NA Digest Sunday, May 21, 2006 Volume 06 : Issue 21

Today's Editor:
Tamara G. Kolda
Sandia National Labs

Submissions for NA Digest:

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Information via email about NA-NET:

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From: "Stephen Vavasis" <>
Date: Tue, 16 May 2006 13:17:27 -0400
Subject: Microsoft's "physics API"

Dear NA digest readers,

I saw a small news item that Microsoft is developing a "physics API" that
will allow computer games to compute directly on fast graphics hardware.
According to my colleagues who are in computer graphics, "physics API" in
this context means an API to routines that solve the differential equations
associated with fluid and solid motion. It seems like Microsoft could
benefit from expertise in our community on numerical solvers for ODE's, DAE's
and PDE's. Conversely, numerical analysts could benefit if the API is
written in a way that allows the best concepts and novel ideas from our field
to be incorporated into solvers.

I sent email to one person at Microsoft Research to try to get answers to
very basic questions-- Does the "physics API" involve numerical solvers for
PDEs and ODEs? Is there a numerical analyst on the development team? --but
I didn't get any information. Is this API something that our community ought
to care about?

-- Steve Vavasis


From: Richard Barrett <>
Date: Fri, 19 May 2006 11:10:13 -0400
Subject: Re: Gene's proposition and less boring digest

Last week Louis Komzsik of UGS claimed, "I believe the Numerical Analysis
and Linear Algebra community has a much bigger problem and that is the
problem of disconnect from real life." As a graduate student I stumbled upon
the following quote in a paper in a SIAM Journal ("Progress in Numerical
Analysis", Beresford N. Parlett, vol 3, pp 443-456, 1978) being thrown out,
and have carried it with me ever since:

Iterative methods for solving Ax=b: This was the most fashionable research
topic from 1950-1965. Has the point of diminished returns been researched?
Here is a quotation from a sophisticated user:

"Iterative techniques for processing large sparse linear systems were
popular in the late 1950's and early 1960's (and there decaying remains
still pollute some computational circles). When iterative methods
finally departed from the finite element scene in the mid 1960's - having
been replaced by direct sparse-matrix methods - the result was a quantum
leap in the reliability of linear analysis packages, which contributed
significantly to the rapid acceptance of FE analysis at the engineering
group level. (This effect, it should be noted, had nothing to do with the
relative computational efficiency; in fact iterative methods can run
faster on many problems if the user happens to know the optimal
acceleration parameters.) Presently, FE analyzers are routinely exercised
as black box devices;..."

Our own view of the situation is different. By their training, the experts
in iterative methods expect to collaborate with users. Indeed, the
combination of user, numerical analyst, and iterative method can be
incredibly effective. Of course, by the same token, inept use can make any
iterative method not only slow but prone to failure. Gaussian elimination,
in contrast, is a classical black box algorithm demanding no cooperation
from the user.

Surely the moral of the story is not that iterative methods are dead,
but that too little attention has been paid to the user's current needs?

-- End quote.

I attend conferences where I listen to many fascinating talks presented by
talented nla researchers, but frustrated in knowing that the vast majority
of the work will never be fairly tested within the context for which it was
developed. The solution? Researchers must find way to dive deep into the
world of applications. Rather than soil the mathematical beauty of linear
algebra, I believe the alert participant will find his or her mind
overwhelmed with new research ideas, and will have the thrill of seeing some
of their work lead to important scientific discoveries.

Richard Barrett
Future Technologies Group
Oak Ridge National Laboratory


From: "Komzsik, Louis" <>
Date: Tue, 16 May 2006 14:05:28 -0700
Subject: Re: Gene's proposition and Pandora's box

Dear Colleagues,

It seems like I have opened up Pandora's box. I have received so many
responses that I feel obligated to issue a general reply. Many of the
direct e-mails I received echoed my sentiments and surprisingly a lot of
them were from academia. Then there were a few defensive academic
e-mails not recognizing that their arguments actually strengthened my
point. A couple of open minded academic e-mails also came from people
who are interested in improving the situation. I have no permission from
the senders of those e-mails to post them somewhere on the web (as one
suggested) and I am not well equipped to do such activity, being in the
industry and all that. I encourage you, however, to send your e-mails
directly to the digest. This should at least liven up the digest (the
original point of Gene's proposition), and we might just be able to
lessen the problem I stated.

