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 4 (approximately April 30, 1995) Today's topics: Change of URL for Marcus Speh's Multigrid Algorithm Library MGNet papers of Z. Cai, T. A. Manteuffel, and S. F. McCormick List of Copper Mountain 95 Papers/Abstracts International Linear Algebra Year at CERFACS ICCI95 (Int. Conf. Control & Info. Hong Kong) Some Recent Bibliography Additions ------------------------------------------------------- Date: Mon, 10 Apr 1995 21:46:12 +0200 (MESZ) From: Ulrich RuedeSubject: Change of URL for Marcus Speh's Multigrid Algorithm Library The Multigrid Algorithm Library on the World Wide Web, formerly maintained by Marcus Speh at DESY in Hamburg, with URL http://info.desy.de/pub/www/projects/MG.html has been updated and moved to Technische Universitaet Muenchen. It is now accessible at URL http://www5.informatik.tu-muenchen.de/MG/MG.html The page provides access to various multigrid related information, including references, preprints, and conferences. Please update existing links to the new location. ------------------------------------------------------- Date: Thu, 27 Apr 1995 13:08:44 -0600 From: stevem@boulder.colorado.edu (steve mccormick) Subject: MGNet papers of Z. Cai, T. A. Manteuffel, and S. F. McCormick Editor's Note: Steve sent 3 papers to MGNet. They are in the directory ------------- mgnet/papers/CaiZ-Manteuffel-McCormick. Steve promises more papers in May. FIRST-ORDER SYSTEM LEAST SQUARES FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS: PART I Z. Cai R. Lazarov T. A. Manteuffel S. F. McCormick Abstract This paper develops ellipticity estimates and discretization error bounds for elliptic equations (with lower order terms) that are reformulated as a least-squares problem for an equivalent first-order system. The main result is the proof of ellipticity, which is used in a companion paper to establish optimal convergence of multiplicative and additive solvers of the discrete systems. Editor's Note: in mgnet/papers/CaiZ_Manteuffel_McCormick/elliptic1.abs ------------- and mgnet/papers/CaiZ_Manteuffel_McCormick/elliptic1.ps.gz FIRST-ORDER SYSTEM LEAST SQUARES FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS: PART II Z. Cai T. A. Manteuffel S. F. McCormick Abstract This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial differential equations in n = 2 or 3 dimensions as a system of first-order equations. In part I [11] a similar functional was developed and shown to be elliptic in the H(div)xH^1 norm and to yield optimal convergence for finite element subspaces of H(div)xH^1. In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. The resulting functional is proved to be elliptic in the (H^1)^{n+1} norm. This immediately implies optimal error estimates for finite element approximation by standard subspaces of (H^1)^{n+1}. Another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent. As an alternative to perturbation based approaches, the least-squares approach developed here applies directly to convection-diffusion-reaction equations in a unified way and also admits a fast multigrid solver, historically a missing ingredient in least-squares methodology. Editor's Note: in mgnet/papers/CaiZ_Manteuffel_McCormick/elliptic2.abs ------------- and mgnet/papers/CaiZ_Manteuffel_McCormick/elliptic2.ps.gz FIRST-ORDER SYSTEM LEAST SQUARES FOR THE STOKES EQUATIONS, WITH APPLICATION TO LINEAR ELASTICITY Z. Cai T. A. Manteuffel S. F. McCormick Abstract Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H^1 product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for m ultiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity, with