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 Current editor:  Craig Douglas       
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Volume 3, Number 10 (October 31, 1993)

Today's topics:

     Convergence Rates of Multigrid Methods for Poisson Equations
     Paper about Gauss adaptive relaxation
     Hackbusch's Iterative Method Book
     The bibliography database on MGNet
     Multigrid solvers for problems with complex data
     Eighth Domain Decomposition Meeting


Date: Tue, 05 Oct 93 15:10:53 EDT
From: "Shang-Hong Lai" 
Subject: Convergence Rates of Multigrid Methods for Poisson Equations

In the following paper
"The convergence rate of a multigrid method with
Gauss-Seidel relaxation for the Poisson equation" 
by Dietrich Braess 
in Mathematics of Computation, vol.42, no.166, pp.505-519, 1984,

it said that the convergence rates of multigrid methods are bounded
away from 1 for the Poisson equation independent of the size of
the problem as long as the domain is convex or the boundary of
the domain is smooth.  

I would like to know if there is any further results which can
relax the conditions of convex domain or smooth boundary and 
retain the convergence rate of the multigrid method bounded 
away from 1.  

Any references or comments related to this question will be
greatly appreciated.  

Shang-Hong Lai

    Editor's Note:  In a paper in the Feb.  1993 issue of SINUM (pp. 136-158)
    -------------   is a very different approach to multigrid theory.  There
                    are 2 theorems in there that may be useful to you.  One is
                    quite simple and at a first glance appears to only say
                    that the convergence rate is bounded by 1.  However, one
                    of the parameters (delta) in the theorem can be
                    approximated almost exactly on each correction iteration.
                    Normally this is << 1, which says that the multigrid
                    convergence rate is also << 1 (see the example on pp.

                    The other theorem uses an affine space decomposition to
                    get a sharper bound.  While much more complicated, it
                    could be used to relax the conditions you are interested
                    in and get a useful bound.

                    I would guess that several readers of this digest will
                    have suggestions, too.  Please copy me.  Thanks.


Date:   Wed, 20 Oct 1993 17:38:58 +0100
From: Christoph Pflaum 
Subject: Paper about Gauss adaptive relaxation

Our paper 

Gauss adaptive relaxation for the multilevel solution of partial differential 
equations on sparse grids

has just been downloaded to the mgnet ftp repository. The full paper
is available in a compressed postscrip format, the abstract is also available
as a plain ascii file.

The paper will be published in "J. of Computing and Information". We
gave a talk about this topic at the 2nd Gauss Symposium in Munich.


Gauss adaptive relaxation for the multilevel solution of partial differential 
equations on sparse grids

C. Pflaum  U. Ruede

Institut fuer Informatik, Technische Universitaet
D-80290 Muenchen, Germany
e-mail: pflaum/

In combination with the multilevel principle, relaxation methods are
among the most efficient numerical solution techniques for elliptic
partial differential equations. Typical methods used today are
derivations of the Gauss -Seidel or Gauss -Jacobi method. Recently it
has been recognized that in the context of multilevel algorithms, the
original  method suggested by Gauss has specific advantages. For
this method the iteration is concentrated on unknowns where fast
convergence can be obtained by intelligently monitoring the residuals.
We will present this algorithm in the context of a sparse grid
multigrid algorithm. Using sparse grids the dimension of the discrete
approximation space can be reduced additionally.

Keywords: adaptivity, multilevel techniques, elliptic PDE,
finite elements 

AMS Classifications: 65N22, 65N30, 65N50, 65N55

    Editor's Note:  in mgnet/papers/Pflaum-Ruede/gauss.abstract and
    -------------      mgnet/papers/Pflaum-Ruede/


Date: Wed, 20 Oct 1993 16:31:12 PDT
From:  (Tony Chan)
Subject: Hackbusch's Iterative Methods Book

I recall that you announced Hackbusch's book in a digest a while ago.  Could
you send me the information again, please?

    Editor's Note:  Actually, I did not announce it.  I saw it listed in a
    -------------   mailing a few months ago, but I do not recall from which
                    publisher.  I also do not know if the English translation
                    is out, yet.  It was not the last time I asked Hackbusch
                    about it, but it was threatening to be.

                    Could someone please e-mail Tony this information and
                    copy me, please?  I will add it to the bibliography
                    database.  Thanks.


Date: Sun, 31 Oct 1993 07:14:32 -0400
From: (Craig Douglas)
Subject: The bibliography database on MGNet

The BibTeX oriented bibliography database on MGNet has received a very large
infusion of citations lately.  This is in part because someone else has been
entering many hundreds of entries (soon to be thousands) in the multigrid and
domain decomposition fields.  The relevant files are on in
the directory mgnet/bib:

    mg.bib      All of the entries sorted (hopefully) in alphabetical order
                by authors and years.
    mgbib.tex   The LaTeX file to generate the bibliography in paper form.
    mgbib.dvi   The paper form of the bibliography.    The PostScript form of the .dvi file.
    siam.bst    A modified form of SIAM's BibTeX style file that does not
                convert van's to V.'s.  You need this if you plan on using

When this was first started, several people asked for a particular change in
the keys used for each entry.  Recently, another change was requested that I
have implemented today.  I will now guarantee that any entry in the database
will never get a new key again.  Period.

The final scheme is


For example, my dissertation's key from 1982 is


The key for a book by Braess, Hackbusch, and Trottenberg from 1984 is


The change today was to include the century and to put a letter after every
year, not just when I happen to know about multiple ones in a year already.

You can help keep this accurate and up to date by spending a few seconds of
your time now.  Just send me by e-mail or by your postal system your own
publication list in these areas (bibtex, ascii, plain paper, vitae style,
etc.).  Your naming convention will be changed for you; please just send the
entries in now.

E-mail to or via the post office to me at
8 South Street, Cos Cob, CT 06807-1618, USA.



Date: Wed, 26 Oct 1993 13:09:01 -0400
From: (Craig Douglas)
Subject: Multigrid solvers for problems with complex data

I am looking for people who are solving (or trying to solve) problems with
complex data.  I am trying to determine how well the latest abstract
multilevel solver I inspired to be written works on such problems.  (The code
works on real data, too.)


Date: Sun, 31 Oct 1993 07:20:01 -0400
From: (Craig Douglas)
Subject: Eighth Domain Decomposition Meeting

At the recent 7th Domain Decomposition Meeting at Penn State, it was announced
that the next one will be in Beijing, May 15-19, 1995.  This is a tentative
date still.


End of MGNet Digest