Send mail to:    mgnet@cs.yale.edu             for the digests or bakeoff
                  mgnet-requests@cs.yale.edu    for comments or help
 Current editor:  Craig Douglas                 douglas-craig@cs.yale.edu
Anonymous ftp repository:    casper.cs.yale.edu (128.36.12.1)
                             ftp.cerfacs.fr     (138.63.200.33)

World Wide Web:  http://na.cs.yale.edu/mgnet/www/mgnet.html or
                 http://www.cerfacs.fr/~douglas/mgnet.html

Today's editor:  Craig Douglas (douglas-craig@cs.yale.edu)

Volume 5, Number 5 (approximately May 31, 1995)

Today's topics:

     Best V-cycle Algorithm for 2D Poisson Equation
     New paper on MGNet (Ruede)
     Announcement of WWW multigrid tutorial
     Paper on cache based multigrid
     Oberwolfach
     An MPI based multigrid solver for 1 or more processors
     Some of the new entries in the bibliography

-------------------------------------------------------

Date: Sun, 28 May 1995 03:38:05 -0400
From: Jun Zhang 
Subject: Best V-cycle Algorithm for 2D Poisson Equation

         Best V-cycle algorithm to Solve 2D Poisson Equation

I'm interested in knowing the best V-cycle algorithm to solve 2D
Poisson equation 
       - (Uxx + Uyy) = f(x,y)       (x,y) in OMEGA
          u(x,y)     = g(x,y)       (x,y) on OMEGA
For simplicity, OMGEA is a unit square. I have 3 test problems (courtesy 
of Dr. S. Fulton)

Test 1.   f(x,y) = -x^2(1-x^2)(2-12y^2) - y^2(1-y^2)(2-12x^2),
          g(x,y) =  x^2*y^2(1-x^2)(1-y^2).
Test 2.   f(x,y) = -(x^2 + y^2)*exp(xy),
          g(x,y) =  exp(xy).
Test 3.   f(x,y) =  52*cos(2x + 6y),
          g(x,y) =  cos(4x + 6y).

Conditions and Cost: 5-point finite difference discretization. V(1,1)-cycle 
algorithm with point Red-Black Gauss-Seidel, injection and bi-linear 
interpolation.  Fortran 77, double precision, stop when L2-error < 10^(-9). 
Meshsize h= 1/N. Initial guess U(x,y)=0. Computer is a SUN  SPARCstation. 
Below are the best results I got:  

  N     Test 1      Test 2       Test 3
 16       9           11           10
 32       9           11           11            These are V(1,1) cycle
 64       9           12           12            numbers, i.e. the iteration
 128      9           13           12            numbers.
 256      9           13           13
 512      9           14           13

If you know better results with comparable cost, would you please send
me an email? I'm also interested in knowing the limit of improving these
results,  so if you know better results with slightly higher cost, I
will appreciate your sending me a message. (I can modify the code to give
slightly better data to some entry, but the above is the best on average.)

Jun Zhang                 zhang@math.gwu.edu
Department of Mathematics
George Washington University
Washington, DC 20052

    Editor's Note: I would have used a nested iteration V cycle myself, but
    -------------  there is an algorithm using nonstandard grids which is a
                   direct method effectively (1 iteration, 0 contraction
                   factor) by (I think) Ruge and Stuben (correction?).  Also,
                   some of the multiple coarse space methods are similar to
                   this (e.g., see a paper by Brezzi, me, and Marini in the
                   MGNet bibliography for an 8 coarse grid decomposition of
                   this problem).

              -->  Please cc mgnet@cs.yale.edu if you send comments to the
                   requestor.

-------------------------------------------------------

Date: Wed, 31 May 1995 17:33:57 +0200 (MESZ)
From: Ulrich Ruede 
Subject: New paper on MGNet

My paper

    Stability of implicit extrapolation methods

has been downloaded to the mgnet ftp repository.  This paper has been
submitted to the proceedings of the 8th International Conference on Domain
Decomposition, May 16-20, Beijing, China.

