Send mail to: mgnet@cs.yale.edu for the digests or bakeoff mgnet-requests@cs.yale.edu for comments or help Current editor: Craig Douglas douglas-craig@cs.yale.edu Anonymous ftp repository: casper.cs.yale.edu (128.36.12.1) ftp.cerfacs.fr (138.63.200.33) World Wide Web: http://na.cs.yale.edu/mgnet/www/mgnet.html or http://www.cerfacs.fr/~douglas/mgnet.html Today's editor: Craig Douglas (douglas-craig@cs.yale.edu) Volume 5, Number 5 (approximately May 31, 1995) Today's topics: Best V-cycle Algorithm for 2D Poisson Equation New paper on MGNet (Ruede) Announcement of WWW multigrid tutorial Paper on cache based multigrid Oberwolfach An MPI based multigrid solver for 1 or more processors Some of the new entries in the bibliography ------------------------------------------------------- Date: Sun, 28 May 1995 03:38:05 -0400 From: Jun ZhangSubject: Best V-cycle Algorithm for 2D Poisson Equation Best V-cycle algorithm to Solve 2D Poisson Equation I'm interested in knowing the best V-cycle algorithm to solve 2D Poisson equation - (Uxx + Uyy) = f(x,y) (x,y) in OMEGA u(x,y) = g(x,y) (x,y) on OMEGA For simplicity, OMGEA is a unit square. I have 3 test problems (courtesy of Dr. S. Fulton) Test 1. f(x,y) = -x^2(1-x^2)(2-12y^2) - y^2(1-y^2)(2-12x^2), g(x,y) = x^2*y^2(1-x^2)(1-y^2). Test 2. f(x,y) = -(x^2 + y^2)*exp(xy), g(x,y) = exp(xy). Test 3. f(x,y) = 52*cos(2x + 6y), g(x,y) = cos(4x + 6y). Conditions and Cost: 5-point finite difference discretization. V(1,1)-cycle algorithm with point Red-Black Gauss-Seidel, injection and bi-linear interpolation. Fortran 77, double precision, stop when L2-error < 10^(-9). Meshsize h= 1/N. Initial guess U(x,y)=0. Computer is a SUN SPARCstation. Below are the best results I got: N Test 1 Test 2 Test 3 16 9 11 10 32 9 11 11 These are V(1,1) cycle 64 9 12 12 numbers, i.e. the iteration 128 9 13 12 numbers. 256 9 13 13 512 9 14 13 If you know better results with comparable cost, would you please send me an email? I'm also interested in knowing the limit of improving these results, so if you know better results with slightly higher cost, I will appreciate your sending me a message. (I can modify the code to give slightly better data to some entry, but the above is the best on average.) Jun Zhang zhang@math.gwu.edu Department of Mathematics George Washington University Washington, DC 20052 Editor's Note: I would have used a nested iteration V cycle myself, but ------------- there is an algorithm using nonstandard grids which is a direct method effectively (1 iteration, 0 contraction factor) by (I think) Ruge and Stuben (correction?). Also, some of the multiple coarse space methods are similar to this (e.g., see a paper by Brezzi, me, and Marini in the MGNet bibliography for an 8 coarse grid decomposition of this problem). --> Please cc mgnet@cs.yale.edu if you send comments to the requestor. ------------------------------------------------------- Date: Wed, 31 May 1995 17:33:57 +0200 (MESZ) From: Ulrich Ruede Subject: New paper on MGNet My paper Stability of implicit extrapolation methods has been downloaded to the mgnet ftp repository. This paper has been submitted to the proceedings of the 8th International Conference on Domain Decomposition, May 16-20, Beijing, China. Keywords: Implicit extrapolation, multigrid, multilevel subspace splitting AMS Classification: 65N22, 65N50, 65N55} Abstract Multilevel methods are generally based on a splitting of the solution space associated with a nested sequence of coarser grids. Besides the straightforward application of extrapolation on the grid system, we propose to use extrapolation implicitly, similar to multigrid tau-extrapolation. This implicit extrapolation, when applied to linear finite elements is related to the p-version of the finite element method. The method depends on a stability condition which must be enforced by a suitable modification of the problem. This (and others of may papers) are available from mgnet and in the WWW through URL http://www5.informatik.tu-muenchen.de/persons/ruede/info/refs.html Ulrich Ruede Institut fuer Informatik, Technische Universitaet, D-80290 Muenchen, Germany, e-mail: ruede@informatik.tu-muenchen.de Tel: +49 89 21058238, Fax: +49 89 21052022 URL: http://www5.informatik.tu-muenchen.de/persons/ruede.html Editor's Note: in mgnet/papers/Ruede/stable_extra.ps.gz and ------------- mgnet/papers/Ruede/stable_extra.abstract. ------------------------------------------------------- Date: