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) Today's editor: Craig Douglas (douglas-craig@cs.yale.edu) Volume 4, Number 4 (April 30, 1994) Today's topics: Correction To Randy Bank's Book Announcement (V4N03) Papers by Kornhuber et al LPARX system + MG codes MGGHAT thesis PostScript Files of J. P. Shao Recent Additions to the Bibliography Database and a Question ------------------------------------------------------- Date: Wed, 06 Apr 94 14:46:20 EST From: bdilisi@siam.org Subject: Correction To Randy Bank's Book Announcement (V4N03) The announcement regarding Randy Bank's book, PLTMG Users' Guide 7.0 list SIAM's address incorrectly. The correct address for SIAM is: SIAM 3600 University City Science Center Philadelphia, PA 19104-2688 (215) 382-9800 Fax: (215) 386-7999 E-Mail: service@siam.org Thank you, B. DiLisi, SIAM ------------------------------------------------------- Date: Fri, 8 Apr 94 09:42:05 +0200 From: kornhuber@sc.ZIB-Berlin.DE (Dr. Ralf Kornhuber) Subject: Papers by Kornhuber et al A Posteriori Error Estimates for Elliptic Problems in Two and Three Space Dimensions Folkmar Bornemann Freie Universit\"at Berlin, Arnimalle 2--6, D-14195 Berlin Bodo Erdmann and Ralf Kornhuber Konrad--Zuse--Zentrum Berlin, Heilbronner Str. 10, D-10711 Berlin Abstract: Let u in H be the exact solution of a given self--adjoint elliptic boundary value problem, which is approximated by some u_S in S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error || u - u_S ||, measuring the (local) quality of u_S, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well--known class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations. Editor's Note: in mgnet/papers/Kornhuber/error.{ps,abs}. ------------- Monotone Multigrid Methods for Variational Inequalities I Ralf Kornhuber Konrad--Zuse--Zentrum Berlin, Heilbronner Str. 10, D-10711 Berlin Abstract: We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify well--known relaxation methods by extending the set of search directions. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, the standard monotone multigrid method and a truncated version. For the considered model problems, the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse--grid transport, while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case. Editor's Note: in mgnet/papers/Kornhuber/obstacle.{ps,abs}. ------------- Monotone Multigrid Methods for Variational Inequalities II Ralf Kornhuber Konrad--Zuse--Zentrum Berlin, Heilbronner Str. 10, D-10711 Berlin Abstract: We derive fast solvers for discrete elliptic variational inequalities of the second kind as resulting from the approximation by piecewise linear finite elements. Following the first part of this paper, monotone multigrid methods are considered as extended underrelaxations. Again, the coarse grid corrections are localized by suitable constraints, which in this case are fixed by fine grid smoothing. We consider the standard monotone multigrid method induced by the multilevel nodal basis and a truncated version. Global convergence results and asymptotic estimates for the convergence rates are given. The numerical results indicate a significant improvement in efficiency compared with previous multigrid approaches. Editor's Note: in mgnet/papers/Kornhuber/stefan.