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 Current editor:  Craig Douglas       
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Today's editor:  Craig Douglas (

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
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:

3600 University City Science Center
Philadelphia, PA  19104-2688
(215) 382-9800
Fax: (215) 386-7999

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

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

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

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: (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

The LPARX distribution is available via anonymous ftp on site
Look in directory pub/sdsc/parallel/lparx.  The distribution is also available
on anonymous ftp site in directory pub/baden/LPARX.  Technical
reports are also available in subdirectory "reports."

Technical questions should be sent to Scott Kohn at

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: (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                            |
Applied and Computational Mathematics Division |
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,

Report No. UIUCDCS-R-88-1436, Dept. of Computer Science, University of
Illinois at Urbana-Champaign

                        FOR ELLIPTIC PROBLEMS

                         William F. Mitchell
                         Bldg 101 Rm A238
                         Gaithersburg, MD 20899

(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 Shao 
Subject: 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

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).

Excerpts from the MGNet Bibliogrpahy.



{\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.

{\sc M.~P. Allen}, {\em Simulation of condensed phases using the distributed
  array processor}, Theor. Chim. Acta, 84 (1993), pp.~399--411.

{\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.

{\sc A.~Arnone, M.~S. Liou, and L.~A. Povinelli}, {\em Multigrid calculation of
  three-dimensional viscous cascade flows}, J. Propul. Power, 9 (1993),

{\sc M.~Baker, G.~Mack, and M.~Speh}, {\em Multigrid meets neural nets}, Nucl.
  Phys. B, Proc. Suppl., 30 (1993), pp.~269--272.

{\sc C.~Basler and W.~Tornig}, {\em On monotone including nonlinear multigrid
  methods and applications}, Comput., 50 (1993), pp.~51--67.

{\sc J.~Belak}, {\em Harnessing the killer micros: applications from {LLNL}'s
  massively parallel computing initiative}, Theor. Chim. Acta, 84 (1993),

{\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.

{\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.

{\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.

{\sc F.~Bornemann and H.~Yserentant}, {\em A basic norm equivalence for the
  theory of multilevel methods}, Numer. Math., 64 (1993), pp.~455--476.

{\sc A.~Borzi and A.~Koubek}, {\em Multi--grid method for the resolution of
  thermodynamic {B}ethe ansatz equations}, Comput. Phys. Commun., 75 (1993),

{\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.

{\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.

{\sc A.~Brandt, W.~Joppich, J.~Linden, G.~Lonsdale, and A.~Schueller}, {\em
  Multigrid Course}, GMD--690, St. Augustin, 1992.

{\sc M.~Breuer and D.~Hanel}, {\em A dual time stepping method for 3--{D},
  viscous, incompressible vortex flows}, Comput. Fluids, 22 (1993),

{\sc W.~L. Briggs and V.~E. Henson}, {\em Wavelets and multigrid}, SIAM J. Sci.
  Comput., 14 (1993), pp.~506--510.

{\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.

{\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.

{\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),

{\sc D.~A. Caughey}, {\em Implicit multigrid techniques for compressible
  flows}, Comput. Fluids, 22 (1993), pp.~117--124.

{\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.

{\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.

{\sc Z.~Chen}, {\em Projection finite element methods for semiconductor device
  equations}, Comput. Math. Appl., 25 (1993), pp.~81--88.

{\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.

{\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.

{\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),

{\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.

{\sc G.~Dahlquist}, {\em A 'multigrid' extension of the {FFT} for the numerical
  inversion of {F}ourier and {L}aplace transforms}, BIT, 33 (1993),

{\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.

{\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.

{\sc A.~O. Demuren}, {\em Characteristics of three--dimensional turbulent jets
  in crossflow}, Int. J. Engng. Sci., 31 (1993), pp.~899--913.

{\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,

{\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.

{\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.

{\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,

{\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,

{\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.

{\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.

{\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.

{\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.

{\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.

{\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.

{\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),

{\sc F.~Grasso and M.~Marini}, {\em Multigrid techniques for hypersonic viscous
  flows}, AIAA J., 31 (1993), pp.~1729--1731.

{\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.

{\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.

{\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.

{\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.

{\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.

{\sc B.~Heise}, {\em Nonlinear field calculations with multigrid {N}ewton
  methods}, Impact Comput. Sci. Eng., 5 (1993), pp.~75--110.

{\sc M.~Holstm and F.~Saied}, {\em Multigrid solution of the {P}oisson
  {B}oltzmann equation}, J. Comput. Chem., 14 (1993), pp.~105--113.

{\sc A.~Hulsebos}, {\em Gribov copies and other gauge fixing beasties on the
  lattice}, Nucl. Phys. B, Proc. Suppl., 30 (1993), pp.~539--542.

{\sc S.~H. Hwang and S.~U. Lee}, {\em An optical flow estimation algorithm
  using the spatio temporal hierarchical structure}, IEICE Trans. Info. Sys.,
  E76--D (1993), pp.~507--514.

{\sc T.~Hwang and I.~D. Parsons}, {\em Multigrid method for the generalized
  symmetric eigenvalue problem. {P}art {I}. {A}lgorithm and implementation}, J.
  Numer. Meth. Engrg., 35 (1992), pp.~1663--1676.

\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Multigrid method for
  the generalized symmetric eigenvalue problem. {P}art {II}. {A}lgorithm and
  implementation}, J. Numer. Meth. Engrg., 35 (1992), pp.~1677--1696.

\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Multigrid solution
  procedures for structural dynamics eigenvalue problems}, Comput. Mech., 10
  (1992), pp.~247--262.

{\sc A.~C. Irving}, {\em Finite size effects, scaling and algorithms for
  lattice {CP$^{N4-1}$}}, Nucl. Phys. B, Proc. Suppl., 30 (1993), pp.~823--826.

{\sc M.~Israeli, L.~Vozovoi, and A.~Averbuch}, {\em Parallelizing implicit
  algorithms for time dependent problems by parabolic domain decomposition}, J.
  Sci. Comput., 8 (1993), pp.~151--166.

\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em Spectral multidomain
  technique with local {F}ourier basis}, J. Sci. Comput., 8 (1993),

{\sc W.~Joppich and R.~A. Lorentz}, {\em High order positive, monotone and
  convex multigrid interpolations}, COMPEL, Int. J. Comput. Math. Electr.
  Electron. Eng., 12 (1993), pp.~59--79.



End of MGNet Digest