Wed, 31 May 1995 17:59:39 +0200 (MESZ) Subject: Announcement of WWW multigrid tutorial From: Ulrich Ruede On the WWW an Online Multigrid Tutorial, the Multigrid Workbench is available. The start page is accessible through URL http://www5.informatik.tu-muenchen.de/sci-comp/xwb/xwb.html The workbench features an active image showing a standard multigrid V-cycle algorithm. By clicking on parts of this algorithm, the corresponding status of the iteration is displayed graphically. In contrast to conventional text, there is no natural sequence, how the different pages should be read. All information is accessible by hypertext links and can be read in any order. I'd appreciate any feedback and comments that would help me to improve this resource. I'd also be interested in hearing about technical problems, e.g. whether the current network speed (in Germany) is still acceptable for using the workbench in its present form. ------------------------------------------------------- Date: Wed, 31 May 1995 13:48:39 -0400 From: douglas@watson.ibm.com (Craig Douglas) Subject: Paper on cache based multigrid Caching in with Multigrid Algorithms: Problems in Two Dimensions Craig C. Douglas IBM T. J. Watson Research Center, Yorktown Heights, NY, USA and Computer Science Department, Yale University, New Haven, CT, USA Abstract Multigrid methods combine a number of standard sparse matrix techniques. Usual implementations separate the individual components (e.g., an iterative methods, residual computation, and interpolation between grids) into nicely structured routines. However, many computers today employ quite sophisticated and potentially large caches whose correct use are instrumental in gaining much of the peak performance of the processors. We investigate when it makes sense to combine several of the multigrid components into one, using block oriented algorithms. We determine how large (or small) the blocks must be in order for the data in the block to just fit into the processor's primary cache. By re-using the data in cache several times, a potential savings in run time can be predicted. This is analyzed for a set of examples. Key words: multigrid, cache, sparse matrix, iterative methods, domain decomposition. Note: This paper will be in the proceedings of the International Conference on Parallel Algorithms, October 15-19, Wuhan, China. Editor's Note: in mgnet/papers/Douglas/cache1.dvi.gz and ------------- mgnet/papers/Douglas/cache1.abs. ------------------------------------------------------- Date: Wed, 31 May 1995 12:24:30 +0200 From: P.W.Hemker@cwi.nl Subject: Oberwolfach Dear Craig, Please find below the titles of the talks, as announced in Oberwolfach. Sue Brenners's titles (between brackets) are possibly more informative. I also add a list of participants. Pieter. Monday AM P. Wesseling "Multigrid solution of the incompressible Navier-Stokes equations in general coordinates" ("Krylov subspace and multigrid methods applied to the incompressible Navier-Stokes equations" (with C. Vuik, S. Zeng)) K. Oosterlee "Multigrid and defect correction for the 3D incompressible Navier-Stokes equations in general coordinates" ("A GMRES-based plane smoother in multigrid for solving 3D anisotropic fluid flow problems") E. Dick "Multigrid methods of Navier-Stokes equations coupled to k-$\epsilon$- turbulence equations" ("Modelling transitional fronts") D. Haenel "Application of multi-sequence Runge-Kutta methods to adaptive solutions of the Navier-Stokes equations" ("Computation of compressible strongly unsteady flows on algebraic unstructured grids") PM U. Ruede "Implicit multilevel extrapolation methods" J. Dendy "Variants of the frequency decomposition MG-method" ("Grandchild of the frequency decomposition method") R. Stevenson "Frequency decomposition multilevel methods" ("Multilevel methods based on space decompositions orthogonal w.r.t. discrete scalar products") Tuesday AM S. Vandewalle "Acceleration of multilevel domain decomposition and multigrid by Krylov subspace methods" ("Schwarz methods: to symmetrize or not to symmetrize") A. Meyer "Preconditioning the pseudo-Laplacian for CFD-simulation" M. Jung "On the parallelization of multigrid methods using a nonoverlapping DD data structure" G. Haase "Dirichlet DD vs. global multigrid methods" (with U. Langer) PM S. Brenner "Convergence of nonconforming or nonnested multigrid methods without