{ps,abs}. ------------- ------------------------------------------------------- Date: Sun, 10 Apr 94 21:53:24 -0700 From: baden@cs.ucsd.edu (Scott B. Baden) Subject: LPARX system + MG codes I'm happy to announce that v1.0 of the LPAR-X system is now available. Included are C++ application codes, among them, multigrid. LPARX provides efficient run-time support for dynamic, non-uniform scientific calculations running on MIMD distributed memory architectures, and is intended for block structured and multilevel applications involving structured meshes, and for particle methods. LPARX applications are portable across a diversity of MIMD machines, and may be written in a form that is partially independent of the problem dimension. They may be debugged on a workstation simplifying code development. The LPARX software is implemented as a C++ class library. It currently run on the Intel Paragon, the CM-5, KSR-1, nCube/2, networks of workstations under PVM, single processor workstations, and on the Cray C-90 (single processor at the moment). LPAR-X will soon run on the C-90 in multitasked mode, and on the T3D. The LPARX distribution is available via anonymous ftp on site ftp.sdsc.edu. Look in directory pub/sdsc/parallel/lparx. The distribution is also available on anonymous ftp site cs.ucsd.edu in directory pub/baden/LPARX. Technical reports are also available in subdirectory "reports." Technical questions should be sent to Scott Kohn at skohn@cs.ucsd.edu. Scott Baden Scott Kohn Editor's Note: in mgnet/lparx. What about an IBM SP-1/SP-2 version??? ------------- author's reply>> Craig, we haven't yet ported LPARX over to author's reply>> the IBM. This is on our list of things to author's reply>> do (near the top). ------------------------------------------------------- Date: Tue, 19 Apr 94 15:43:45 EDT From: mitchell@cam.nist.gov (William_F._Mitchell x3808) Subject: MGGHAT thesis A year ago I released MGGHAT, my adaptive multilevel high order finite element program for elliptic PDEs, to netlib and mgnet. Since then I have received over 40 requests for a copy of my PhD thesis "Unified multilevel adaptive finite element methods for elliptic problems". I guess it's time I put it on an ftp server, rather than emailing it all the time. William F. Mitchell | mitchell@cam.nist.gov Applied and Computational Mathematics Division | na.wmitchell@na-net.ornl.gov National Institute of Standards and Technology | Voice: (301) 975-3808 Gaithersburg, MD 20899 | Fax: (301) 990-4127 Editor's Note: in mgnet/papers/Mitchell/thesis.{abs,ps}. Below is the ------------- abstract. The PostScript file comes in a gzip'ed flavor, too. Report No. UIUCDCS-R-88-1436, Dept. of Computer Science, University of Illinois at Urbana-Champaign UNIFIED MULTILEVEL ADAPTIVE FINITE ELEMENT METHODS FOR ELLIPTIC PROBLEMS William F. Mitchell Bldg 101 Rm A238 NIST Gaithersburg, MD 20899 mitchell@cam.nist.gov (Work performed while at the University of Illinois at Urbana-Champaign and partially funded by DOE grant DEFG02-87ER25026) Many elliptic partial differential equations can be solved numerically with near optimal efficiency through the uses of adaptive refinement and multigrid solution techniques. It is our goal to develop a more unified approach to the combined process of adaptive refinement and multigrid solution which can be used with high order finite elements. The basic step of the refinement process is the bisection of a pair of triangles, which corresponds to the addition of one or more basis functions to the approximation space. An approximation of the resulting change in the solution is used as an error indicator to determine which triangles to divide. The multigrid iteration uses a red-black Gauss- Seidel relaxation in which the black relaxations are used only locally. The grid transfers use the change between the nodal and hierarchical bases. This multigrid iteration requires only O(N) operations, even for highly nonuniform grids, and is defined for any finite element space. The full multigrid method is an optimal blending of the processes of adaptive refinement and