Keywords: Implicit extrapolation, multigrid, multilevel subspace splitting
AMS Classification:     65N22, 65N50, 65N55}

Abstract

Multilevel methods are generally based on a splitting of the solution
space associated with a nested sequence of coarser grids.  Besides the
straightforward application of extrapolation on the grid system, we
propose to use extrapolation implicitly, similar to multigrid
tau-extrapolation.  This implicit extrapolation, when applied to linear
finite elements is related to the p-version of the finite element
method.  The method depends on a stability condition which must be
enforced by a suitable modification of the problem.


This (and others of may papers) are available from mgnet and in the WWW
through URL

    http://www5.informatik.tu-muenchen.de/persons/ruede/info/refs.html

Ulrich Ruede
Institut fuer Informatik, Technische Universitaet,
D-80290 Muenchen, Germany, e-mail: ruede@informatik.tu-muenchen.de
Tel: +49 89 21058238, Fax: +49 89 21052022
URL: http://www5.informatik.tu-muenchen.de/persons/ruede.html

    Editor's Note: in mgnet/papers/Ruede/stable_extra.ps.gz and
    -------------     mgnet/papers/Ruede/stable_extra.abstract.

-------------------------------------------------------

Date:   Wed, 31 May 1995 17:59:39 +0200 (MESZ)
Subject: Announcement of WWW multigrid tutorial
From: Ulrich Ruede 

On the WWW an

    Online Multigrid Tutorial,

the

    Multigrid Workbench

is available. The start page is accessible through URL

    http://www5.informatik.tu-muenchen.de/sci-comp/xwb/xwb.html

The workbench features an active image showing a standard multigrid
V-cycle algorithm. By clicking on parts of this algorithm, the
corresponding status of the iteration is displayed graphically. In
contrast to conventional text, there is no natural sequence, how the
different pages should be read.  All information is accessible by
hypertext links and can be read in any order.

I'd appreciate any feedback and comments that would help me to improve
this resource. I'd also be interested in hearing about technical
problems, e.g. whether the current network speed (in Germany) is still
acceptable for using the workbench in its present form.

-------------------------------------------------------

Date: Wed, 31 May 1995 13:48:39 -0400
From: douglas@watson.ibm.com (Craig Douglas)
Subject: Paper on cache based multigrid

       Caching in with Multigrid Algorithms: Problems in Two Dimensions

                               Craig C. Douglas

                      IBM T. J. Watson Research Center,
                          Yorktown Heights, NY, USA
                                     and
                         Computer Science Department,
                               Yale University,
                              New Haven, CT, USA


                                   Abstract

Multigrid methods combine a number of standard sparse matrix techniques.
Usual implementations separate the individual components (e.g., an iterative
methods, residual computation, and interpolation between grids) into nicely
structured routines.  However, many computers today employ quite sophisticated
and potentially large caches whose correct use are instrumental in gaining
much of the peak performance of the processors.

We investigate when it makes sense to combine several of the multigrid
components into one, using block oriented algorithms.  We determine how large
(or small) the blocks must be in order for the data in the block to just fit
into the processor's primary cache.  By re-using the data in cache several
times, a potential savings in run time can be predicted.  This is analyzed for
a set of examples.

Key words:  multigrid, cache, sparse matrix, iterative methods, domain
decomposition.

Note:  This paper will be in the proceedings of the International Conference
on Parallel Algorithms, October 15-19, Wuhan, China.

    Editor's Note: in mgnet/papers/Douglas/cache1.dvi.gz and
    -------------     mgnet/papers/Douglas/cache1.abs.

-------------------------------------------------------

Date: Wed, 31 May 1995 12:24:30 +0200
From: P.W.Hemker@cwi.nl
Subject: Oberwolfach

Dear Craig,

Please find below the titles of the talks, as announced in Oberwolfach.  Sue
Brenners's titles (between brackets) are possibly more informative.

I also add a list of participants.

Pieter.