estimates that are uniform in the Lam{\'e} constants. Editor's Note: in mgnet/papers/CaiZ_Manteuffel_McCormick/stokes.abs ------------- and mgnet/papers/CaiZ_Manteuffel_McCormick/stokes.ps.gz ------------------------------------------------------- Date: Sun, 30 Apr 1995 11:11:11 -0400 (EST) From: Craig Douglas Subject: List of Copper Mountain 95 Papers/Abstracts On MGNet, in the mgnet/Conferences/CopperMtn95 directory are numerous papers from the conference, held during April 2-7. The papers are mostly gzipped PostScript files. The abstracts are mostly in plain text in the Abstracts subdirectory. Attendees should update their papers as they revise them for the forthcoming printed proceedings. What is currently there is the following: Loyce Adams A multigrid algorithm for immersed interface problems Abstracts/adams.abs adams.ps.gz Eyal Arian and Shlomo Ta'asan Smoothers for optimization problems Abstracts/arian_taasan.abs arian_taasan.ps.gz arian_taasan.tex.gz Dana M. Bedivan A finite element method for inhomogeneous problems using a multigrid algorithm for eliminating the boundary condition Abstracts/bedivan.abs bedivan.tex.gz M. Berndt and K. Witsch Multigrid With Overlapping Patches Abstracts/berndt_witsch.abs berndt_witsch.ps.gz P. Bochev, Z. Cai, T. A. Manteuffel, and S. F. McCormick First-order system least squares for the Navier-Stokes equations Abstracts/bochev_cai_manteuffel_mccormick.abs bochev_cai_manteuffel_mccormick.tex.gz James Bordner and Faisal Saied MGLab: An Interactive Environment for Multigrid Methods Abstracts/bordner_saied.abs bordner_saied.ps.gz A. Borz\`{\i}, K.W. Morton, E. S\"uli and M. Vanmaele A full multi-grid method for the solution of the cell vertex finite volume Cauchy--Riemann equations Abstracts/borzi_morton_suli_vanmaele.abs borzi_morton_suli_vanmaele.ps.gz J. H. Bramble, J. E. Pasciak, and A. T. Vassilev Analysis of the inexact Uzawa algorithm for saddle point problems Abstracts/bramble_pasciak_vassilev-abs.ps.gz bramble_pasciak_vassilev.ps.gz A. Brandt and L. Zaslavsky Multilevel algorithm for atmospheric data assimilation Abstracts/brandt_zaslavsky-abs.ps.gz brandt_zaslavsky.ps.gz Jan Broeze, Bernard Geurts, Hans Kuerten, Mand artin Streng Multigrid acceleration of time-accurate DNS of compressible turbulent flows Abstracts/broeze_geurts_kuerten_streng.abs broeze_geurts_kuerten_streng.ps.gz Z. Cai, T. A. Manteuffel, and S. F. McCormick First-order system least squares for the Stokes equations, with application to linear elasticity Abstracts/cai_manteuffel_mccormick.abs cai_manteuffel_mccormick.ps.gz Mario Casarin Quasi-optimal Schwarz methods for the conforming spectral element discretization Abstracts/casarin.abs casarin.ps.gz Zhangxin Chen and Richard E. Ewing Recent development of multigrid algorithms for mixed and nonconforming methods for second order elliptic problems Abstracts/chen_ewing.abs chen_ewing.ps.gz M. B. Davis and G. F. Carey Parallel element-by-element spectral multilevel techniques for finite elements Abstracts/davis_carey.abs davis_carey.ps.gz J. E. Dendy, Jr. Revenge of the semicoarsening frequency decomposition multigrid method Abstracts/dendy.abs dendy.ps.gz Qingping Deng An optimal order nonnested mixed multigrid method for generalized Stokes problems Abstracts/deng.abs deng.tex.gz Craig C. Douglas, Jim Douglas, Jr., and David E. Fyfe A Unified Multigrid Theory for Non-Nested Grids and/or Quadrature Abstracts/douglas_douglas_fyfe.abs douglas_douglas_fyfe.ps.gz Eugene G. D'yakonov Effective numerical methods for solving elliptic problems in strengthened Sobolev spaces Abstracts/dyakonov.abs dyakonov.tex.gz