multigrid iteration. So as to minimize the number of operations required, the duration of the refinement phase is based on increasing the dimension of the approximation space by some fixed factor which is determined to be the largest possible for the given error-reducing power of the multigrid iteration. The result is an algorithm which (i) uses only O(N) operations with a reasonable constant of proportionality, (ii) solves the discrete system to the accuracy of the discretization error, (iii) is able to achieve the optimal order of convergence of the discretization error in the presence of singularities. Numerical experiments confirm this for linear, quadratic and cubic elements. It is believed that the method can also be applied to more practical problems involving systems of PDE's, time dependence, and three spacial dimensions. ------------------------------------------------------- Date: Tue, 26 Apr 1994 18:41:13 EDT From: Jian Ping ShaoSubject: PostScript Files Thanks for your interesting on my applications. Here is my thesis abstract. I will send you two postscript files of my recent papers in the following two e-mails. Jian Ping Shao Editor's Note: mgnet/papers/Shao/jpshao{1,2,3}.ps with jpshao1.abs. ------------- (abstract below). Domain Decomposition Algorithms Dissertation Abstract Jian Ping Shao Advisor: Tony F. Chan Department of Mathematics University of California, at Los Angeles Los Angeles, CA 90024 Approval Date: September 1993 Domain decomposition (DD) has been widely used to design parallel efficient algorithms for solving elliptic problems. In this thesis, we focus on improving the efficiency of DD methods and applying them to more general problems. Specifically, we propose efficient variants of the vertex space DD method and minimize the complexity of general DD methods. In addition, we apply DD algorithms to coupled elliptic systems, singular Neumann boundary problems and linear algebraic systems. We successfully improve the vertex space DD method of Smith by replacing the exact edge, vertex dense matrices by approximate sparse matrices. It is extremely expensive to calculate, invert and store the exact vertex and edge Schur complement dense sub-matrices in the vertex space DD algorithm. We propose several approximations for these dense matrices, by using {\em Fourier approximation} and an algebraic {\em probing} technique. Our numerical and theoretical results show that these variants retain the fast convergence rate and greatly reduce the computational cost. We develop a simple way to reduce the overall complexity of domain decomposition methods through choosing the coarse grid size. For sub-domain solvers with different complexities, we derive the optimal coarse grid size $H_{opt},$ which asymptotically minimizes the total computational cost of DD methods under the sequential and parallel environments. The overall complexity of DD methods is significantly reduced by using this optimal coarse grid size. We apply the additive and multiplicative Schwarz algorithms to solving coupled elliptic systems. Using the Dryja-Widlund framework, we prove that their convergence rates are independent of both the mesh and the coupling parameters. We also construct several approximate interface sparse matrices by using Sobolev inequalities, Fourier analysis and probe technique. We further discuss the application of DD to the singular Neumann boundary value problems. We extend the general framework to these problems and show how to deal with the null space in practice. Numerical and theoretical results show that these modified DD methods still have optimal convergence rate. By using the DD methodology, we propose algebraic