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 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 DouglasSubject: 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 http://www.ica.uni-stuttgart.de/formular/EMG96_formular.html Send the abstracts by e-mail to emg96@ica.uni-stuttgart.de Ninth International Conference on Domain Decomposition Methods June 3-8, 1996, by the Hardanger Fjord, near Bergen, Norway. See the URL http://www.ii.uib.no/dd9/ International Linear Algebra Year at CERFACS, Workshop on Iterative Methods, June 11-14, 1996, UNESCO Centre, Toulouse, France. See the URL http://www.cerfacs.fr/~wlay/LAY/lay.html Conference on Algebraic Multilevel Iteration Methods with Applications June 13-15, 1996, University of Nijmegen, The Netherlands. Send e-mail to amli96@sci.kun.nl 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: Trade@cup.cam.ac.uk. Features * Complete descriptions of most standard domain decomposition, multilevel, and multigrid algorithms. * All algorithms carefully presented using easy-to-understand matrix notation. * Numerical results indicating the fundamental behavior of the algorithms. * 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 ABSTRACT 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/mg-pre.ps.gz 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 COPPER MOUNTAIN CONFERENCE ON MULTIGRID METHODS TO BE HELD APRIL 6-11, 1997 IN COPPER MOUNTAIN, COLORADO TENTATIVE SHEDULE IS AS FOLLOWS: 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 Equations - 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 http://www.tc.cornell.edu/Events/Advanced.Aug96/ 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) Topics: 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: - http://www.numerik.uni-kiel.de/gamm.html 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 corrections. [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- 117. [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- 499. [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- 191. [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, 1980. [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- 446. [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- 308. [102] P. H. Worley, Information Requirements and the Implica- tiosn for Parallel Computation, PhD thesis, Stanford Uni- versity, Department of Computer Science, Stanford, CA, 1989. [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 **************************