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Volume 6, Number 5 (approximately May 31, 1996)

Today's topics:

     Important Dates
     Domain Decomposition Book
     Paper on Auxiliary Grid Multilevel Preconditioners
     Virtual Proceedings for ParMGM96 (Strobl, Austria)
     Virtual Preproceedings for CMCIM96 (Copper Mountain, CO, USA)
     1997 Copper Mountain Multigrid Conference
     Workshop on Advanced Topics in High Performance Computing 
     Numerical Treatment of Multi-Scale Problems
     Some of the new entries in the bibliography


Date: Thu, 30 May 1996 13:23:20 -0500
From: Craig Douglas 
Subject: Important Dates

Abstracts are due *** June 1 *** for the European Multigrid Conference (EMG96)
(October 1-4) which will be held at ICA at the University of Stuttgart.  For
further information and registration, see the URL
Send the abstracts by e-mail to

Ninth International Conference on Domain Decomposition Methods
June 3-8, 1996, by the Hardanger Fjord, near Bergen, Norway.  See the URL

International Linear Algebra Year at CERFACS, Workshop on Iterative Methods,
June 11-14, 1996, UNESCO Centre, Toulouse, France.  See the URL

Conference on Algebraic Multilevel Iteration Methods with Applications
June 13-15, 1996, University of Nijmegen, The Netherlands.  Send e-mail to

Also, see the 3 conference announcements later in this issue!


Date: Fri, 31 May 1996 08:22:51 -0500 (CDT)
From: Barry Smith 
Subject: Domain Decomposition Book

Domain Decomposition

Parallel Multilevel Methods for Elliptic Partial Differential Equations

Barry Smith, Petter Bjorstad, and William Gropp

The emergence of parallel computers and their potential for the numerical
solution of Grand Challenge problems has led to a large amount of research
in domain decomposition methods. This book presents an easy-to-read
discussion of domain decomposition algorithms, their implementation and
analysis. This book is ideal for graduate students about to embark on a
career in computational science. It will also be a valuable resource for all
those interested in parallel computing and numerical computational methods.

Cambridge University Press: Europe- North America

ISBN 0-521-49589-X

Domain Decomposition will be available at your favorite bookstore or order
directly from Cambridge.

Ordering Information

   * North America: Price $34.95. Telephone orders 1-800-872-7423.
     Electronic orders.
   * Australia: Cambridge University Press, Australian Branch, 10 Stamford
     Rd (or P.O. Box 85), Oakleigh, Victoria. 3166. Phone: 03 568 0322. Fax:
     03 568 1517.
   * Europe and elsewhere: Customer Services Dept, Cambridge University
     Press, The Edinburgh Building, Cambridge CB2 2RU, UK. Phone: +44 (1223)
     325970. Fax: +44 (1223) 325959. Email:


   * Complete descriptions of most standard domain decomposition,
     multilevel, and multigrid algorithms.
   * All algorithms carefully presented using easy-to-understand matrix
   * Numerical results indicating the fundamental behavior of the
   * Careful development of the convergence theory for these methods written
     for a wide audience.
   * Discussion of parallel implementations of these methods.