Peter Eliasson and Bj\"{o}rn Engquist The effects of dissipation and coarse grid resolution for multigrid in flow problems Abstracts/eliasson_engquist.abs eliasson_engquist.tex.gz Howard C. Elman Multigrid and Krylov subspace methods for the discrete Stokes equations Abstracts/elman.abs elman.ps.gz Thor Gjesdal A cell-centred multigrid algorithm for all grid sizes Abstracts/gjesdal.abs gjesdal.ps.gz W. L. Golik Numerical study of multigrid methods with various smoothers for the elliptic grid generation equations Abstracts/golik.abs golik.tex.gz Herve Guillard Some aspects of multigrid methods on non-structured meshes Abstracts/guillard.abs guillard.ps.gz Mike Holst and Stefan Vandewalle Schwarz methods: to symmetrize or not to symmetrize Abstracts/holst_vandewalle.abs holst_vandewalle.ps.gz Hong Wang An ELLAM-based domain decomposition and local refinement algorithm for second-order hyperbolic equations with interfaces Abstracts/hwang.abs hwang.ps.gz Jim E. Jones A mixed finite volume element method for flow calculations in porous media Abstracts/jones.abs jones.ps.gz M. Jung and U. Ruede Implicit extrapolation methods for variable coefficient problems Abstracts/jung_ruede.abs jung_ruede.ps.gz R. Jyotsna and S. P. Vanka A pressure based multigrid procedure for the Navier-Stokes on unstructured grids Abstracts/jyotsna_vanka.abs jyotsna_vanka Hwar-Ching Ku and Aleksander S. Popel The multigrid-mask numerical method for solution of incompressible Navier-Stokes equations Abstracts/ku_popel.abs ku_popel.tex.gz Chen-Y ao G. Lai Implementation of hybrid V-cycle algebraic multilevel methods for mixed finite element systems with penalty Abstracts/lai.abs lai.ps.gz Chang-Ock Lee Multigrid methods for the pure traction problem of linear elasticity: mixed formulation Abstracts/lee.abs lee.ps.gz Chaoqun Liu and Zhining Liu Multiple scale simulation for transitional and turbulent flow Abstracts/liu_liu.abs liu_liu.ps.gz Serguei Maliassov Multilevel substructuring preconditioning of nonconforming finite element approximations of second order elliptic problems Abstracts/maliassov.abs maliassov.ps.gz Jan Mandel and Radek Tezaur Convergence of a substructuring method with Lagrange multipliers Abstracts/mandel_tezaur.abs mandel_tezaur.ps.gz T. A. Manteuffel, S. F. McCormick, and G. Starke First order system least-squares for second-order elliptic problems with discontinuous coefficients Abstracts/manteuffel_mccormick_starke-abs.ps.gz manteuffel_mccormick_starke.ps.gz Thomas A. Manteuffel and Klaus J. Ressel Least-squares finite-element solution of the neutron transport equation in diffusive regimes Abstracts/manteuffel_ressel.abs manteuffel_ressel.ps.gz A. J. Meir On DGS relaxation: the Stokes problem Abstracts/meir.abs meir.ps.gz meir.tex.gz N. Duane Melson and Mark D. Sanetrik Multigrid acceleration of time-accurate Navier-Stokes calculations Abstracts/melson_sanetrik.abs melson_sanetrik H. Molenaar Multigrid methods for fully implicit black-oil reservoir simulation Abstracts/molenaar.abs molenaar.ps.gz E. Morano, D. Mavriplis, and V. Venkatakrishnan Coarsening strategies for unstructured multigrid techniques with applications to anisotropic problems Abstracts/morano_mavriplis_venkatakrishnan-abs.ps.gz morano_mavriplis_venkatakrishnan.ps.gz S. V. Nepomnyaschikh Preconditioning operators on unstructured grids Abstracts/nepomnyaschikh.abs nepomnyaschikh.tex.gz nepomnyaschikh.ps.gz Elyas Nurgat and Martin Berzins Multigrid methods for EHD problems Abstracts/nurgat_berzins.abs nurgat_berzins.ps.gz Suely Oliveira Multigrid and Krylov subspace methods for transport equations: absorption case Abstracts/oliveira.abs