full elliptic regularity" S. Turek "On robust and efficient multilevel Schur-complement solvers for incompressible and compressible Stokes and Navier-Stokes equations" D. Braess "Efficient smoothing of the Navier-Stokes equations by u-dominant iterations" (with R. Sarazin) Wednesday AM M. Feistauer "Numerical solution of nonlinear convection-diffusion problems and applications to compressible fluid flow" A. Reusken "Analysis of multigrid methods for convection-diffusion problems" J. Fuhrmann "On algebraic multigrid methods for PDEs" N. Neuss "Homogenization and multigrid" PM hike Thursday AM P. Oswald "Multilevel preconditioners for nonconforming elements" J. Junkherr "Multigrid methods for weakly singular integral equations of the first kind" R. Kornhuber "A posteriori error estimates for linear and nonlinear elliptic problems" PM R. Bank "An algorithm for coarsening unstructured meshes" (with J. Xu) S. Sauter "Composite finite element spaces, coarsening and MG methods" ("A new finite element space for the approx. of PDEs on domains with complicated microstructures") (with W. Hackbusch) R. Hoppe "Adaptive multilevel methods for mixed FEM for 2nd order elliptic BVPs" (with B. Wohlmuth) Friday AM F. Bornemann "Cascadic multigrid methods" (with P. Deuflhard) C. Wagner "Filtering decompositions for asymmetric and heterogeneous problems" ("Frequency filtering decomposition") P. Vanek "Algebraic multigrid for thin elastic problems" ("Algebraic multigrid by smoothed aggregation for 2nd and 4th order elliptiv BVPs" (with J. Mandel and M. Brezina) W. Mulder "Application of multigrid to porous media flow" ======================================================================== Participants: rbank@ucsd.edu (Randy Bank), peter@ica3.uni-stuttgart.de (Peter Bastian), juergen@na.mathematik.uni-tuebingen.de (Juergen Bey), bornemann@zib-berlin.de (Folkmar Bornemann), braess@num.ruhr-uni-bochum.de (Dietrich Braess), brenner@math.scarolina.edu (Susanne Brenner), dahmen@igpm.rwth-aachen.de (Wolfgang Dahmen), jed@lanl.gov (J.E. Dendy), deuflhard@zib-berlin.de (Peter Deuflhard), Erik.Dick@rug.ac.be (Erik Dick), FEIST@MS.MFF.CUNI.CZ (Miloslav Feistauer), fuhrmann@iaas-berlin.d400.de (Juergen Fuhrmann), griebel@informatik.tu-muenchen.de (Michael Griebel), ghaase@numa.uni-linz.ac.at (Gundolf Haase), wh@informatik.uni-kiel.de (Wolfgang Hackbusch), hj454ha@vug.uni-duisburg.de (Dieter Haenel), pieth@cwi.nl (Piet Hemker), rohop@mathematik.tu-muenchen.de (Ronald Hoppe), dr.michael.jung@mathematik.tu-chemnitz.de (Michael Jung), jj@informatik.uni-kiel.de (Joerg Junkherr), kornhuber@iaas-berlin.d400.de (Ralf Kornhuber), ulanger@miraculix.numa.uni-linz.ac.at (Ulrich Langer), maitre@cc.ec-lyon.fr (Jean-Francois Maitre), a.meyer@mathematik.tu-chemnitz.de (Arndt Meyer), mittelmann@math.la.asu.edu (Hans D. Mittelmann), mulderw@ksepl.nl (W. A. Mulder), neuss@iwr.uni-heidelberg.de (Nikolas Neuss), KEES.OOSTERLEE@GMD.DE (K. Oosterlee), Peter.Oswald@math.tamu.edu (Peter Oswald), WSANAR@WIN.TUE.NL (Arnold Reusken), ruede@informatik.tu-muenchen.de (Ulrich Ruede), sas@informatik.uni-kiel.de (Stefan Sauter), stevenso@win.tue.nl (Rob Stevenson), ture@gaia.iwr.uni-heidelberg.de (Stefan Turek), STEFAN@AMA.CALTECH.EDU (Stefan Vandewalle), rv@silly.num1.ruhr-uni-bochum.de (Ruediger Verfuerth), chris@ica3.uni-stuttgart.de (Christian Wagner), p.wesseling@math.tudelft.nl (Pieter Wesseling), wieners@ica3.uni-stuttgart.de (Wieners), wittum@icasun.ica.uni-stuttgart.de (Gabriel Wittum), harry@na.uni-tuebingen.de (Harry Yserentant) ------------------------------------------------------- Date: Wed, 31 May 1995 14:20:32 -0400 From: douglas-craig@CS.YALE.EDU (Craig Douglas) Subject: An MPI based multigrid solver for 1 or more processors At the Copper Mountain meeting, I talked very briefly one evening at a workshop about what I thought was new and interesting for parallel multigrid. I pointed out that MPI is a standard, developed by academics and computer companies, that is actually appearing to make headway as a very nice parallel computer message passing system. I even ran a demonstration on an IBM ThinkPad (running Linux, not DOS/Windows) of how it works on a single processor, but fakes the system into simulating (quasi-randomly) a multiprocessor. I volunteered to put a demonstration code on MGNet, which I have done, in Codes/douglas. There are several examples in 2D (Poisson's equation and variable