Table of Contents

   * Chapter 1: One Level Algorithms
        o 1.1 Classical Alternating Schwarz Method
        o 1.2 Approximate Solvers
        o 1.3 Many Subdomains
        o 1.4 Convergence Behavior
        o 1.5 Implementation Issues
        o 1.6 Variational Formulation
   * Chapter 2: Two Level Algorithms
        o 2.1 Subdomain Solves Are Not Sufficient
        o 2.2 A Simple Two Level Method
        o 2.3 General Two Level Methods
        o 2.4 Coarse Grid Corrections
        o 2.5 Convergence Behavior
        o 2.6 Implementation Issues
        o 2.7 Fourier Analysis of Two Level Methods
        o 2.8 Variational Formulation
   * Chapter 3: Multilevel Algorithms
        o 3.1 Additive Multilevel Schwarz Methods
        o 3.2 Multiplicative Multilevel Schwarz Methods
        o 3.3 Full Multigrid
        o 3.4 Practical Multilevel Methods
        o 3.5 Multilevel Methods as Classical Jacobi and Gauss-Seidel
        o 3.6 Complexity Issues
        o 3.7 Implementation Issues
        o 3.8 Variational Formulation
   * Chapter 4: Substructuring Methods
        o 4.1 Direct Substructuring Methods
        o 4.2 The Two Subdomain Case
        o 4.3 Many Subdomains
        o 4.4 Inexact Subdomain Solves
        o 4.5 Implementation Issues
        o 4.6 Variational Formulation
   * Chapter 5: A Convergence Theory
        o 5.1 Abstract Convergence Analysis
        o 5.2 Analysis of Standard Methods
        o 5.3 Indefinite and Nonsymmetric Problems
   * Appendix 1: Preconditioners and Accelerators
   * Appendix 2: Software for Numerical Parallel Computing


Date: Mon, 20 May 96 10:36:22 -0400
From: Sachit Malhotra 
Subject: Paper on Auxiliary Grid Multilevel Preconditioners

       A Characterization of Mapping Unstructured Grids onto Structured
                Grids and Using Multigrid as a Preconditioner

                                 S. Malhotra
                   Yale Center for Parallel Supercomputing
                        Department of Computer Science
                               Yale University
                         New Haven, CT 06520-8285 USA

                                C. C. Douglas
             IBM Research Division, T. J. Watson Research Center
                     Yorktown Heights, NY 10598-0218 and
                        Department of Computer Science
                               Yale University
                         New Haven, CT 06520-8285 USA

                                M. H. Schultz
                   Yale Center for Parallel Supercomputing
                        Department of Computer Science
                               Yale University
                         New Haven, CT 06520-8285 USA


Many problems based on unstructured grids provide a natural multigrid
framework due to using an adaptive gridding procedure.  When the grids are
saved, even starting from just a fine grid problem poses no serious
theoretical difficulties in applying multigrid.

A more difficult case occurs when a highly unstructured grid problem is to be
solved with no hints how the grid was produced.  Here, there may be no natural
multigrid structure and applying such a solver may be quite difficult to do.

Since unstructured grids play a vital role in scientific computing, many
modifications have been proposed in order to apply a fast, robust multigrid
solver.  One suggested solution is to map the unstructured grid onto a
structured grid and then apply multigrid to a sequence of structured grids as
a preconditioner.

In this paper, we derive both general upper and lower bounds on the condition
number of this procedure in terms of computable grid parameters.

We provide examples to illuminate when this preconditioner is a useful (e.g.,
p or h-p formulated finite element problems on semi-structured grids) or
should be avoided (e.g., typical computational fluid dynamics (CFD) or
boundary layer problems).  We show that unless great care is taken, this
mapping can lead to a system with a high condition number which eliminates the
advantage of the multigrid method.

    Editor's Note: in mgnet/papers/Malhotra-Douglas-Schultz/ and
    -------------                                       .../mg-pre.abs


Date: Thu, 30 May 1996 14:42:11 -0500
From: Craig Douglas 
Subject: Virtual Proceedings for ParMGM96 (Strobl, Austria)

The recent, highly successful 9th GAMM Workshop on Parallel Multigrid Methods
will have an electronic proceedings, which will be at the Johannes Kepler
University in Linz, Austria and mirrored on MGNet.

The program, some nice pictures, the abstracts, and the first of the papers
are already on MGNet.  These can be best accessed by your web viewer.

    Editor's Note: in mgnet/Conferences/ParMGM96/...


Date: Thu, 30 May 1996 14:45:39 -0500
From: Craig Douglas 
Subject: Virtual Proceedings for ParMGM96 (Strobl, Austria)

A number of papers from the recent, highly successful 4th Copper Mountain
Conference on Iterative Methods are on MGNet.  I have been (slowly) getting
the web page interface up.  It will improve further over the next month or so,
but you can look at all of the papers that are there now with your web viewer.