oliveira.ps.gz Mary Ellen Oman Fast multigrid techniques in total variation-based image reconstruction Abstracts/oman.abs oman.ps.gz Christoph Pflaum A multi-level-algorithm for the solution of second order elliptic differential equations on sparse grids Abstracts/pflaum.abs pflaum.ps.gz J. R. Phillips Error and complexity analysis for a collocation-grid-projection plus precorrected-FFT algorithm for solving potential integral equations with Laplace or Helmholtz kernels Abstracts/phillips.abs phillips.ps.gz Yair Shapira Analysis of red-black relaxation in multigrid Abstracts/shapira-circus.abs shapira-circus.ps.gz Yair Shapira Multigrid techniques for highly indefinite equations Abstracts/shapira.abs shapira.ps.gz A. Stathopoulos, Y. Saad, and C. F. Fischer A Schur complement method for eigenvalue problems Abstracts/stathopoulos_saad_fischer-abs.ps.gz stathopoulos_saad_fischer.ps.gz Petr Van\v{e}k, Jan Mandel, and Marian Brezina Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems Abstracts/vanek_mandel_brezina.abs vanek_mandel_brezina.ps.gz vanek_mandel_brezina.tex.gz C. Vuik, P. Wesseling, and S. Zeng Krylov subspace and multigrid methods applied to the incompressible Navier-Stokes equations Abstracts/vuik_wesseling_zeng.abs vuik_wesseling_zeng.ps.gz R. Webster An algebraic multigrid solver for the Navier-Stokes problems in the discrete second-order approximation Abstracts/webster_robinson.abs webster.ps.gz Shengyou Xiao and David Young Multiple coarse grid multigrid methods for solving elliptic problems Abstracts/xiao_young.abs xiao_young.ps.gz Dexuan Xie New nonlinear multigrid analysis Abstracts/xie.abs xie.ps.gz Jinchao Xu The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids Abstracts/xu.abs xu.ps.gz Xiaoqing Zheng, Chaoqun Liu, Changming Liao, Zhining Liu, and S. F. McCormick Multigrid method for modeling multi-dimensional combustion with detailed chemistry Abstracts/zheng_liu_liao_mccormick-abs.ps.gz zheng_liu_liao_mccormick.ps.gz ------------------------------------------------------- Date: Fri, 28 Apr 1995 18:41:50 +0200 From: wlay@cerfacs.fr Subject: International Linear Algebra Year at CERFACS As announced in our previous message, information about the International Linear Algebra Year is now available on WWW. Our home page is http://www.cerfacs.fr/~wlay/LAY/lay.html You will find informations about the workshops, the registration, the accomodation, the visitor programme ... There will soon be detailed programmes for the first two workshops with the possibility of reading and downloading the available abstracts. For those of you who may have trouble in accessing WWW, the same information is accessible via anonymous ftp. Our server is ftp.cerfacs.fr (138.63.200.33). The directory /pub/algo/workshops/ILAY/Info contains the postscript versions of the information available through WWW. Requests for further information should be made to wlay@cerfacs.fr. We remind you of the main information about ILAY. 1995 September 26-29 Direct methods October 17-20 Eigenvalues and beyond 1996 April 22-25 Optimization and linear algebra June 11-14 Iterative methods The two workshops in 1995 will be held at the UNESCO Centre in Toulouse. Each three-day workshop will be preceded by a tutorial or industrially oriented first day. Because of support, we are able to keep the cost to 1500 FFR (for registration, documentation, lunches, teas and coffees) for the three-day workshop and tutorial, with a reduction to 1000 FFR for full-time students, and a fee of 3000 FFR for non-academics. Living costs in the Toulouse region are quite low, normally about 250FFR per day for bed, breakfast