Several of you specifically asked about the origin and type of the large
industrial problems I mentioned. They are generalized, symmetric
eigenvalue problems mostly and come from structural analysis in various
industries. Some specific examples from my experience:

1. The current trend in engine and transmission departments of car
manufacturers is to use only one model for both geometry and analysis
work. This tendency results in 10-20 million node finite element models
and 60 to a 120 million degree of freedom global finite element

2. In the car body analysis at the same companies there is a desire to
use a single finite element model for both crash and vibration analyses.
This results in models with 5 to 25 million degrees of freedom, however,
with a frequency range of interest up to several thousand Hz, hence
thousands of mode shapes.

3. In the airframe industry, simulating the wing break scenario requires
well over 100 million degrees of freedom in the matrices to capture both
the local and global behavior of the skin of the wing. We have run an
industry leading 200 million degree of freedom analysis in this area.

4. Finally, in the aerospace industry (note the distinction from
airframe), the analysis of space craft and satellite components require
models to be analyzed often up to ten thousand Hz, resulting in
potentially over ten thousand mode shapes and all of them are equally

Here are some statistics gathered from a 75 million global degree of
freedom normal modes finite element analysis from the first class above:

Number of non-zeroes in the stiffness matrix: 1,384,305,995 terms (64 bit each)
Maximum front-size: 30,310
Memory used to factorize: 14 Gigabytes
Nonzero terms in factor matrix: 43,827,004,000 (an ~340 Gigabyte single file)
Lanczos recurrence time: 3,361 CPU seconds
Spectral transformation (shift and factor) time: 57,496 CPU seconds
Block solve time: 12 vectors per block, 15 solves totaling 30,645 CPU seconds
Complete modes analysis finding 20 modes: 680 elapsed minutes
IBM P5-575 with 8 POWER (1.95 GHz) workstation server in parallel

The practical eigenvalue analysis technique we use to solve such
problems is the block, shifted Lanczos method with partial and selective
orthogonalization. We have introduced this into NASTRAN more than 20
years ago. Since then, we have implemented many industry specific
proprietary improvements with both numerical (dealing with and
exploiting the zero subspaces of the matrices) and computational
(frequency, geometry domain decompositions and their parallel execution)
capabilities. We have been, of course, following the academic activities
in hopes of a new, better method.

The academia meanwhile has been mainly focused on preconditioned
conjugate gradient based eigenvalue methods, inner-outer iterations,
algebraic multi-grid approaches and so on, none of which have panned out
so far in the industry. They all strive for some sort of avoidance of
the sparse direct solve and the spectral transformation. Based on my
quoted numbers above, in itself this is a reasonable goal. However, the
evaluation of the new algorithms cannot be done on a different
(exterior) playing field and with an academic problem (the Laplacian
mentioned before).

I hope this answers some of the questions raised and no, unfortunately I
cannot disseminate above model as it is one of our client's production

Louis Komzsik
Chief Numerical Analyst
NASTRAN Development


From: "Ivan Slapnicar" <>
Date: Mon, 15 May 2006 16:52:00 +0200 (CEST)
Subject: Less boring digest

In the last Digest Louis Komzsik wrote the following observation:

"I believe the Numerical Analysis and Linear Algebra community has a much
bigger problem and that is the problem of disconnect from real life.
We, in the industry, solve 100 million order eigenvalue problems for tens of
thousands of eigenvectors (in feasible time). On the other hand, the
numerical and linear algebra conference presentations and publications in
the refereed journals (by the same clique) are discussing finding an
extreme eigenvalue of Laplacian matrices and comparing each other's
methods that are completely useless in the industry."

This remark has some truth in it. The best way to remedy this grievance
and improve relationship between academia and industry is to pose a
concrete question, that is to give the matrix. It is, namely, obvious,
that the industry is not interested in solving any 100 million order
problem, but a problem with a given structure. Even if just a matrix is
supplied without its history it may not help. So go ahead and pose your
problem to the community.

Kresimir Veselic and Ivan Slapnicar
University of Split


From: "Omar Ghattas" <>
Date: Fri, 19 May 2006 20:51:43 -0500
Subject: Live Webcast of ICES/TACC Lecture: David Keyes, 5/23 @ 3:30pm (CDT)

The Distinguished Lecture Series in Petascale Simulation, hosted by
the Institute for Computational Engineering and Sciences (ICES) and
the Texas Advanced Computing Center (TACC) at the University of Texas
at Austin, will be webcast live beginning with the first lecture on
Tuesday May 23, 2006.