As with the multigrid conferences, participants can update their papers
anytime they like.  People who did not contribute a paper to this area who
still want to are welcome to do so.

    Editor's Note: in mgnet/Conferences/CMCIM96/...


Date: Fri, 10 May 1996 13:46:34 -0600
From: Catherine Rachwalski  
Subject: 1997 Copper Mountain Multigrid Conference

                       ANNOUNCING THE 1997
                       ON MULTIGRID METHODS
                            TO BE HELD
                         APRIL 6-11, 1997
Sunday, April 6    - Reception & Registration 7:00 p.m. - 9:00 p.m.
Monday, April 7    - 8:00 - 12:00 meeting  4:00 - 7:00 meeting
Tuesday, April 8   - 8:00 - 12:00 meeting  4:00 - 7:00 meeting
Wednesday, April 9 - 8:00 - 12:00 meeting  4:00 - 6:00 meeting
                                        7:00 - 9:00 p.m. Banquet
Thursday, April 10 - 8:00 - 12:00 meeting  4:00 - 6:00 meeting
Friday, April 11   -  8:00 - 12:00 meeting  4:00 - 6:00 meeting 
                                          conference adjourns
This schedule is subject to change.
Further information will be available later this summer.


Date: Mon, 20 May 1996 15:26:28 -0400
From: Jeanne C. Butler  
Subject: Workshop on Advanced Topics in High Performance Computing 

Workshop on Advanced Topics in High Performance Computing

Cornell Theory Center
Cornell University, Ithaca, NY
Monday, August 19 - Wednesday, August 21, 1996
Registration deadline: July 15, 1996

The Cornell Theory Center (CTC), a nationally funded high performance
computing and communications center, is offering three days of lecture and
discussion that take an in-depth look at specific topics in high performance
computing.  This workshop is intended for intermediate and expert parallel
programmers who are actively involved in research that will benefit from the
topics presented.  The planned session titles are:

- Multigrid Methods
- Object-Oriented Methods for the Solution of Partial Differential
- A Parallel Partial Differential Equation Solver for Fluid
  Dynamics Computations
- Special Topics in HPF Programming
- Domain Decomposition and Parallel Code Optimization
- Quantum Monte Carlo Methods for Continuum Systems
- Data Explorer for Scientific Visualization
- Iterative Methods

For more information on this workshop and access to the registration form,
please see

    Editor's Note: in mgnet/conferences/cornell-0896.txt


Date: Fri, 24 May 1996 12:53:36 +0200 (MET DST)
From: Jens Burmeister 
Subject: Numerical Treatment of Multi-Scale Problems

First Announcement                            

13th GAMM-Seminar Kiel on 
Numerical Treatment of Multi-Scale Problems
January 24th to 26th, 1997.

Chairmanship: W. Hackbusch (Kiel), G. Wittum (Stuttgart)


Numerical Treatment and Implementation Aspects of 
  - problems 
      - defined on complicated geometries, 
      - with highly varying coefficients, 
  - coarsening strategies for 
      - multi-level methods, 
      - finite element spaces, 
  - (discrete) homogenisation techniques, 
  - multi-scale discretisations. 

See for more information: 


    Editor's Note: in mgnet/conferences/gamm-0197.txt


Date: Tue, 30 Apr 1996 10:10:10 -0500
From: Craig Douglas 
Subject: Some of the new entries in the bibliography