Monday
AM

P. Wesseling "Multigrid solution of the incompressible Navier-Stokes equations
              in general coordinates"
             ("Krylov subspace and multigrid methods applied to the
              incompressible Navier-Stokes equations" (with C. Vuik, S. Zeng))
K. Oosterlee "Multigrid and defect correction for the 3D incompressible
              Navier-Stokes equations in general coordinates"
             ("A GMRES-based plane smoother in multigrid for solving
             3D anisotropic fluid flow problems")
E. Dick      "Multigrid methods of Navier-Stokes equations coupled to
              k-$\epsilon$- turbulence equations"
             ("Modelling transitional fronts")
D. Haenel    "Application of multi-sequence Runge-Kutta methods to
              adaptive solutions of the Navier-Stokes equations"
             ("Computation of compressible strongly unsteady flows on algebraic
              unstructured grids")

PM
U. Ruede     "Implicit multilevel extrapolation methods"
J. Dendy     "Variants of the frequency decomposition MG-method"
             ("Grandchild of the frequency decomposition method")
R. Stevenson "Frequency decomposition multilevel methods"
             ("Multilevel methods based on space decompositions orthogonal
              w.r.t. discrete scalar products")

Tuesday
AM
S. Vandewalle "Acceleration of multilevel domain decomposition and multigrid
               by Krylov subspace methods"
              ("Schwarz methods: to symmetrize or not to symmetrize")
A. Meyer      "Preconditioning the pseudo-Laplacian for CFD-simulation"
M. Jung       "On the parallelization of multigrid methods using a
               nonoverlapping DD data structure"
G. Haase      "Dirichlet DD vs. global multigrid methods" (with U. Langer)

PM
S. Brenner    "Convergence of nonconforming or nonnested multigrid methods
               without full elliptic regularity"
S. Turek      "On robust and efficient multilevel Schur-complement solvers
               for incompressible and compressible Stokes and Navier-Stokes
               equations"
D. Braess     "Efficient smoothing of the Navier-Stokes equations by
               u-dominant iterations" (with R. Sarazin)

Wednesday
AM
M. Feistauer  "Numerical solution of nonlinear convection-diffusion problems
               and applications to compressible fluid flow"
A. Reusken    "Analysis of multigrid methods for convection-diffusion problems"
J. Fuhrmann   "On algebraic multigrid methods for PDEs"
N. Neuss      "Homogenization and multigrid"

PM
hike

Thursday
AM
P. Oswald    "Multilevel preconditioners for nonconforming elements"
J. Junkherr  "Multigrid methods for weakly singular integral equations of the
             first kind"
R. Kornhuber "A posteriori error estimates for linear and nonlinear elliptic
              problems"

PM
R. Bank   "An algorithm for coarsening unstructured meshes" (with J. Xu)
S. Sauter "Composite finite element spaces, coarsening and MG methods"
          ("A new finite element space for the approx. of PDEs on domains
           with complicated microstructures") (with W. Hackbusch)
R. Hoppe  "Adaptive multilevel methods for mixed FEM for 2nd order
           elliptic BVPs"  (with B. Wohlmuth)

Friday
AM
F. Bornemann "Cascadic multigrid methods"  (with P. Deuflhard)
C. Wagner    "Filtering decompositions for asymmetric and heterogeneous
              problems"
             ("Frequency filtering decomposition")
P. Vanek     "Algebraic multigrid for thin elastic problems"
             ("Algebraic multigrid by smoothed aggregation for 2nd and
              4th order elliptiv BVPs" (with J. Mandel and M. Brezina)
W. Mulder    "Application of multigrid to porous media flow"

========================================================================

Participants:

rbank@ucsd.edu                          (Randy Bank),
peter@ica3.uni-stuttgart.de             (Peter Bastian),
juergen@na.mathematik.uni-tuebingen.de  (Juergen Bey),
bornemann@zib-berlin.de                 (Folkmar Bornemann),
braess@num.ruhr-uni-bochum.de           (Dietrich Braess),
brenner@math.scarolina.edu              (Susanne Brenner),
dahmen@igpm.rwth-aachen.de              (Wolfgang Dahmen),
jed@lanl.gov                            (J.E. Dendy),
deuflhard@zib-berlin.de                 (Peter Deuflhard),
Erik.Dick@rug.ac.be                     (Erik Dick),
FEIST@MS.MFF.CUNI.CZ                    (Miloslav Feistauer),
fuhrmann@iaas-berlin.d400.de            (Juergen Fuhrmann),
griebel@informatik.tu-muenchen.de       (Michael Griebel),
ghaase@numa.uni-linz.ac.at              (Gundolf Haase),
wh@informatik.uni-kiel.de               (Wolfgang Hackbusch),
hj454ha@vug.uni-duisburg.de             (Dieter Haenel),
pieth@cwi.nl                            (Piet Hemker),
rohop@mathematik.tu-muenchen.de         (Ronald Hoppe),
dr.michael.jung@mathematik.tu-chemnitz.de  (Michael Jung),
jj@informatik.uni-kiel.de               (Joerg Junkherr),
kornhuber@iaas-berlin.d400.de           (Ralf Kornhuber),
ulanger@miraculix.numa.uni-linz.ac.at   (Ulrich Langer),
maitre@cc.ec-lyon.fr                    (Jean-Francois Maitre),
a.meyer@mathematik.tu-chemnitz.de       (Arndt Meyer),
mittelmann@math.la.asu.edu              (Hans D. Mittelmann),
mulderw@ksepl.nl                        (W. A. Mulder),
neuss@iwr.uni-heidelberg.de             (Nikolas Neuss),
KEES.OOSTERLEE@GMD.DE                   (K. Oosterlee),
Peter.Oswald@math.tamu.edu              (Peter Oswald),
WSANAR@WIN.TUE.NL                       (Arnold Reusken),
ruede@informatik.tu-muenchen.de         (Ulrich Ruede),
sas@informatik.uni-kiel.de              (Stefan Sauter),
stevenso@win.tue.nl                     (Rob Stevenson),
ture@gaia.iwr.uni-heidelberg.de         (Stefan Turek),
STEFAN@AMA.CALTECH.EDU                  (Stefan Vandewalle),
rv@silly.num1.ruhr-uni-bochum.de        (Ruediger Verfuerth),
chris@ica3.uni-stuttgart.de             (Christian Wagner),
p.wesseling@math.tudelft.nl             (Pieter Wesseling),
wieners@ica3.uni-stuttgart.de           (Wieners),
wittum@icasun.ica.uni-stuttgart.de      (Gabriel Wittum),
harry@na.uni-tuebingen.de               (Harry Yserentant)

-------------------------------------------------------

Date: Wed, 31 May 1995 14:20:32 -0400
From: douglas-craig@CS.YALE.EDU (Craig Douglas)
Subject: An MPI based multigrid solver for 1 or more processors

At the Copper Mountain meeting, I talked very briefly one evening at a
workshop about what I thought was new and interesting for parallel multigrid.
I pointed out that MPI is a standard, developed by academics and computer
companies, that is actually appearing to make headway as a very nice
parallel computer message passing system.  I even ran a demonstration on an
IBM ThinkPad (running Linux, not DOS/Windows) of how it works on a single
processor, but fakes the system into simulating (quasi-randomly) a
multiprocessor.  I volunteered to put a demonstration code on MGNet, which I
have done, in Codes/douglas.  There are several examples in 2D (Poisson's
equation and variable coefficients) with a variety of matrix storage formats
(including matrix-free).  They can be modified quite easily.

I will put some 3D codes out when they are cleaned up sufficiently.

These codes are not meant for high speed, just to demonstrate how MPI can be
used in interesting ways (such as transferring non-stride 1 data without
having to pack and unpack data yourself).

One word of warning.  I used the experimental version (1.0.9) of mpich that is
in pub/mpi/misc on info.mcs.anl.gov and also IBM's MPI product for the SP2 in
developing this.  With mpich, I found that one of the routines (the
simulataneous send and receive routine) did not quite work correctly on a mesh
of processors (it worked with the IBM product).  Bill Gropp fixed this within
2 hours of hearing about it and pointed me to a newer version (many thanks,
Bill).