The goal of the Lecture Series is to identify the scientific
breakthroughs that will be enabled by petascale computing, and the
modeling, algorithmic, and architectural challenges that must be
tackled to achieve science at the petascale level. Details of the
first lecture are as follows:

Speaker: Prof. David Keyes, Columbia University
Title: Petaflops, Seriously
Date: Tuesday, May 23, 2006
Time: 3:30pm CDT (UTC -5 hours)

Please visit to view
the abstract of the lecture, and to download the required browser
plug-in (which will take just a few seconds to install). Future
lectures in the series are listed at the website as well.


From: dana <>
Date: Sun, 14 May 2006 13:34:27 -0600
Subject: NA_WorkSheet Tool

NA_WorkSheet Tool

A basic tool that I created while taking NA classes here is Montana is
available as an open source Java applet/application at The
worksheet provides the basic algorithms that might be used in a first course
in NA. Parameters may be varied and detailed output is provided for most
algorithms. Any feedback on the accuracy, usefulness or lack there in is
greatly appreciated. I would like to take the NA_WorkSheet from beta to
production status.

Dana Proctor

NA_WorkSheet Web Location:


From: Ruediger Seydel <>
Date: Thu, 18 May 2006 10:50:02 +0200 (MEST)
Subject: New book, Computational Finance

Dear Collegues,

my book "Tools for Computational Finance" (Springer)
has been thoroughly revised and updated, and significantly
more material has been added. The new third edition is
available now, with 299 pages, 68 exercises, and 74 figures.
For more information see

Ruediger Seydel


From: Julie Haenisch <>
Date: Fri, 19 May 2006 11:20:30 -0400
Subject: New books from Princeton Univ Press

Google's PageRank and Beyond: The Science of Search Engine Rankings
Amy N. Langville and Carl D. Meyer

An Invitation to Modern Number Theory
Steven J. Miller and Ramin Takloo-Bighash

Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills
Paul J. Nahin

The Structure and Dynamics of Networks
Mark Newman, Albert-László Barabási, and Duncan J. Watts

Visit our math site at:


From: Lothar Reichel <>
Date: Fri, 19 May 2006 14:25:19 -0400
Subject: Bill Gragg Fest, Nov 06


We are planning an informal meeting on Friday and Saturday, November
3-4, 2006, at the Naval Postgraduate School in Monterey, California,
on the occasion of Bill Gragg's 70th birthday. The following speakers
have agreed to present talks:

Zhaojun Bai, Maribel Bueno, Adhemar Bultheel, Jim Bunch, Ralph Byers,
Annie Cuyt, Les Foster, Roland Freund, Gene Golub, Ming Gu, Ilse Ipsen,
Carl Jagels, Ren-Cang Li, Anders Lindquist, Aaron Melman, Vadim Olshevsky,
Beresford Parlett, Axel Ruhe, Ed Saff, Fiorella Sgallari, Danny Sorensen,
Herbert Stahl, Frank Stenger, Pete Stewart, Michael Stewart, Qiang Ye.

There will be a conference banquet on the evening of November 3.
People interested in attending the conference and/or the banquet should
send e-mail to a member of the organizing committee. Due to access
restrictions at the Naval Postgraduate School, we must have a list of
all attendees before September 1.

From the organizing committee:

Greg Ammar
Carlos Borges
Lothar Reichel
Marc Van Barel


From: gerhardwilhelm weber <>
Date: Sat, 20 May 2006 10:41:25 +0300
Subject: Workshop on "Networks in Computational Biology", Turkey, Sep 06

Workshop on
Ankara, Turkey, September 10-12, 2006

The Workshop on Networks in Computational Biology 2006 aims at giving an
overview and presenting most recent and important interdisciplinary
contributions by applied mathematics to the wealth of biological networks
focusing on theory, methodology, and applications. Based on the variety of
scientific backgrounds and experience combined for this special scientific
event, the theory of optimization, both continuous and discrete, dynamical
systems, data mining procedures, and the theory relating to "inverse problems"
will be addressed.

