From: jzhang@cs.engr.uky.edu (Jun Zhang) Subject: Re: A question related to Multigrid... Date: 18 Oct 1999 19:33:22 GMT Newsgroups: sci.math.num-analysis Keywords: different approaches in algebraic and geometric multigrid methods Classical multigrid method, geometric multigrid method, and standard multigrid method(s) mean that a series of grids are generated and the partial differential equation is discretized on all of these grid. The approximate solutions are computed on all grids and are used to correct the approximate solutions on a next finer grids; @article{Brandt77a, author = "A.~Brandt", title = "Multi-level adaptive solutions to boundary-value problems", journal = "Math. Comp.", volume = "31", number = "138", year = "1977", pages = "333--390" } Algebraic multigrid method means only the finest grid is generated. The so-called coarse (levels) grids are constructed by algebraic means, usually through Schur complement approaches, or their equivalent ones. Strictly speaking, they are not multigrid methods, because there is no multiple grids generated. @incollection{Ruge-Stuben87a, author = "J.~W. Ruge and K.~St{\"u}ben", title = "Algebraic multigrid", booktitle = "Multigrid Methods", series = "Frontiers in Appl. Math.", chapter = "4", editor = "S.~McCormick", publisher = "{SIAM}", address = "Philadelphia, PA", year = "1987", pages = "73--130" } The above two references are easies to understand than the books, except the short book @book{Briggs87a, author = "W.~L. Briggs", title = "A Multigrid Tutorial", publisher = "{SIAM}", address = "Philadelphia, PA", year = "1987" } Hope this helps. Jun Zhang ----------------- In article , Nobat wrote: >Hi; > >I am new in the Multigrid techniques, and I have difficulty to undrestand >what are the differences between algebraic and geometrical multigrids. I >appreciate if somebody point me to appropriate references ( I am an >engineer). > >Thank you in advance; > >Nobat, > > > > -- ********************************************************************** * Jun Zhang * E-mail: jzhang@cs.uky.edu * * Department of Computer Science * URL:http://www.cs.uky.edu/~jzhang * * University of Kentucky * Tel:(606)257-3892 * ============================================================================== From: jzhang@cs.engr.uky.edu (Jun Zhang) Subject: Re: A question related to Multigrid... Date: 18 Oct 1999 20:38:51 GMT Newsgroups: sci.math.num-analysis Hi, Nobat: My explanations are based on my understanding of multigrid method and algebraic multigrid (multilevel) method. First of all, geometric multigrid methods are used to solve partial differential equations, or something like that (linear or nonlinear). You have the freedom to use different grid structures (finite difference, finite elements, different mesh size, etc). Even you are solving a discretized linear system, the multigrid method is tied to the underlying partial differential equations and the discretization strategies. For algebraic multigrid method, you are essentially solving a linear system, you do not even have to know where the linear system come from. There are some methods between the purely algebraic and purely geometric, the so-called matrix-dependent interpolation and coarse grid operator approaches. However, so long as there is no geometric coarse grid generated, I would not consider them as (classical) multigrid methods. The approaches used in geometric multigrid methods, such as coarse-grid correction, full approximation scheme, or nested iterations are not fully defined in algebraic multigrid methods (with the same meanings as in geometric methods). For example, the nested iteration means starting iteration from the coarsest grid, which is not available if you only have a linear system arising from discretizing the finest grid problems. By the way, if you have a problem close to a Poisson equation on a rectangular domain, you should use geometric multigrid methods. Otherwise you may want to download a blackbox solver to solve your linear systems. Geometric multigrid methods are very efficient, when they work. They are not very robust with respect to parameter variation. Algebraic multigrid methods are usually much more robust, but may cost more for those problems that can be solved by geometric multigrid methods. If you want to look at some solvers and papers, please visit the MGNET web page at www.mgnet.org, maintained by Craig Douglas. Best regards, Jun Zhang ----------- In article , Nobat wrote: >Dear Jun Zhang > >If you let me I ask another question: > >If I undrestand well: different approaches can be used in geometric >multigrid methods for example: coarse-grid correction, full approximation >scheme or nested iterations... > >I am wondering if the same approaches are available for algebraic multigrid >methods or not? if not how can we categorize them? > >Now I am definitively going to library... > >I appreciate if you help me again (with your brief and efficient >explanations) > > >Nobat > > > > -- ********************************************************************** * Jun Zhang * E-mail: jzhang@cs.uky.edu * * Department of Computer Science * URL:http://www.cs.uky.edu/~jzhang * * University of Kentucky * Tel:(606)257-3892 *