coefficients) with a variety of matrix storage formats (including matrix-free). They can be modified quite easily. I will put some 3D codes out when they are cleaned up sufficiently. These codes are not meant for high speed, just to demonstrate how MPI can be used in interesting ways (such as transferring non-stride 1 data without having to pack and unpack data yourself). One word of warning. I used the experimental version (1.0.9) of mpich that is in pub/mpi/misc on info.mcs.anl.gov and also IBM's MPI product for the SP2 in developing this. With mpich, I found that one of the routines (the simulataneous send and receive routine) did not quite work correctly on a mesh of processors (it worked with the IBM product). Bill Gropp fixed this within 2 hours of hearing about it and pointed me to a newer version (many thanks, Bill). Editor's Note: in mgnet/Codes/douglas/2d.tgz. ------------- ------------------------------------------------------- Date: Wed, 31 May 1995 14:19:28 -0400 From: douglas-craig@CS.YALE.EDU (Craig Douglas) Subject: Some of the new entries in the bibliography This is somewhat repetitive (my apologies). The next batch is in next month's digest. \begin{thebibliography}{100} \bibitem{VIAgoshkov_1994a} {\sc V.~I. Agoshkov}, {\em Domain decomposition methods using modified basis functions}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~3--15. \bibitem{GAguilar_FLisbona_1994a} {\sc G.~Aguilar and F.~Lisbona}, {\em Interface conditions for a kinf of non linear elliptic--hyperbolic problems}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~89--95. \bibitem{RArina_CCanuto_1994a} {\sc R.~Arina and C.~Canuto}, {\em A {X}--formulation of the viscous--inviscid domain decomposition for the {E}uler/{N}avier--{S}tokes equations}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~453--458. \bibitem{SFAshby_CTKelley_PESaylor_JSScroggs_1994a} {\sc S.~F. Ashby, C.~T. Kelley, P.~E. Saylor, and J.~S. Scroggs}, {\em Preconditioning via asymptotically--defined decomposition}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~139--150. \bibitem{MAzaiez_AQuarteroni_1994a} {\sc M.~Azaiez and A.~Quarteroni}, {\em A spectral {S}tokes solver in domain decomposition methods}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~151--156. \bibitem{NSBakhvalov_AVKnyazev_1994a} {\sc N.~S. Bakhvalov and A.~V. Knyanzev}, {\em Preconditioned iterative methods in a subspace for linear algebraic equations with large jumps in the coefficients}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~157--162. \bibitem{REBank_JXu_1994a} {\sc R.~E. Bank and J.~Xu}, {\em The hierarchical basis multigrid method and incomplete {LU} decompostion}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~163--173. \bibitem{JDBenamou_YBrenier_1994a} {\sc J.-D. Benamou and Y.~Brenier}, {\em A domain decomposition method for the polar factorization of vector fields}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~231--236. \bibitem{BBialecki_XCCai_MDryja_GFairweather_1994a} {\sc B.~Bialecki, X.-C. Cai, M.~Dryja, and G.~Fairweather}, {\em An additive {S}charz algorithm for piecewise {H}ermite bicubic orthogonal spline collocation}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~237--244. \bibitem{PEBjorstad_WMCoughran_EGrosse_1994a} {\sc P.~E. Bj{\o}rstad, W.~M. Coughran, and E.~Gross}, {\em Parallel domain decomposition applied to coupled transport equations}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~369--380. \bibitem{FABornemann_1994a} {\sc F.~A. Bornemann}, {\em Interpolation spaces and optimal multilevel preconditioners}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~3--8. \bibitem{JFBourgat_PLeTallec_BPerthame_YQiu_1994a} {\sc J.~F. Bourgat, P.~LeTallec, B.~Perthame, and Y.~Qiu}, {\em Coupling {B}oltzmann and {E}uler equations without overlapping}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~377--398. \bibitem{ABrambilla_CCarlenzoli_GGazzaniga_PGervasio_GSacchi_1994a} {\sc A.~Brambilla, C.~Carlenzoli, G.~Gazzaniga, P.~Gervasio, and G.~Sacchi}, {\em Implementation of domain decomposition techniques on n{CUBE}2 parallel machine}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~345--351. \bibitem{JHBramble_JEPasciak_1994a} {\sc J.~H. Bramble and J.~E. Pasciak}, {\em Uniform convergence estimates for multigrid {V}--cycle algorithms with less than full elliptic regularity}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~17--26. \bibitem{ABrandt_BDiskin_1994a} {\sc A.