and evening meal. Reduced price accommodation is available at UNESCO. Editor's Note: There will be domain decomposition and multigrid days ------------- during the Iterative methods workshops. ------------------------------------------------------- Date: Mon, 1 May 95 19:10:52 -0600 From: Floyd Hanson Subject: ICCI95 (Int. Conf. Control & Info. Hong Kong) ICCI95 will be held 5-9 June 1995, The Institute of Mathematical Sciences, Chinese University of Hong Kong, Shatin NT, Hong Kong. Technical Co-Sponsored by the IEEE Control System Society and the IEEE Robotics and Automation Society. The Conference will include invited keynote speakers, contributed sessions and invited minisymposia, in broad areas of automatic control, system theory, information theory and applications. Keynote Invited Speakers: Jian Song, Jagdish Chandra, Eugene Wong, Y.-C. Larry Ho, S. K. Mitter, J. C. Willems, B. A. Francis, B. D. O. Anderson and Suguru Arimoto. Minisymposia: Integrated Design for High Performance Systems, Parallel and Related Methods in Control, and Stochastic Theory-Adaptive Control. There are about 15 contributed sessions. The conference is also sponsored by the US Army Research Office, the Army Research Office-Far East, US Office of Naval Research, the K. C. Wong Education Foundation, the American GNC Corporation, Varitronix Limited, Hong Kong Baptist University, as well as other foundations. the Institute of Mathematical Sciences of the Chinese University of Hong Kong is the host of the conference. For local information, advance program, or registration forms can be found by Anonymous FTP using: ftp euler.math.cuhk.hk; login: anonymous; password: ``your-email-address''; cd /conf/icci95; get program.tex; get local-info.tex; get regis.frm; quit. For more information, contact: Dr. Kung Fu Ng, ICCI95 Local Arrangements Committee, Department of Mathematics, Chinese University of Hong Kong Shatin NT, Hong Kong, E-mail: ngkf@cuhk.hk ------------------------------------------------------- Date: Mon, 1 May 1995 15:57:44 -0400 From: douglas@noisy.watson.ibm.com (Craig Douglas) Subject: Some Recent Bibliography Additions \begin{thebibliography}{10} \bibitem{REBank_JMandel_SFMcCormick_1985a} {\sc R.~E. Bank, J.~Mandel, and S.~F. McCormick}, {\em Variational multigrid theory}, in Multigrid Methods, SIAM, Philadelphia, 1985, pp.~131--178. \bibitem{CBennet_RSharpley_1988a} {\sc C.~Bennet and R.~Sharpley}, {\em Interpolation of Operators}, Academic Press, New York, 1988. \bibitem{CBernardi_YMaday_1992a} {\sc C.~Bernardi and Y.~Maday}, {\em Approximations spectrales de probl{\`e}mes aux limites elliptiques}, vol.~Math{\`e}matiques \& Applications, Springer--Verlag, Paris, 1992. \bibitem{JHBramble_JEPasciak_JXu_1992a} {\sc J.~H. Bramble, J.~E. Pasciak, and J.~Xu}, {\em A multilevel preconditioner for domain decomposition boundary systems}, in Proceedings of the Tenth International Conference on Computer Methods in Applied Science and Engineering, New York, 1992, Nova Sciences. \bibitem{JHBramble_JXu_1989a} {\sc J.~H. Bramble and J.~Xu}, {\em A local post--processing technique for improvising the accuracy in mixed finite element approximations}, SIAM J. Numer. Anal., 24 (1989), pp.~1267--1275. \bibitem{ABrandt_AALubrecht_1990a} {\sc A.~Brandt and A.~A. Lubrecht}, {\em Multilevel matrix multiplication and fast solution of integral equations}, J. Comp. Phys., 90 (1990), pp.~348--370. \bibitem{SCBrenner_1994a} {\sc S.~C. Brenner}, {\em A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity}, Math. Comp., 63 (1994), pp.~435--460 and S1--S5. \bibitem{SCBrenner_LRScott_1994a} {\sc S.~C. Brenner and L.