additive and multiplicative Schwarz methods to solve general sparse linear algebraic systems. We analyze the eigenvalue distribution of the iterative matrix of each each algebraic DD method to study the convergence behavior. ------------------------------------------------------- Date: Sat, 30 Apr 94 20:12:63 EST From: Craig Douglas Subject: Recent Additions to the Bibliography Database and a Question I have had several requests to break the database up into a number of smaller files. In fact, I keep it in 27 files and produce the large one using a trivial makefile. If enough people ask, I will unprotect the directory containing the partial database. Here is a simple LaTeX file which you can print out or preview. The entries are all in mgnet/bib/mgnet.bib (now up to 2,000 entries and still growing). \documentstyle[12pt]{siam} \sloppy \begin{document} \bibliographystyle{siam} Excerpts from the MGNet Bibliogrpahy. \nocite{JCAdams_1993a} \nocite{MPAllen_1993a} \nocite{KAmaratunga_JRWilliams_1993a} \nocite{AArnone_MSLiou_LAPovinelli_1993a} \nocite{MBaker_GMack_MSpeh_1993a} \nocite{CBasler_WTornig_1993a} \nocite{JBelak_1993a} \nocite{GBerkooz_ESTiti_1993a} \nocite{RBhogeswara_JEKillough_1993a} \nocite{IPBoglaev_VVSirotkin_1993a} \nocite{FBornemann_HYserentant_1993a} \nocite{ABorzi_AKoubek_1993a} \nocite{CBouman_KSauer_1992a} \nocite{JHBramble_JEPasciak_1993a} \nocite{ABrandt_WJoppich_JLinden_GLonsdale_ASchueller_1992a} \nocite{MBreuer_DHanel_1993a} \nocite{WLBriggs_VEHenson_1993a} \nocite{DBrown_JHRClarke_MOkuda_TYamazaki_1993a} \nocite{ZCai_IGoldstein_JEPasciak_1993a} \nocite{XCCai_OBWidlund_1993a} \nocite{DACaughey_1993a} \nocite{HMChen_FCBerry_1993a} \nocite{MChen_RTemam_1993a} \nocite{ZChen_1993a} \nocite{ARClare_DPStevens_1993a} \nocite{CRCollins_1993a} \nocite{GBCook_MWChoptuik_MRDubal_SKlasky_RAMatzner_SROliveira_1993a} \nocite{LCrivelli_CFarhat_1993a} \nocite{GDahlquist_1993a} \nocite{MBDavis_GFCarey_1993a} \nocite{FDefaux_IMoccagatta_BRouchouze_TEbrahimi_MKunt_1993a} \nocite{AODemuren_1993a} \nocite{JEDendy_1993b} \nocite{CCDouglas_1994a} \nocite{DDrikakis_ESchreck_1993a} \nocite{FDufaux_MKunt_1992a} \nocite{ADupuy_JEKillough_1993a} \nocite{BGErsland_RTeigland_1993a} \nocite{MAFallavollita_JDMcDonald_DBaganoff_1992a} \nocite{CFarhat_MLesoinne_1993a} \nocite{DAField_YPressburger_1993a} \nocite{PGburzynski_JMaitan_1993a} \nocite{MGrabenstein_BMikeska_1993a} \nocite{MGrabenstein_KPinn_1993a} \nocite{FGrasso_MMarini_1993a} \nocite{MFGuest_PSherwood_JHVLenthe_1993a} \nocite{SNGupta_MZubair_CEGrosch_1992a} \nocite{WHackbusch_1992a} \nocite{YRHakopian_YAKuznetsov_1991a} \nocite{MHarmatz_PLauwers_SSolomon_TWittlich_1993a} \nocite{BHeise_1993a} \nocite{MHolst_FSaied_1993a} \nocite{AHulsebos_1993a} \nocite{SHHwang_SULee_1993a} \nocite{THwang_IDParsons_1992a} \nocite{THwang_IDParsons_1992b} \nocite{THwang_IDParsons_1992c} \nocite{ACIrving_1993a} \nocite{MIsraeli_LVozovoi_AAverbuch_1993a} \nocite{MIsraeli_LVozovoi_AAverbuch_1993b} \nocite{WJoppich_RALorentz_1993a} \begin{thebibliography}{10} \bibitem{JCAdams_1993a} {\sc J.~C. Adams}, {\em {MUDPACK} 2: multigrid software for approximating elliptic partial differential equations on uniform grids with any resolution}, Appl. Math. Comput., 53 (1993), pp.~235--249. \bibitem{MPAllen_1993a} {\sc M.~P. Allen}, {\em Simulation of condensed phases using the distributed array processor}, Theor. Chim. Acta, 84 (1993), pp.~399--411. \bibitem{KAmaratunga_JRWilliams_1993a} {\sc K.~Amaratunga and J.~R. Williams}, {\em Wavelet based {G}reen's function approach to 2{D PDE}s}, Eng. Comput., 10 (1993), pp.~349--367. \bibitem{AArnone_MSLiou_LAPovinelli_1993a} {\sc A.~Arnone, M.~S. Liou, and L.