~Brandt and B.~Diskin}, {\em Multigrid solvers on decomposed domains}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~135--155. \bibitem{SCBrenner_1994b} {\sc S.~C. Brenner}, {\em Two--level additive {S}chwarz preconditioners for nonconforming finite elements}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~9--14. \bibitem{FBrezzi_LDMarini_1994a} {\sc F.~Brezzi and L.~D. Marini}, {\em A three--fold domain decomposition method}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~27--34. \bibitem{MOBristeau_RGlowinski_JPeriaux_1994a} {\sc M.~O. Bristeau, R.~Glowinski, and J.~P{\'e}riaux}, {\em On the numerical solution of the {H}elmholtz equations at large wave numbers using exact controllability methods. {A}pplication to scattering}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~399--419. \bibitem{HJBungartz_MGriebel_DRoschke_CZenger_1994a} {\sc H.-J. Bungartz, M.~Griebel, D.~R{/"}oschke, and C.~Zenger}, {\em Two proofs of convergence for the combination technique for the efficient solution of sparse grid problems}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~15--20. \bibitem{WCai_1994a} {\sc W.~Cai}, {\em Domain decomposition and computation of two dimensional detonation waves}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~459--464. \bibitem{XCCai_MDryja_1994a} {\sc X.-C. Cai and M.~Dryja}, {\em Domain decomposition methods for monotone nonlinear eliptic problems}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~21--27. \bibitem{XCCai_WDGropp_DEKeyes_MDTidriri_1994a} {\sc X.-C. Cai, W.~D. Gropp, D.~E. Keyes, and M.~D. Tidriri}, {\em Parallel implicit methods for aerodynamics}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~465--470. \bibitem{YCai_IMNavon_1994a} {\sc Y.~Cai and I.~M. Navon}, {\em Parallel domain--decomposed preconditioners in finite element shallow water flow modeling}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~471--476. \bibitem{FCamilli_MFalcone_PLanucara_ASeghini_1994a} {\sc F.~Camilli, M.~Falcone, P.~Lanucara, and A.~Seghini}, {\em A domain decomposition method for {B}ellman equations}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~477--483. \bibitem{CCanuto_ARusso_1994a} {\sc C.~Canuto and A.~Russo}, {\em Self--adaptive coupling of mathematical models and/or numerical methods}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~35--44. \bibitem{TFChan_TPMathew_1994b} {\sc T.~F. Chan and T.~P. Mathew}, {\em Doamin decomposition preconditioners for convection diffusion problems}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~157--175. \bibitem{TFChan_BFSmith_1994a} {\sc T.~F. Chan and B.~F. Smith}, {\em Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~175--189. \bibitem{JGChefter_CKChu_DEKeyes_1994a} {\sc J.~G. Chefter, C.~K. Chu, and D.~E. Keyes}, {\em Domain decomposition for the shallow water equations}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~485--490. \bibitem{NChrisochoides_GFox_JThompson_1994a} {\sc N.~Chrisochoides, G.~Fox, and J.~Thompson}, {\em {MENUS--PGG}: {A} mapping environment for unstructured numerical parallel grid generation}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~381--386. \bibitem{PGCiarletJr_1994b} {\sc P.~G. Ciarle{t,~Jr.}}, {\em A comparison of three iterative algorithms based on domain decomposition methods}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~387--393. \bibitem{PGCiarletJr_GMeurant_1994a} {\sc P.~G. Ciarle{t,~Jr.} and G.~Meurant}, {\em A class of domian decomposition preconditioners for massively parallel computers}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~353--359. \bibitem{MCCiccoli_JADesideri_JPeriaux_1994a} {\sc M.~C. Ciccoli, J.~A. Desideri, and J.~P{\'e}riaux}, {\em Introduction of domain decomposition techniques in time-- dependent flow problems}, in Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, vol.~157 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~433--439. \bibitem{RKCoomer_IGGraham_1994a} {\sc R.~K. Coomer and I.~G. Graham}, {\em Domain decomposition methods for device modelling}, in Domain Decomposition Methods in Scientific and Engineering Computing: Proceedings of the Seventh International Conference on Domain Decomposition, vol.~180 of Contemporary Mathematics, Providence, Rhode Island, 1994, American Mathematical Society, pp.~491--496. \bibitem{MCCurran_1994a} {\sc M.~C. 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