    Editor's Note: in mgnet/Codes/douglas/2d.tgz.
    -------------

-------------------------------------------------------

Date: Wed, 31 May 1995 14:19:28 -0400
From: douglas-craig@CS.YALE.EDU (Craig Douglas)
Subject: Some of the new entries in the bibliography

This is somewhat repetitive (my apologies).  The next batch is in next month's
digest.


\begin{thebibliography}{100}

\bibitem{VIAgoshkov_1994a}
{\sc V.~I. Agoshkov}, {\em Domain decomposition methods using modified basis
  functions}, in Domain Decomposition Methods in Science and Engineering: The
  Sixth International Conference on Domain Decomposition, vol.~157 of
  Contemporary Mathematics, Providence, Rhode Island, 1994, American
  Mathematical Society, pp.~3--15.

\bibitem{GAguilar_FLisbona_1994a}
{\sc G.~Aguilar and F.~Lisbona}, {\em Interface conditions for a kinf of non
  linear elliptic--hyperbolic problems}, in Domain Decomposition Methods in
  Science and Engineering: The Sixth International Conference on Domain
  Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode
  Island, 1994, American Mathematical Society, pp.~89--95.

\bibitem{RArina_CCanuto_1994a}
{\sc R.~Arina and C.~Canuto}, {\em A {X}--formulation of the viscous--inviscid
  domain decomposition for the {E}uler/{N}avier--{S}tokes equations}, in Domain
  Decomposition Methods in Scientific and Engineering Computing: Proceedings of
  the Seventh International Conference on Domain Decomposition, vol.~180 of
  Contemporary Mathematics, Providence, Rhode Island, 1994, American
  Mathematical Society, pp.~453--458.

\bibitem{SFAshby_CTKelley_PESaylor_JSScroggs_1994a}
{\sc S.~F. Ashby, C.~T. Kelley, P.~E. Saylor, and J.~S. Scroggs}, {\em
  Preconditioning via asymptotically--defined decomposition}, in Domain
  Decomposition Methods in Scientific and Engineering Computing: Proceedings of
  the Seventh International Conference on Domain Decomposition, vol.~180 of
  Contemporary Mathematics, Providence, Rhode Island, 1994, American
  Mathematical Society, pp.~139--150.

\bibitem{MAzaiez_AQuarteroni_1994a}
{\sc M.~Azaiez and A.~Quarteroni}, {\em A spectral {S}tokes solver in domain
  decomposition methods}, in Domain Decomposition Methods in Scientific and
  Engineering Computing: Proceedings of the Seventh International Conference on
  Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode
  Island, 1994, American Mathematical Society, pp.~151--156.

\bibitem{NSBakhvalov_AVKnyazev_1994a}
{\sc N.~S. Bakhvalov and A.~V. Knyanzev}, {\em Preconditioned iterative methods
  in a subspace for linear algebraic equations with large jumps in the
  coefficients}, in Domain Decomposition Methods in Scientific and Engineering
  Computing: Proceedings of the Seventh International Conference on Domain
  Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode
  Island, 1994, American Mathematical Society, pp.~157--162.

\bibitem{REBank_JXu_1994a}
{\sc R.~E. Bank and J.~Xu}, {\em The hierarchical basis multigrid method and
  incomplete {LU} decompostion}, in Domain Decomposition Methods in Scientific
  and Engineering Computing: Proceedings of the Seventh International
  Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics,
  Providence, Rhode Island, 1994, American Mathematical Society, pp.~163--173.

\bibitem{JDBenamou_YBrenier_1994a}
{\sc J.-D. Benamou and Y.~Brenier}, {\em A domain decomposition method for the
  polar factorization of vector fields}, in Domain Decomposition Methods in
  Science and Engineering: The Sixth International Conference on Domain
  Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode
  Island, 1994, American Mathematical Society, pp.~231--236.

\bibitem{BBialecki_XCCai_MDryja_GFairweather_1994a}
{\sc B.~Bialecki, X.-C. Cai, M.~Dryja, and G.~Fairweather}, {\em An additive
  {S}charz algorithm for piecewise {H}ermite bicubic orthogonal spline
  collocation}, in Domain Decomposition Methods in Science and Engineering: The
  Sixth International Conference on Domain Decomposition, vol.~157 of
  Contemporary Mathematics, Providence, Rhode Island, 1994, American
  Mathematical Society, pp.~237--244.