- Nihat Ay, Max-Planck Inst for Mathematics in the Sciences, Germany
- Fatihcan Atay, Max-Planck Inst for Mathematics in the Sciences, Germany
- Turker Biyikoglu, Max-Planck Inst for Mathematics in the Sciences, Germany
- Jerry L.R. Chandler, Krasnow Inst for Advanced Study, George Mason Univ, USA
- Jurgen Jost, Max Planck Inst for Mathematics in the Sciences, Germany

- Andreas Dress, Max-Planck Inst, Germany, and
CAS-MPG Partner Institute on Computational Biology, Shanghai, China
- Peter F. Stadler, Institut for Informatik and IZBI, Univ Leipzig, Germany

Please send an abstract of your talk in LATEX format (1/2-1 page, including
title and address) to {bulent, gweber}
Submission of Abstracts: May 15, 2006, but submissions are still welcome!
Notification of Acceptance: until June 15, 2006


From: El Mostafa DAOUDI <>
Date: Wed, 17 May 2006 19:31:22 +0200
Subject: CFP, Intl. Symp. on Computer Science and Its Applications

First Call for Papers
International Symposium on Computer Science and its Applications (IA’2006)
Oujda (Morocco), October 31 and November 1-2, 2006
Web page:
Email: and

The university Mohammed 1st, with the support of the CUD (Belgium), organizes
at the School of Applied Sciences (ENSAO), an International Symposium on
Computer Sciences and its Applications.

Topics (this list is non exhaustive)
- Software engineering
- Databases
- Computer security
- Internet technologies
- Networks and systems
- Real time systems
- Parallel and distributed computing systems
- Information technologies for teaching
- Scientific computing

We invite researchers to submit papers highlighting the most recent state of
their works, in English or in French, 8 pages maximum (.pdf, .ps or .doc), A4
format, size of characters 11 points. All the papers will be reviewed by the
symposium scientific committee. The selected ones will be published in the
proceedings the symposium. All the submissions must be sent before June 30,
2006 to the following address:

The notification of acceptance will be sent to the authors on Sep 15, 2006.

El Mostafa Daoudi: ENSA, Oujda, Morocco
Pierre Manneback, FPMS, Mons, Belgium


From: Klaus Frick <>
Date: Fri, 19 May 2006 11:47:35 +0200
Subject: Workshop: Variational and PDE Level Set Methods, Sep 06

Workshop: Variational and PDE Level Set Methods
Obergurgl, Tyrol, Austria
September 1st - 3rd, 2006

Aim and Scope:
Within the Forschungsschwerpunktprogramm "Industrial Geometry" founded
by the Austrian Science Foundation (FWF) we are organizing a workshop on
"Variational, PDE, and Level Set Methods" in Obergurgl, Tyrol, Austria
(September 1st - 3rd, 2006). The focus of this workshop will be on PDE
and variational methods on manifolds, as well as level set methods.
Theoretical as well as numerical aspects should be covered. We encourage
you to participate. You can enrol at the workshop homepage and if you
plan to give a talk or present a poster please submit an abstract there
or mail us.

Otmar Scherzer (University of Innsbruck)
Klaus Frick (University of Innsbruck)
Matthias Fuchs (University of Innsbruck)

Invited Speakers:
Martino Bardi (University of Padova)
Martin Burger (Johannes Kepler University, Linz)
Vicent Caselles (Balearic Islands University)
Gerhard Dziuk (Albert Ludwig University, Freiburg)
Stanley Osher (University of California at Los Angeles)
Martin Rumpf (Rheinische Friedrich-Wilhelms University, Bonn)
Christoph Schnoerr (University of Mannheim)
Fiorella Sgallari (University of Bologna)
Gabriele Steidl (University of Mannheim)
Joachim Weickert (Saarland University)
Jean-Paul Zolesio (INRIA)

Deadline for registration and abstract submission: July 30, 2006.


From: Subhashini Sivagnanam <>
Date: Wed, 17 May 2006 10:17:04 -0700 (PDT)
Subject: Summer Institute at the San Diego Supercomputer Center

Registration is now open for the 12th Annual Summer Institute

Taking Your Science to the Next Level: On the Path to Petascale Computing

SDSC is now accepting applications to attend its 12th Annual Summer
Institute. This year, the Summer Institute will focus on providing solutions
for the common limitations faced by our high-end users. The week-long program
includes lectures and hands-on laboratory sessions led by experts from SDSCs
staff as well as from other institutions who will introduce researchers to
technologies that can help them reach their goals faster and more efficiently.

