Journal ArticleDOI
On the Robustness of ILU Smoothing
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In the present paper, a detailed analysis of a multigrid method with an ILU smoother applied to a singularly perturbed problem is given, and a variant of the usual incomplete LU factorization is introduced, which is especially suited as robust smoother.Abstract:
In the present paper, a detailed analysis of a multigrid method with an ILU smoother applied to a singularly perturbed problem is given. Based on the analysis of a simple anisotropic model problem, a variant of the usual incomplete LU factorization is introduced, which is especially suited as robust smoother. For this variant a detailed analysis and a proof of robustness is given. Furthermore, some contradictions between the smoothing rates predicted by local Fourier analysis and the practically observed convergence factors are explained (see [W. Hackbusch, Multi-grid Methods and Applications, Springer-Verlag, Berlin, Heidelberg, 1985; R. Kettler, “Analysis and comparison of relaxation schemes in robust multi-grid and preconditioned conjugate gradient methods,” in Multi-grid Methods, Lecture Notes in Math. 960, Springer-Verlag, Berlin, 1982; C. A. Thole, Beitrage zur Fourieranalyse von Mehrgitterver fahren, Diplomarbeit, Universitat Bonn, 1983]. The theoretical results are confirmed by numerical tests.read more
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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.
Journal ArticleDOI
Preconditioning techniques for large linear systems: a survey
TL;DR: This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices, including progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions.
Journal ArticleDOI
Experimental study of ILU preconditioners for indefinite matrices
Edmond Chow,Yousef Saad +1 more
TL;DR: A better practical understanding is gained of ILU preconditioners and how these problems can sometimes be circumvented through pivoting, reordering, scaling, perturbing diagonal elements, and preserving symmetric structure.
Journal ArticleDOI
Old and new convergence proofs for multigrid methods
TL;DR: The convergence theory of multigrid methods for self-adjoint and coercive linear elliptic boundary value problems reached a mature, if not its final state, in the early 1990s as mentioned in this paper.
Journal ArticleDOI
Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier-Stokes Equations
Per-Olof Persson,Jaime Peraire +1 more
TL;DR: In this paper, a coarse scale correction with postsmoothing based on a block incomplete LU factorization with zero fill-in (ILU0) of the Jacobian matrix is proposed.
References
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Matrix iterative analysis
TL;DR: In this article, the authors propose Matrix Methods for Parabolic Partial Differential Equations (PPDE) and estimate of Acceleration Parameters, and derive the solution of Elliptic Difference Equations.
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.
Journal ArticleDOI
An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix
TL;DR: A particular class of regular splittings of not necessarily symmetric M-matrices is proposed, if the matrix is symmetric, this splitting is combined with the conjugate-gradient method to provide a fast iterative solution algorithm.