\bibitem{PEBjorstad_WMCoughran_EGrosse_1994a}
{\sc P.~E. Bj{\o}rstad, W.~M. Coughran, and E.~Gross}, {\em Parallel domain
  decomposition applied to coupled transport equations}, in Domain
  Decomposition Methods in Scientific and Engineering Computing: Proceedings of
  the Seventh International Conference on Domain Decomposition, vol.~180 of
  Contemporary Mathematics, Providence, Rhode Island, 1994, American
  Mathematical Society, pp.~369--380.

\bibitem{FABornemann_1994a}
{\sc F.~A. Bornemann}, {\em Interpolation spaces and optimal multilevel
  preconditioners}, in Domain Decomposition Methods in Scientific and
  Engineering Computing: Proceedings of the Seventh International Conference on
  Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode
  Island, 1994, American Mathematical Society, pp.~3--8.

\bibitem{JFBourgat_PLeTallec_BPerthame_YQiu_1994a}
{\sc J.~F. Bourgat, P.~LeTallec, B.~Perthame, and Y.~Qiu}, {\em Coupling
  {B}oltzmann and {E}uler equations without overlapping}, in Domain
  Decomposition Methods in Science and Engineering: The Sixth International
  Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics,
  Providence, Rhode Island, 1994, American Mathematical Society, pp.~377--398.

\bibitem{ABrambilla_CCarlenzoli_GGazzaniga_PGervasio_GSacchi_1994a}
{\sc A.~Brambilla, C.~Carlenzoli, G.~Gazzaniga, P.~Gervasio, and G.~Sacchi},
  {\em Implementation of domain decomposition techniques on n{CUBE}2 parallel
  machine}, in Domain Decomposition Methods in Science and Engineering: The
  Sixth International Conference on Domain Decomposition, vol.~157 of
  Contemporary Mathematics, Providence, Rhode Island, 1994, American
  Mathematical Society, pp.~345--351.

\bibitem{JHBramble_JEPasciak_1994a}
{\sc J.~H. Bramble and J.~E. Pasciak}, {\em Uniform convergence estimates for
  multigrid {V}--cycle algorithms with less than full elliptic regularity}, in
  Domain Decomposition Methods in Science and Engineering: The Sixth
  International Conference on Domain Decomposition, vol.~157 of Contemporary
  Mathematics, Providence, Rhode Island, 1994, American Mathematical Society,
  pp.~17--26.

\bibitem{ABrandt_BDiskin_1994a}
{\sc A.~Brandt and B.~Diskin}, {\em Multigrid solvers on decomposed domains},
  in Domain Decomposition Methods in Science and Engineering: The Sixth
  International Conference on Domain Decomposition, vol.~157 of Contemporary
  Mathematics, Providence, Rhode Island, 1994, American Mathematical Society,
  pp.~135--155.

\bibitem{SCBrenner_1994b}
{\sc S.~C. Brenner}, {\em Two--level additive {S}chwarz preconditioners for
  nonconforming finite elements}, in Domain Decomposition Methods in Scientific
  and Engineering Computing: Proceedings of the Seventh International
  Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics,
  Providence, Rhode Island, 1994, American Mathematical Society, pp.~9--14.

\bibitem{FBrezzi_LDMarini_1994a}
{\sc F.~Brezzi and L.~D. Marini}, {\em A three--fold domain decomposition
  method}, in Domain Decomposition Methods in Science and Engineering: The
  Sixth International Conference on Domain Decomposition, vol.~157 of
  Contemporary Mathematics, Providence, Rhode Island, 1994, American
  Mathematical Society, pp.~27--34.

\bibitem{MOBristeau_RGlowinski_JPeriaux_1994a}
{\sc M.~O. Bristeau, R.~Glowinski, and J.~P{\'e}riaux}, {\em On the numerical
  solution of the {H}elmholtz equations at large wave numbers using exact
  controllability methods. {A}pplication to scattering}, in Domain
  Decomposition Methods in Science and Engineering: The Sixth International
  Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics,
  Providence, Rhode Island, 1994, American Mathematical Society, pp.~399--419.

