component of the residue for flat elements, and this gives rise to deterioration in the convergence. Moreover, the multigrid method with conjugate gradient (CG) smoother is shown to be more robust against mesh distortion than that with Gauss-Seidel smoother.

Robust Multigrid Algorithms for the Navier–Stokes Equations

Anisotropies occur naturally in computational fluid dynamics where the simulation of small-scale physical phenomena, such as boundary layers at high Reynolds numbers, causes the grid to be highly stretched, leading to a slowdown in convergence of multigrid methods. Several approaches aimed at making multigrid a robust solver have been proposed and analyzed in the literature using the scalar diffusion equation. However, they have rarely been applied to solving more complicated models, such as the incompressible Navier–Stokes equations. This paper contains the first published numerical results of the behavior of two popular robust multigrid approaches (alternating-plane smoothers combined with standard coarsening and plane-implicit smoothers combined with semi-coarsening) for solving the 3-D incompressible Navier–Stokes equations in the simulation of the driven-cavity and a boundary layer over a flat plate on a stretched grid. Grid size, grid stretching, and Reynolds number are the factors considered in evaluating the robustness of the multigrid methods. Both approaches yield large increases in convergence rates over cell-implicit smoothers on stretched grids. The combination of plane-implicit smoothers and semi-coarsening was found to be fully robust in the flat-plate simulation up to Reynolds numbers 106 and the best alternative in the driven-cavity simulation for Reynolds numbers above 103. The alternating-plane approach exhibits a better behavior for lower Reynolds numbers (below 103) in the driven-cavity simulation.

On the convergence of multi-grid methods with transforming smoothers

In the present paper we give a convergence theory for multi-grid methods with transforming smoothers as introduced in [31] applied to a general system of partial differential equations. The theory follows Hackbusch's approach for scalar pde and allows a convergence proof for some well-known multi-grid methods for Stokes- and Navier-Stokes equations as DGS by Brandt-Dinar, [5], TILU from [31] and the SIMPLE-methods by Patankar-Spalding, [23].