~A. Povinelli}, {\em Multigrid calculation of three-dimensional viscous cascade flows}, J. Propul. Power, 9 (1993), pp.~605--614. \bibitem{MBaker_GMack_MSpeh_1993a} {\sc M.~Baker, G.~Mack, and M.~Speh}, {\em Multigrid meets neural nets}, Nucl. Phys. B, Proc. Suppl., 30 (1993), pp.~269--272. \bibitem{CBasler_WTornig_1993a} {\sc C.~Basler and W.~Tornig}, {\em On monotone including nonlinear multigrid methods and applications}, Comput., 50 (1993), pp.~51--67. \bibitem{JBelak_1993a} {\sc J.~Belak}, {\em Harnessing the killer micros: applications from {LLNL}'s massively parallel computing initiative}, Theor. Chim. Acta, 84 (1993), pp.~315--323. \bibitem{GBerkooz_ESTiti_1993a} {\sc G.~Berkooz and E.~S. Titi}, {\em Galerkin projections and the proper orthogonal decomposition for equivariant equations}, Phys. Lett. A, 174 (1993), pp.~94--102. \bibitem{RBhogeswara_JEKillough_1993a} {\sc R.~Bhogeswara and J.~E. Killough}, {\em Parallel linear solvers for reservoir simulation: A generic approach for existing and emerging computer architectures}, in Proceedings of the SPE Symposium on Reservoir Simulation 1993, Richardson, TX, 1993, Soc of Petroleum Engineers of AIME, pp.~71--82. \bibitem{IPBoglaev_VVSirotkin_1993a} {\sc I.~P. Boglaev and V.~V. Sirotkin}, {\em Computational method for a singular perturbation problem via domain decomposition and its parallel implementation}, Appl. Math. Comput., 56 (1993), pp.~71--95. \bibitem{FBornemann_HYserentant_1993a} {\sc F.~Bornemann and H.~Yserentant}, {\em A basic norm equivalence for the theory of multilevel methods}, Numer. Math., 64 (1993), pp.~455--476. \bibitem{ABorzi_AKoubek_1993a} {\sc A.~Borzi and A.~Koubek}, {\em Multi--grid method for the resolution of thermodynamic {B}ethe ansatz equations}, Comput. Phys. Commun., 75 (1993), pp.~118--126. \bibitem{CBouman_KSauer_1992a} {\sc C.~Bouman and K.~Sauer}, {\em Nonlinear multigrid methods of optimization in {B}ayesian tomographic image reconstruction}, Proc. SPIE - Int. Soc. Opt. Eng., 1766 (1992), pp.~296--306. \bibitem{JHBramble_JEPasciak_1993a} {\sc J.~H. Bramble and J.~E. Pasciak}, {\em New estimates for multilevel algorithms including the v cycle}, Math. Comp., 60 (1993), pp.~447--471. \bibitem{ABrandt_WJoppich_JLinden_GLonsdale_ASchueller_1992a} {\sc A.~Brandt, W.~Joppich, J.~Linden, G.~Lonsdale, and A.~Schueller}, {\em Multigrid Course}, GMD--690, St. Augustin, 1992. \bibitem{MBreuer_DHanel_1993a} {\sc M.~Breuer and D.~Hanel}, {\em A dual time stepping method for 3--{D}, viscous, incompressible vortex flows}, Comput. Fluids, 22 (1993), pp.~467--484. \bibitem{WLBriggs_VEHenson_1993a} {\sc W.~L. Briggs and V.~E. Henson}, {\em Wavelets and multigrid}, SIAM J. Sci. Comput., 14 (1993), pp.~506--510. \bibitem{DBrown_JHRClarke_MOkuda_TYamazaki_1993a} {\sc D.~Brown, J.~H.~R. Clarke, M.~Okuda, and T.~Yamazaki}, {\em A domain decomposition parallelization strategy for molecular dynamics simulations on distributed memory machines}, Comput. Phys. Commun., 74 (1993), pp.~67--80. \bibitem{XCCai_OBWidlund_1993a} {\sc X{.--C.} Cai and O.~B. Widlund}, {\em Multiplicative {S}chwarz algorithms for some nonsymmetric and indefinite problems}, SIAM J. Numer. Anal., 30 (1993), pp.~936--952. \bibitem{ZCai_IGoldstein_JEPasciak_1993a} {\sc Z.~Cai, I.~Goldstein, and J.~E. Pasciak}, {\em Multilevel iteration for mixed finite element systems with penalty}, SIAM J. Sci. Comput., 14 (1993), pp.~1072--1088. \bibitem{DACaughey_1993a} {\sc D.~A. Caughey}, {\em Implicit multigrid techniques for compressible flows}, Comput. Fluids, 22 (1993), pp.~117--124. \bibitem{HMChen_FCBerry_1993a} {\sc H.~M. Chen and F.~C. Berry}, {\em Parallel load--flow algorithm using a decomposition method for space--based power systems}, IEEE Trans. Aero. Electron. Sys., 29 (1993), pp.