\bibitem{HJBungartz_MGriebel_DRoschke_CZenger_1994a}
{\sc H.-J. Bungartz, M.~Griebel, D.~R{/"}oschke, and C.~Zenger}, {\em Two
  proofs of convergence for the combination technique for the efficient
  solution of sparse grid problems}, in Domain Decomposition Methods in
  Scientific and Engineering Computing: Proceedings of the Seventh
  International Conference on Domain Decomposition, vol.~180 of Contemporary
  Mathematics, Providence, Rhode Island, 1994, American Mathematical Society,
  pp.~15--20.

\bibitem{WCai_1994a}
{\sc W.~Cai}, {\em Domain decomposition and computation of two dimensional
  detonation waves}, in Domain Decomposition Methods in Scientific and
  Engineering Computing: Proceedings of the Seventh International Conference on
  Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode
  Island, 1994, American Mathematical Society, pp.~459--464.

\bibitem{XCCai_MDryja_1994a}
{\sc X.-C. Cai and M.~Dryja}, {\em Domain decomposition methods for monotone
  nonlinear eliptic problems}, in Domain Decomposition Methods in Scientific
  and Engineering Computing: Proceedings of the Seventh International
  Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics,
  Providence, Rhode Island, 1994, American Mathematical Society, pp.~21--27.

\bibitem{XCCai_WDGropp_DEKeyes_MDTidriri_1994a}
{\sc X.-C. Cai, W.~D. Gropp, D.~E. Keyes, and M.~D. Tidriri}, {\em Parallel
  implicit methods for aerodynamics}, in Domain Decomposition Methods in
  Scientific and Engineering Computing: Proceedings of the Seventh
  International Conference on Domain Decomposition, vol.~180 of Contemporary
  Mathematics, Providence, Rhode Island, 1994, American Mathematical Society,
  pp.~465--470.

\bibitem{YCai_IMNavon_1994a}
{\sc Y.~Cai and I.~M. Navon}, {\em Parallel domain--decomposed preconditioners
  in finite element shallow water flow modeling}, in Domain Decomposition
  Methods in Scientific and Engineering Computing: Proceedings of the Seventh
  International Conference on Domain Decomposition, vol.~180 of Contemporary
  Mathematics, Providence, Rhode Island, 1994, American Mathematical Society,
  pp.~471--476.

\bibitem{FCamilli_MFalcone_PLanucara_ASeghini_1994a}
{\sc F.~Camilli, M.~Falcone, P.~Lanucara, and A.~Seghini}, {\em A domain
  decomposition method for {B}ellman equations}, in Domain Decomposition
  Methods in Scientific and Engineering Computing: Proceedings of the Seventh
  International Conference on Domain Decomposition, vol.~180 of Contemporary
  Mathematics, Providence, Rhode Island, 1994, American Mathematical Society,
  pp.~477--483.

\bibitem{CCanuto_ARusso_1994a}
{\sc C.~Canuto and A.~Russo}, {\em Self--adaptive coupling of mathematical
  models and/or numerical methods}, in Domain Decomposition Methods in Science
  and Engineering: The Sixth International Conference on Domain Decomposition,
  vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994,
  American Mathematical Society, pp.~35--44.

\bibitem{TFChan_TPMathew_1994b}
{\sc T.~F. Chan and T.~P. Mathew}, {\em Doamin decomposition preconditioners
  for convection diffusion problems}, in Domain Decomposition Methods in
  Science and Engineering: The Sixth International Conference on Domain
  Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode
  Island, 1994, American Mathematical Society, pp.~157--175.

\bibitem{TFChan_BFSmith_1994a}
{\sc T.~F. Chan and B.~F. Smith}, {\em Domain decomposition and multigrid
  algorithms for elliptic problems on unstructured meshes}, in Domain
  Decomposition Methods in Scientific and Engineering Computing: Proceedings of
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\end{thebibliography}

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