Closed-form multigrid smoothing factors for lexicographic Gauss–Seidel

@article{Hocking2012ClosedformMS,
  title={Closed-form multigrid smoothing factors for lexicographic Gauss–Seidel},
  author={L. Robert Hocking and Chen Greif},
  journal={Ima Journal of Numerical Analysis},
  year={2012},
  volume={32},
  pages={795-812}
}
  • L. R. Hocking, C. Greif
  • Published 1 July 2012
  • Computer Science, Mathematics
  • Ima Journal of Numerical Analysis
This paper aims to present a unified framework for deriving analytical formulas for smoothing factors in arbitrary dimensions, under certain simplifying assumptions. To derive these expressions we rely on complex analysis and geometric considerations, using the maximum modulus principle and Möbius transformations. We restrict our attention to pointwise and block lexicographic Gauss–Seidel smoothers on a d-dimensional uniform mesh, where the computational molecule of the associated discrete… 

Figures and Tables from this paper

Efficient relaxed-Jacobi smoothers for multigrid on parallel computers
Optimized sparse approximate inverse smoothers for solving Laplacian linear systems
TLDR
New efficient sparse approximate inverse smoothers for solving the two-dimensional and three-dimensional Laplacian linear system with geometric multigrid methods with the advantage of inherent parallelism are proposed.
Compact High Order Accurate Schemes for the Three Dimensional Wave Equation
TLDR
A family of compact fourth order accurate finite difference schemes for the three dimensional scalar wave (d’Alembert) equation with constant or variable propagation speed with inversion at the upper time level is constructed.

References

SHOWING 1-10 OF 16 REFERENCES
Multigrid for High-Dimensional Elliptic Partial Differential Equations on Non-equidistant Grids
TLDR
This work presents techniques for building the general d-dimensional adaptations of the multigrid components and proposes grid coarsening strategies to handle anisotropies that occur due to discretization on a non-equidistant grid.
Multigrid smoothing factors for red-black Gauss-Seidel relaxation applied to a class of elliptic operators
Analytic formulae are obtained for the smoothing factors yielded by Gauss–Seidel relaxation in two-color ordering for a class of scalar elliptic operators. Block and point relaxations, in conjunction
Multigrid Smoothing Factors For Red-Black Gauss-Seidel Applied To A Class Of Elliptic Operators
Analytic formulae are obtained for the smoothing factors yielded by Gauss-Seidel relaxation in two-color ordering for a class of scalar elliptic operators. Block and point relaxation, in conjunction
Multi-level adaptive solutions to boundary-value problems math comptr
TLDR
The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes, enabling us to conveniently adapt the discretization to the evolving solution in a nearly optimal way, obtaining "°°-order" approximations and low n, even when singularities are present.
The Convergence Rate of Multi-Level Algorithms Applied to the Convection-Diffusion Equation
We consider the solution of the convection-diffusion equation in two dimensions by various multi-level algorithms (MLAs). We study the convergence rate of the MLAs and the stability of the
On Red-Black SOR Smoothing in Multigrid
  • I. Yavneh
  • Computer Science
    SIAM J. Sci. Comput.
  • 1996
TLDR
The resulting relaxation schemes are found to retain very high efficiency over an appreciable range of coefficients of the elliptic differential operator, yielding simple, inexpensive, and fully parallelizable smoothers in many situations where less cost-effective block- and alternating-direction schemes are commonly used.
INTRODUCTION TO MULTIGRID METHODS
For any integer N , we consider a uniform grid, denoted by Th, of the interval [0, 1] as follows: 0 = x0 < x1 < . . . xN < xN+1 = 1, xj = jh, j = 0 : N + 1, where h = 1/(N + 1) is the length of each
Accelerated Multigrid Convergence and High-Reynolds Recirculating Flows
TLDR
Techniques are developed for accelerating multigrid convergence in general, and for advection-diffusion and incompressible flow problems with small viscosity in particular, showing very significant improvement in convergence rates at little cost.
Matrix Iterative Analysis
Matrix Properties and Concepts.- Nonnegative Matrices.- Basic Iterative Methods and Comparison Theorems.- Successive Overrelaxation Iterative Methods.- Semi-Iterative Methods.- Derivation and
...
...