~1024--1030. \bibitem{MChen_RTemam_1993a} {\sc M.~Chen and R.~Temam}, {\em Nonlinear {G}alerkin method in the finite difference case and wavelet like incremental unknowns}, Numer. Math., 64 (1993), pp.~271--294. \bibitem{ZChen_1993a} {\sc Z.~Chen}, {\em Projection finite element methods for semiconductor device equations}, Comput. Math. Appl., 25 (1993), pp.~81--88. \bibitem{ARClare_DPStevens_1993a} {\sc A.~R. Clare and D.~P. Stevens}, {\em Implementing finite difference ocean circulation models on {MIMD}, distributed memory computers}, Future Gen. Comput. Sys., 9 (1993), pp.~11--18. \bibitem{CRCollins_1993a} {\sc C.~R. Collins}, {\em Computations of twinning in shape--memory materials}, in Proceedings of SPIE -- The International Society for Optical Engineering, vol.~1919, Bellingham, WA, 1993, Society of Photo-Optical Instrumentation Engineers, pp.~30--37. \bibitem{GBCook_MWChoptuik_MRDubal_SKlasky_RAMatzner_SROliveira_1993a} {\sc G.~B. Cook, M.~W. Choptuik, M.~R. Dubal, S.~Klasky, R.~A. Matzner, and S.~R. Oliveira}, {\em Three dimensional initial data for the collision of two black holes}, Phys. Rev. D, Part. Fields Gravit. Cosmol., 47 (1993), pp.~1471--1490. \bibitem{LCrivelli_CFarhat_1993a} {\sc L.~Crivelli and C.~Farhat}, {\em Implicit transient finite element structural computations on {MIMD} systems: {FETI} v.s. direct solvers}, in 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Collection of Technical Papers -- AIAA/ASME Structures, Structural Dynamics and Materials Conference, vol.~1, Washington, DC, 1993, AIAA, pp.~118--130. \bibitem{GDahlquist_1993a} {\sc G.~Dahlquist}, {\em A 'multigrid' extension of the {FFT} for the numerical inversion of {F}ourier and {L}aplace transforms}, BIT, 33 (1993), pp.~85--112. \bibitem{MBDavis_GFCarey_1993a} {\sc M.~B. Davis and G.~F. Carey}, {\em Iterative solution of the stream function vorticity equations using a multigrid solver with finite elements}, Comm. Numer. Meth. Engrg., 9 (1993), pp.~587--594. \bibitem{FDefaux_IMoccagatta_BRouchouze_TEbrahimi_MKunt_1993a} {\sc F.~Defaux, I.~Moccagatta, B.~Rouchouze, T.~Ebrahimi, and M.~Kunt}, {\em Motion compensated generic coding of video based on a multiresolution data structure}, Optical Engng., 32 (1993), pp.~1559--1570. \bibitem{AODemuren_1993a} {\sc A.~O. Demuren}, {\em Characteristics of three--dimensional turbulent jets in crossflow}, Int. J. Engng. Sci., 31 (1993), pp.~899--913. \bibitem{JEDendy_1993b} {\sc J.~E. Dendy}, {\em Multigrid methods for petroleum reservoir simulation on {SIMD} machines}, in Proceedings of the SPE Symposium on Reservoir Simulation, Richardson, TX, 1993, Soc of Petroleum Engineers of AIME, pp.~97--104. \bibitem{CCDouglas_1994a} {\sc C.~C. Douglas}, {\em Some remarks on completely vectorizing point {G}auss--{S}eidel while using the natural ordering}, Advances Comput. Math., 2 (1994), pp.~215--222. \bibitem{DDrikakis_ESchreck_1993a} {\sc D.~Drikakis and E.~Schreck}, {\em Parallel multi--level calculations for viscous compressible flows}, in CFD Algorithms and Applications for Parallel Processors American Society of Mechanical Engineers, Fluids Engineering Division (Publication) FED, vol.~156, ASME, New York, NY, 1993, pp.~9--23. \bibitem{FDufaux_MKunt_1992a} {\sc F.~Dufaux and M.~Kunt}, {\em Multigrid block matching motion estimation with an adaptive local mesh refinement}, in Proceedings of the SPIE, vol.~1818, The International Society for Optical Engineering, 1992, pp.~97--109. \bibitem{ADupuy_JEKillough_1993a} {\sc A.~Dupuy and J.~E. Killough}, {\em Fully implicit simulation on the connection machine}, in Proceedings of the SPE Symposium on Reservoir Simulation, Richardson, TX, 1993, Soc of Petroleum Engineers of AIME, pp.