Here are some recent new entries.  As usual, please send additions and

[41] O.  Kolp  and  H.  Mierendorff, Performance estimations
         for  SUPRENUM  systems,  Parallel  Comput.,  7  (1988),
         pp. 357-366.
[42] O. Kolp,  H. Mierendorff,  and W. Seidel, Analysis of
         multigrid  methods  for  non-shared  memory  systems  by  a
         simple performance model, in CONPAR 86: Conference on
         Algorithms and Hardware for Parallel Processing, Berlin,
         1986, Springer-Verlag, pp. 95-103.
[43] B. Koren, Upwind discretization of the steady Navier-Stokes
         equations, Int. J. Numer. Meth. Fluids, 11 (1990), pp. 99-
[44] N.  Kukutsu,  N.  Yoshida,  and  I.  Fukai,  Application of
         multigrid  technique  to  the  spatial  network  method,  Elec.
         Comm.  Japan,  Part  2,  74  (1991),  pp.  34-42.   Trans-
         lated from Denshi Joho Tsushin Gakkai Ronbunshi, 73-C-1
         (1990), pp. 493-500.
[45] C.-C. J. Kuo and B. C. Levy, Two-color Fourier analysis of
         the multigrid method with red-black Gauss-Seidel smooth-
         ing, Appl. Math. Comput., 29 (1989), pp. 69-87.
[46] J. K. Lee, Y.-Y. Chen, C. A. Lin, and C. M. Lu, Mod-
         elling three-dimensional gas-turbine-combustor-model flows
         on a parallel machine with distributed memory, in Parallel
         Computational Fluid Dynamics, Elsevier Science Publish-
         ers B.V. (North-Holland), Amsterdam, 1995, pp. 477-484.
[47] X.-J. Li and A. D. Sokal, Rigorous lower bound on the dy-
         namic critical exponent of some multigrid Swensen-Wang
         algorithms, Phys. Rev. Lett., 67 (1991), pp. 1482-1485.
[48] Y. Li, S. Holmberg, A. Paprocki, and Y.-Q. Tang, Sim-
         ulation of room flows with small ventilation openings in a
         local grid-refinement technique, Building Serv. Eng. Res.
         Technol., 15 (1993), pp. 1-10.
[49] C.  Liddiard,  Charge  Integration  and  Multigrid  Techniques
         in  Semiconductor  Simulation,  PhD  thesis,  University  of
         Swansea, Swansea, Wales, UK, 1986.
[50] C. Liddiard and P. Mole, A multigrid approach to solv-
         ing Poisson's equation for a p-n diode,  in Simulation of
         Semiconductor Devices and Processes, vol. 3, Technoprint,
         Bologna, 1989, pp. 485-493.
[51] J.  Linden,  B.  Steckel,  and  K.  St"uben, Parallel multi-
         grid solution of Navier-Stokes equations on general 2D do-
         mains, Parallel Comput., 7 (1988), pp. 461-475.
[52] K. Lust,  J. DeKeyser,  and D. Roose, A parallel block-
         structured Euler/Navier-Stokes code with adaptive refine-
         ment  and  run-time  load  balancing  on  the  iPSC-860  hy-
         percube, in Parallel Computational Fluid Dynamics, Else-
         vier Science Publishers B.V. (North-Holland), Amsterdam,
         1995, pp. 243-250.
[53] D. J. Mavriplis and A. Jameson, Multigrid solution of the
         Navier-Stokes equations on triangular meshes, AIAA J.,
         28 (1990), pp. 1415-1425.
[54] O. A. McBryan, New architectures:  performance highlights
         and new algorithms, Parallel Comput., 7 (1988), pp. 477-
[55] D. R. McCarthy and W. R. Jones, Adaptive domain de-
         composition and parallel CFD, in Parallel Computational
         Fluid Dynamics, Elsevier Science Publishers B.V. (North-
         Holland), Amsterdam, 1995, pp. 31-40.
[56] S.  F.  McCormick  and  J.  W.  Ruge,  Algebraic multigrid
         methods  applied  to  problems  in  computational  structural