~R. Scott}, {\em The Mathematical Theory of Finite Element Methods}, Texts in Applied Mathematics, Springer--Verlag, New York, 1994. \bibitem{HJBungartz_1992a} {\sc H.-J. Bungartz}, {\em An adaptive {P}oisson solver using hierarchical bases and sparse grids}, in Proceedings of the IMACS International Symposium on Iterative Methods in Linear Algebra, Brussels, April, 1991, Amsterdam, 1992, Elsevier. \bibitem{TFChan_1989c} {\sc T.~F. Chan}, {\em Domain decomposition algorithms and computational fluid dynamics}, in Vector and Parallel Computing: issues in applied research and development, Ellis Horwood Ltd. (J. Wiley \& Sons), Chichester, 1989, pp.~65--82. \bibitem{ZChen_1993b} {\sc Z.~Chen}, {\em Analysis of mixed methods using conforming and nonconforming finite element methods}, RAIRO, Math. Mod. Num. Anal., 27 (1993), pp.~9--34. \bibitem{CCDouglas_JDouglas_DEFyfe_1994a} {\sc C.~C. Douglas, J.~Douglas, and D.~E. Fyfe}, {\em A multigrid unified theory for non-nested grids and/or quadrature}, E. W. J. Numer. Math., 2 (1994), pp.~285--294. \bibitem{CCDouglas_AErn_MDSmooke_1994a} {\sc C.~C. Douglas, A.~Ern, and M.~D. Smooke}, {\em Numerical simulation of laminar diffusion flames}, SIAM News, 27 (1994), pp.~1, 12--13, 17. \bibitem{JBDrake_LJGray_1989a} {\sc J.~B. Drake and L.~J. Gray}, {\em Parallel implementation of the boundary element method}, in Vector and Parallel Computing: issues in applied research and development, Ellis Horwood Ltd. (J. Wiley \& Sons), Chichester, 1989, pp.~83--92. \bibitem{MDryja_BFSmith_OBWidlund_1994a} {\sc M.~Dryja, B.~F. Smith, and O.~B. Widlund}, {\em Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions}, SIAM J. Numer. Anal., 31 (1994). \bibitem{MDryja_OBWidlund_1994a} {\sc M.~Dryja and O.~B. Widlund}, {\em Domain decomposition algorithms with small overlap}, SIAM J. Sci. Comput., 15 (1994), pp.~604--620. \bibitem{TFDupont_RPKendall_HHRachford_1968a} {\sc T.~F. Dupont, R.~P. Kendall, and H.~H. Rachford}, {\em An approximate factorization procedure for solving self--adjoint elliptic difference equations}, SIAM J. Numer. Anal., 5 (1968), pp.~559--573. \bibitem{REEwing_1989b} {\sc R.~E. Ewing}, {\em Large scale computing in reservoir simulation}, in Vector and Parallel Computing: issues in applied research and development, Ellis Horwood Ltd. (J. Wiley \& Sons), Chichester, 1989, pp.~123--137. \bibitem{REEwing_YKuznetsov_RLazarov_SMaliassov_1994a} {\sc R.~E. Ewing, Yu. Kuznetsov, R.~Lazarov, and S.~Maliassov}, {\em Preconditioning of nonconforming finite element approximations of second order elliptic problems}, in Proc. of the Third Int. Conf. on Advances in Numerical Methods and Applications, Bulgaria, 1994, pp.~101--110. \bibitem{WDGropp_ELusk_ASkjellum_1994a} {\sc W.~D. Gropp, E.~Lusk, and A.~Skjellum}, {\em Using MPI: Portable Parallel Programming with the Message-Passing Interface}, Scientific and Engineering Computation, MIT Press, Cambridge, MA, 1994. \bibitem{VEHenson_1989a} {\sc V.~E. Henson}, {\em Parallel compact symmetric ffts}, in Vector and Parallel Computing: issues in applied research and development, Ellis Horwood Ltd. (J. Wiley \& Sons), Chichester, 1989, pp.~153--164. \bibitem{YMaday_DMeiron_ATPatera_EMRonquist_1993a} {\sc Y.~Maday, D.~Meiron, A.~T. Patera, and E.~M. R{\o}nquist}, {\em Analysis of iteratifve methods for steady and unsteady {S}tokes problem: {A}pplication of spectral element discretization}, SIAM J. Sc. Comp., 14 (1993), pp.~301--337. \bibitem{MEGOng_1989a} {\sc M.~E.~G. Ong}, {\em The {3D} linear hierarchical basis preconditioner and its shared memory parallel implementation}, in Vector and Parallel Computing: issues in applied research and development, Ellis Horwood Ltd. (J. Wiley \& Sons), Chichester, 1989, pp.