~459--466. \bibitem{BGErsland_RTeigland_1993a} {\sc B.~G. Ersland and R.~Teigland}, {\em Comparison of two cell--centered multigrid schemes for problems with discontinuous coefficients}, Numer. Meth. for PDE, 9 (1993), pp.~265--283. \bibitem{MAFallavollita_JDMcDonald_DBaganoff_1992a} {\sc M.~A. Fallavollita, J.~D. McDonald, and D.~Baganoff}, {\em Parallel implementation of a particle simulation for modeling rarefied gas dynamic flow}, Comput. Syst. Eng., 3 (1992), pp.~283--289. \bibitem{CFarhat_MLesoinne_1993a} {\sc C.~Farhat and M.~Lesoinne}, {\em Automatic partitioning of unstructured meshes for the parallel solution of problems in computational mechanics}, J. Numer. Meth. Engrg., 36 (1993), pp.~745--764. \bibitem{DAField_YPressburger_1993a} {\sc D.~A. Field and Y.~Pressburger}, {\em h--p -- multigrid method for finite element analysis}, J. Numer. Meth. Engrg., 36 (1993), pp.~893--908. \bibitem{PGburzynski_JMaitan_1993a} {\sc P.~Gburzynski and J.~Maitan}, {\em Performance of multigrid network architecture ({MNA}) under uniform load}, in Proceedings of SPIE - The International Society for Optical Engineering, vol.~1784, Bellingham, WA, 1993, Int. Soc. for Optical Engineering, pp.~270--281. \bibitem{MGrabenstein_BMikeska_1993a} {\sc M.~Grabenstein and B.~Mikeska}, {\em Multigrid {M}onte {C}arlo algorithm with higher cycles in the sine {G}ordon model}, Phys. Rev. D (Particles, Fields, Gravitation, and Cosmology), 47 (1993), pp.~3103--3105. \bibitem{MGrabenstein_KPinn_1993a} {\sc M.~Grabenstein and K.~Pinn}, {\em Theoretical analysis of acceptance rates in multigrid {M}onte {C}arlo}, Nucl. Phys. B, Proc. Suppl., 30 (1993), pp.~265--268. \bibitem{FGrasso_MMarini_1993a} {\sc F.~Grasso and M.~Marini}, {\em Multigrid techniques for hypersonic viscous flows}, AIAA J., 31 (1993), pp.~1729--1731. \bibitem{MFGuest_PSherwood_JHVLenthe_1993a} {\sc M.~F. Guest, P.~Sherwood, and J.~H. van Lenthe}, {\em Parallelism in computational chemistry. {I}. {H}ypercube connected multicomputers}, Theor. Chim. Acta, 84 (1993), pp.~423--441. \bibitem{SNGupta_MZubair_CEGrosch_1992a} {\sc S.~N. Gupta, M.~Zubair, and C.~E. Grosch}, {\em A multigrid algorithm for parallel computers: {CPMG}}, J. Sci. Comput., 7 (1992), pp.~263--279. \bibitem{WHackbusch_1992a} {\sc W.~Hackbusch}, {\em The frequency decomposition multi grid method. {II}. {C}onvergence analysis based on the additive {S}chwarz method}, Numer. Math., 63 (1992), pp.~433--453. \bibitem{YRHakopian_YAKuznetsov_1991a} {\sc Yu.~R. Hakopian and Yu.~A. Kuznetsov}, {\em Algebraic multigrid/substructuring preconditioners on triangular grids}, Sov. J. Numer. Anal. Math. Modell., 6 (1991), pp.~453--483. \bibitem{MHarmatz_PLauwers_SSolomon_TWittlich_1993a} {\sc M.~Harmatz, P.~Lauwers, S.~Solomon, and T.~Wittlich}, {\em Visual study of zero modes role in {PTMG} convergence}, Nucl. Phys. B, Proc. Suppl., 30 (1993), pp.~192--199. \bibitem{BHeise_1993a} {\sc B.~Heise}, {\em Nonlinear field calculations with multigrid {N}ewton methods}, Impact Comput. Sci. Eng., 5 (1993), pp.~75--110. \bibitem{MHolst_FSaied_1993a} {\sc M.~Holstm and F.~Saied}, {\em Multigrid solution of the {P}oisson {B}oltzmann equation}, J. Comput. Chem., 14 (1993), pp.~105--113. \bibitem{AHulsebos_1993a} {\sc A.~Hulsebos}, {\em Gribov copies and other gauge fixing beasties on the lattice}, Nucl. Phys. B, Proc. Suppl., 30 (1993), pp.~539--542. \bibitem{SHHwang_SULee_1993a} {\sc S.~H. Hwang and S.~U. Lee}, {\em An optical flow estimation algorithm using the spatio temporal hierarchical structure}, IEICE Trans. Info. 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