         mechanics, in State-of-the-Art Surveys on Computational
         Mechanics, ASME, New York, 1989, pp. 237-270.
[57] P. Mehring and E. Aposporidis, Multi-level simulator for
         VLSI  -  an  overview,  in  PARLE:  Parallel  Architectures
         and Language Europe 1987, vol. 1, Berlin, 1987, Springer-
         Verlag, pp. 446-460.
[58] E.  M'emin,  F.  Heitz,  and  F.  Charot,  Efficient  parallel
         multigrid relaxation algorithms for Markov random field-
         based  low-level  vision  applications,  in  Proceedings  1994
         IEEE Computer Society Conference on Computer Vision
         and Pattern Recognition, Los Alamitos, CA, 1994, IEEE
         Computer Society Press, pp. 644-648.
[59] H. Mierendorff and U. Trottenberg, Performance eval-
         uation for SUPRENUM systems, in Evaluating Supercom-
         puters:  Strategies for Exploiting, Evaluating, and Bench-
         marking Computers with Advanced Architectures, Chap-
         man and Hall, 1990, pp. 95-114.
[60] W. F. Mitchell, Unified Multilevel Adaptive Finite Element
         Methods for Elliptic Problems, PhD thesis, Univ. of Illinois
         at Urbana-Champaign, Urbana, Illinois, 1988.
[61] R.  Morandi  and  A.  Sestini,  Parallel  coputing  multigrid
         methods, Supercomputer, 38 (1990), pp. 39-47.
[62] J. E. Morel and T. A. Manteuffel, An angular multi-
         grid  acceleration  technique  for  the  sn  equations  with
         highly forward-peaked scattering, Trans. ASME, 61 (1990),
         pp. 165-166.
[63] R.  Mu"noz,  Theoretical  analysis  of  some  spectral  multigrid
         methods,  Comput.  Meth.  Appl.  Mech.  Eng.,  80  (1990),
         pp. 287-294.
[64] A. E. Mynett, P. Wesseling, A. Segal, and C. G. M.
         Kassels, The ISNaS incompressible Navier-Stokes solver:
         invariant discretization, Appl. Sci. Res., 48 (1991), pp. 175-
[65] W. Najjar and J.-L. Gaudiot, A hierarchical data-driven
         model  for  multi-grid  problem  solving,  in  High  Perfor-
         mance Computer Systems, Elsevier Science Publishers B.V.
         (North-Holland), Amsterdam, 1988, pp. 67-78.
[66] F.  Nataf  and  F.  Rogier,  A  Schur  type  method  based
         on outflow boundary condition, in Parallel Computational
         Fluid Dynamics, Elsevier Science Publishers B.V. (North-
         Holland), Amsterdam, 1995, pp. 359-361.
[67] H. Nestle and H. Wollnik, FAMA/FAMULAS: Programs
         for the fast calculation of two-dimensional distributions of
         magnetic and electric fields, Nucl. Inst. Meth. Phys. Res.,
         A276 (1989), pp. 568-572.
[68] J.  Nietro,  Multigrid  methods  in  network  optimization:
         Overview and appraisal, Master's thesis, Naval Postgradu-
         ate School, Monterey, CA, 1994.
[69] V. Pan and J. Reif, On the bit-complexity of discrete so-
         lutions of PDEs:  compact multigrid,  in Automata,  Lan-
         guages,  and  Programming,  Springer-Verlag,  New  York,
         1990, pp. 612-625.
[70] I.  D.  Parsons,  The  implementation  of  an  element  level
         multigrid algorithm on the Alliant FX/8, Computer Phys.
         Comm., 53 (1989), pp. 337-348.
[71] I. D. Parsons and J. F. Hall, A finite element investigation
         of the elastostatic state near a three dimensional edge crack,
         Eng. Fract. Mech., 33 (1989), pp. 45-63.
[72] K. Peinze, The SUPRENUM preprototype:  status and expe-
         riences, Parallel Comput., 7 (1988), pp. 297-313.
[73] L. Plank, E. Stein, and D. Bischoff, Accuracy and adap-
