Book ChapterDOI
Multigrid Methods and Finite Difference Schemes for 2D Singularly Perturbed Problems
Francisco J. Gaspar,Francisco J. Lisbona,C. Clavero +2 more
- pp 316-324
Reads0
Chats0
TLDR
This work presents numerical experiments obtained with the multigrid method for this class of linear systems and sees that modifying only the restriction operator in an appropriate form, the algorithm is convergent, the CPU time increases linearly with the discretization parameter and the number of cycles is independent of the mesh sizes.Abstract:
Solving the algebraic linear systems proceeding from the discretization on some condensed meshes of 2D singularly perturbed problems, is a difficult task. In this work we present numerical experiments obtained with the multigrid method for this class of linear systems. On Shishkin meshes, the classical multigrid algorithm is not convergent. We see that modifying only the restriction operator in an appropriate form, the algorithm is convergent, the CPU time increases linearly with the discretization parameter and the number of cycles is independent of the mesh sizes.read more
Citations
More filters
Journal ArticleDOI
Robust Solution of Singularly Perturbed Problems Using Multigrid Methods
Scott MacLachlan,Niall Madden +1 more
TL;DR: In this paper, the authors consider the problem of solving linear systems of equations that arise in the numerical solution of singularly perturbed ordinary and partial differential equations of reaction-diffusion type, and propose and prove optimality of a new block-structured preconditioning approach that is robust for small values of the perturbation parameter.
Journal ArticleDOI
Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type
C. Clavero,José Luis Gracia +1 more
TL;DR: In this work, parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle are considered, in the simplest case that the diffusion parameter is the same for all equations of the system.
Journal ArticleDOI
Boundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems
TL;DR: A boundary layer preconditioner, in the style of that introduced by MacLachlan and Madden for a finite-difference method, is proposed and it is proved the optimality of this preconditionser and establish a suitable stopping criterion for one-dimensional problems.
Journal ArticleDOI
An analysis of diagonal and incomplete Cholesky preconditioners for singularly perturbed problems on layer-adapted meshes
Thái Anh Nhan,Niall Madden +1 more
TL;DR: The analysis shows that when the singularly perturbed problem features no corner layers, an incomplete Cholesky preconditioner performs extremely well, than when the perturbation parameter is small, and numerical evidence is provided that these findings extend to three-dimensional problems.
Journal ArticleDOI
Preconditioning techniques for singularly perturbed differential equations
TL;DR: A careful analysis of the numerical solution of linear systems arising from finite difference and finite element discretizations of singularly perturbed reaction-diffusion problems by giving a concrete formula for the magnitude of the fill-in entries in the Cholesky factors in terms of the perturbation parameter, ε, and the discretization parameter, N.
References
More filters
Journal ArticleDOI
BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems
TL;DR: Numerical experiments indicate that the new variant of Bi-CG, named Bi- CGSTAB, is often much more efficient than CG-S, so that in some cases rounding errors can even result in severe cancellation effects in the solution.
Journal ArticleDOI
Multi-level adaptive solutions to boundary-value problems
TL;DR: In this paper, the boundary value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes, and interactions between these levels enable us to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); and conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining "°°-order" approximations and low n, even when singularities are present.
Book
Multi-Grid Methods and Applications
TL;DR: This paper presents the Multi-Grid Method of the Second Kind, a method for solving Singular Perturbation Problems and Eigenvalue Problems and Singular Equations of the Two-Grid Iteration.
Book
An Introduction to Multigrid Methods
TL;DR: These notes were written for an introductory course on the application of multigrid methods to elliptic and hyperbolic partial differential equations for engineers, physicists and applied mathematicians, restricting ourselves to finite volume and finite difference discretization.
Book
Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions
TL;DR: Fitted mesh methods focus on the appropriate distribution of the mesh points for singularly perturbed problems as mentioned in this paper, and the global errors in the numerical approximations are measured in the pointwise maximum norm.