~273--283. \bibitem{SSPahl_1993a} {\sc S.~S. Pahl}, {\em Schwarz type domain decomposition methods for spectral element discretizations}, Master's thesis, University of Witwatersrand, Johannesburg, South Africa, 1993. \bibitem{JEPasciak_1988b} {\sc J.~E. Pasciak}, {\em Domain decomposition preconditioners for elliptic problems in two and three dimensions}, in Numerical Algorithms for Modern Parallel Computer Architectures, IMA Volumes Math. Appl., Springer--Verlag, New York, 1988, pp.~163--172. \bibitem{LFPavarino_1992a} {\sc L.~F. Pavarino}, {\em Domain decomposition algorithms for the p--version finite element mehtod for elliptic problems}, PhD thesis, New York University, New York, 1992. \bibitem{LFPavarino_1994a} \leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Additive {S}chwarz methods for the p--version finite element method}, Numer. Math., 66 (1994), pp.~493--515. \bibitem{LFPavarino_1994b} \leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Schwarz methods with local refinement for the p-version finite element method}, Numer. Math., 69 (1994), pp.~185--211. \bibitem{LFPavarino_OBWidlund_1994a} {\sc L.~F. Pavarino and O.~B. Widlund}, {\em Iterative substructuring methods for spectral elements in three dimensions}, in The finite element method: fifty years of the Courant element, Marcel Dekker, New York, Basel, Hong Kong, 1994, pp.~345--355. \bibitem{LFPavarino_OBWidlund_1995a} \leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Preconditioned conjugate gradient solvers for spectral elements in 3{D}}, in Solution Techniques for Large-Scale {CFD} Problems, John Wiley \& Sons, New York, 1995, pp.~189--210. \bibitem{CPflaum_1992a} {\sc C.~Pflaum}, {\em Anwendung von {M}ehrgitterverfahren auf d{\"u}nnen {G}ittern}, PhD thesis, Technische Universit{\"a}t M{\"u}nchen, Munich, 1992. \bibitem{AReusken_1988b} {\sc A.~Reusken}, {\em Convergence of the multigrid full approximation scheme for a class of elliptic mildly nonlinear boundary value problems}, Numer. Math., 52 (1988), pp.~251--277. \bibitem{AReusken_1988c} \leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Convergence of the multigrid full approximation scheme including the {V}--cycle}, Numer. Math., 53 (1988), pp.~663--686. \bibitem{LRScott_SZhang_1992a} {\sc L.~R. Scott and S.~Zhang}, {\em Higher--dimensional nonnested multigrid methods}, Math. Comp., 58 (1992), pp.~457--466. \bibitem{RVerfurth_1988a} {\sc R.~Verf{\"u}rth}, {\em Multi--level algorithms for mixed probelms {II}. {T}reatment of the mini--element}, SIAM J. Numer. Anal., 25 (1988), pp.~285--293. \bibitem{RVerfurth_1991a} \leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em A posteriori error estimates for the {S}tokes equations {II} non--conforming discretizations}, Numer. Math., 60 (1991), pp.~235--249. \bibitem{TvonPetersdorff_EPStephan_1990a} {\sc T.~von Petersdorff and E.~P. Stephan}, {\em On the convergence of the multigrid method for a hypersingular integral equation of the first kind}, Numer. Math., 57 (1990), pp.~379--391. \bibitem{DXie_1995a} {\sc D.~Xie}, {\em Nonlinear multigrid analysis for a mildly elliptic nonlinear boundary value problems}, PhD thesis, U. of Houston, Houston, 1995. \bibitem{CZenger_1991a} {\sc C.~Zenger}, {\em Sparse grids}, in Paralle algorithms for partial differential equations: Proceedings of the Sixth GAMM-Seminar, Kiel, Jan. 1990, vol.~Notes on Numerical Fluid Mechanics, vol. 31, Vieweg, Braunschweig, 1991. \end{thebibliography} ------------------------------ End of MGNet Digest **************************