         tivity in the numerical analysis of thin-walled structures,
         Comput. Meth. Appl. Mech. Eng., 82 (1990), pp. 223-256.
[74] W.  H.  Press  and  S.  A.  Teukolsky,  Multigrid  methods
         for boundary value problems II, Comput. Phys., 5 (1991),
         pp. 626-629.
[75] R. Ramamurti and R. L"ohner, Simulation of complex in-
         compressible flows using a finite element solver on MIMD
         machine, in Parallel Computational Fluid Dynamics, Else-
         vier Science Publishers B.V. (North-Holland), Amsterdam,
         1995, pp. 443-450.
[76] R. Rankin, J. P. DeVilliers, and J. C. Swanson, Par-
         allel magnetohydrodynamics on Myrias MIMD computers,
         in Parallel Computational Fluid Dynamics,  Elsevier Sci-
         ence Publishers B.V. (North-Holland), Amsterdam, 1995,
         pp. 117-124.
[77] R.  Rannacher,  Parallel  solution  methods  for  the  Navier-
         Stokes equations, in Parallel Computational Fluid Dynam-
         ics, Elsevier Science Publishers B.V. (North-Holland), Am-
         sterdam, 1995, pp. 61-70.
[78] U. van Rienen and T. Weiland, Impedance calculation with
         URMEL-I using multigrid methods, IEEE Trans. Magnet-
         ics, 26 (1990), pp. 743-746.
[79] G. Robinson, A simple parallel algebraic multigrid, in Occam
         and the Transputer, IOS Press, 1991, pp. 62-75.
[80] G. Rodrigue and T. Ferretta, Coarse grid acceleration of
         some domain decomposition methods on multiprocessors, in
         Aspects of Computation on Asynchronous Parallel Proces-
         sors, Amsterdam, 1989, North Holland, pp. 255-260.
[81] J.  R.  van  Rosendale,  Rapid  Solution  of  Finite  Element
         Equations on Locally Refined Grids by Multi-Level Meth-
         ods, PhD thesis, University of Illinois, Urbana-Champaign,
[82] N. Satofuka, M. Obata, and T. Suzuki, Parallel compu-
         tation of 2-D potential and Euler equations on transputer
         arrays, in Parallel Computational Fluid Dynamics, Else-
         vier Science Publishers B.V. (North-Holland), Amsterdam,
         1995, pp. 525-532.
[83] P. Schiano and A. Matrone, Parallel CFD applications:
         experiences on scalable distributed multicomputers, in Par-
         allel Computational Fluid Dynamics, Elsevier Science Pub-
         lishers B.V. (North-Holland), Amsterdam, 1995, pp. 3-12.
[84] A.  S.-L.  Shieh,  On  the  solution  of  the  advection-diffusion
         equation  on  a  non-uniform  mesh  by  a  RMP-accelerated
         multigrid  method,  in  Advanced  Computational  Methods
         in Heat Transfer, Computational Mechanics Publications,
         1990, pp. 275-281.
[85] T. Simchony and R. Chellappa, Direct analytical methods
         for solving Poisson equations in computer vision problems,
         IEEE Trans. Pattern Machine Intell., 12 (1990), pp. 435-
[86] SMijalkovi'c,  DPanti'c,  and  NStojadinovi'c,  On  effi-
         ciency of multigrid methods in two-dimensional impurity
         redistribution simulation, in Simulation of Semiconductor
         Devices and Processes, vol. 3, Technoprint, Bologna, 1989,
         pp. 463-474.
[87] K.  Solchenbach,  Grid  applications  on  distributed  mem-
         ory architectures: implementation and evaluation, Parallel
         Comput., 7 (1988), pp. 341-356.
[88] K.  Solchenbach  and  U.  Trottengerg,  SUPRENUM:
         system essentials and grid applications, Parallel Comput.,
         7 (1988), pp. 265-281.
[89] J. Steelant and E. Dick, A multigrid method for the com-
         pressible Navier-Stokes equations coupled to the k - ffl tur-
         bulence equations, Int. J. Num. Meth. Heat Fluid Flow, 4
         (1994), pp. 99-113.
[90] P. N. Swarztrauber and R. A. Sweet, Vector and paral-
         lel methods for the direct solution of poisson's equation, J.
         Comput. Appl. Math., 27 (1989), pp. 241-263.
[91] R. Szeliski, Fast surface interpolation using hierarchical ba-
         sis functions, IEEE Trans. Pattern Anal. Mach. Intell., 12
         (1990), pp. 513-528.
[92] S. Ta'asan, Multigrid method for stability problems, J. Super-
         conput., 3 (1988), pp. 261-274.
[93] M. S. Tai, Parallel implicit Navier-Stokes solver on the Intel
         Paragon, in Parallel Computational Fluid Dynamics, Else-
         vier Science Publishers B.V. (North-Holland), Amsterdam,
         1995, pp. 333-340.
[94] R. Teigland and G. E. Fladmark, Multilevel methods in
         porous media flow, in Second European Conference on the
         Mathematics of Oil Recovery, Paris, 1990, E'ditions Tech-
         nip, pp. 355-358.
[95] C.-A. Thole and U. Trottenberg, A short note on stan-
         dard parallel multigrid algorithms for 3D problems, Appl.
         Math. Comput., 27 (1988), pp. 101-115.
[96] J. F. Thompson, The national grid project, Comput. Syst.
         Eng., 3 (1992), pp. 393-399.
[97] D.  Tromeur-Dervout  and  F.-X.  Roux,  Parallelization
         via domain decomposition techniques of multigrid and ADI
         solvers for Navier-Stokes equations, in Parallel Computa-
         tional Fluid Dynamics,  Elsevier Science Publishers B.V.
         (North-Holland), Amsterdam, 1995, pp. 349-356.
[98] D. Vanderstraeten, R. Keunings, and C. Farhat, Op-
         timization of mesh partitions and impact on parallel CFD,
         in Parallel Computational Fluid Dynamics,  Elsevier Sci-
         ence Publishers B.V. (North-Holland), Amsterdam, 1995,
          pp. 233-239.
 [99] Ch.  Walshaw  and  M.  Berzins,  Adaptive time-dependent
          CFD on distributed unstructured meshes, in Parallel Com-
          putational  Fluid  Dynamics,  Elsevier  Science  Publishers
          B.V. (North-Holland), Amsterdam, 1995, pp. 191-198.
[100] S. R. White, J. W. Wilkins, and M. P. Teter, Finite-
          element method for electronic structure, Phys. Rev. B, 39
          (1989), pp. 5819-5833.
[101] C. M. Woods and D. E. Brewe, The solution of the Elrod
          algorithm for a dynamically loaded bearing using multigrid
          techniques, J. Tribol. Trans. ASME, 111 (1989), pp. 302-
[102] P. H. Worley, Information Requirements and the Implica-
          tiosn for Parallel Computation, PhD thesis, Stanford Uni-
          versity, Department of Computer Science, Stanford, CA,
[103] J. W. Yokota, Diagonally inverted lower-upper factored im-
          plicit multigrid scheme for the three-dimensional Navier-
          Stokes equations, AIAA J., 28 (1990), pp. 1642-1649.
[104] J.  W.  Yokota  and  D.  A.  Caughey,  LU  implicit  multi-
          grid algorithm for the three-dimensional Eueler equations,
          AIAA J., 26 (1988), pp. 1061-1069.
[105] S. Yoon and A. Jameson, Lower-upper symmetric-Gauss-
          Seidel method for the Euler and Navier-Stokes equations,
          AIAA J., 26 (1988), pp. 1025-1026.
[106] L.-B. Zhang, Semi-coarsening in multigrid solution of steady
          incompressible Navier-Stokes equations, J. Comput. Math.,
          8 (1990), pp. 92-97.


End of MGNet Digest