The tentative schedule has been posted online at

Sessions include: MPI2/MPI I/O, Performance Characterization of Codes on
Different Machines, Optimization techiques, Profiling, Debugging parallel
codes, SDSC Storage Resource Broker, Data mining, Using Data visualization
tools, Various Case studies, Structured open lab

Deadline for applications is May 25, 2006. But dont wait until the last
minute. Space is limited to a maximum of only 35 attendees. Be sure to fill
out the entire registration form and include as much information as possible
regarding your research. Since we anticipate more applicants than there are
spaces, we will select those who will most benefit from the workshop. All
applicants will be contacted by June 1, 2006 to inform them whether they have
been accepted.

Free accommodations in UCSD student housing will be available to all student
attendees, and meals will be provided for all. Travel stipends will be
available for qualified, matriculating graduate and undergraduate students at
U.S. colleges and universities.


From: Rick Kufrin <>
Date: Tue, 16 May 2006 11:05:24 -0500 (CDT)
Subject: IWOMP 2006 OMPlab, Jun 06

IWOMP 2006 OMPlab Offers Comprehensive Hands-On Exposure, Information,
and Insight to the OpenMP Community

Organizers of the 2006 International Workshop on OpenMP (IWOMP), scheduled to
take place on June 12-15, 2006 in Reims, France, have announced the agenda for
OMPlab, the hands-on laboratory portion of the meeting. OMPlab is to be held
at the University of Reims Champagne-Ardenne during the second half of IWOMP
(June 14-15), following a two-day schedule of keynote speeches, technical
paper presentations, panel discussions, and social events scheduled at nearby
Pommery Castle. The OMPlab agenda, along with latest updates, can be found at
the IWOMP website:

Attendees of OMPlab will have access to a variety of current hardware for
running applications of their choice for activities such as porting and
performance tuning/evaluation for the duration of the event. Several
application-focused proposals for OMPlab have been submitted by OMPlab
participants and are available on the IWOMP website. Additional openings
remain for researchers who would like to bring their own codes to work
with during OMPlab in order to take advantage of the assistance of OpenMP
experts and tool developers who will be in attendance. Prepared lab
examples and exercises will be also be available for those participants
who prefer to gain hands-on experience in a more structured format.
The expected hardware and software offerings supporting OMPlab are
available for viewing at the IWOMP website.

OMPlab registration is included in the price of IWOMP and can be completed
online at The registration deadline for IWOMP is
June 8. IWOMP offers discounts for a limited number of student attendees.


From: gerhardwilhelm weber <>
Date: Sat, 20 May 2006 11:08:08 +0300
Subject: CfP: Special Issue of DAM: "Networks in Computational Biology"

Call for Papers
Journal "Discrete Applied Mathematics" (DAM)
Special Issue on

Discrete Applied Mathematics will publish a special issue on Networks in
Computational Biology, a topic that is also the theme of a workshop to be held
at METU, Ankara, Turkey, September 10-12, 2006.

See the following web page for futher details:


From: Martin Hanke-Bourgeois <>
Date: Fri, 19 May 2006 09:01:54 +0200
Subject: Tenure track position at the University of Mainz

Faculty position availabe:

Junior professor (W1), tenure track,
for Numerical Analysis/Scientific Computing

The Department of Physics, Mathematics, and Computer Science at the
Johannes Gutenberg-University in Mainz, Germany, invites applications
for a tenure track faculty position at the Juniorprofessor level
starting October 2006.

The position is available for candidates with strong expertise and
research interests in scientific computing (numerical solution of
ordinary or partial differential equations) on solid mathematical
grounds. The successful candidate is expected to teach, among other
things, graduate courses in the newly founded master program
"Computational Sciences - Rechnergestutzte Naturwissenschaften".

For additional information, see


From: Notay Y <>
Date: Fri, 19 May 2006 14:01:24 +0200 (CEST)
Subject: Visiting or postdoctoral position at ULB, Brussels

Applications are invited for a one-year visiting or postdoc position
in the "Service de Metrologie Nucleaire" of the University of Brussels,
starting January 1, 2007 or as agreed.
Applicants should have completed their PhD not before 2001.
Because the funding aims at international mobility, persons who
formerly lived, studied, or worked in Belgium are not eligible.

The successful applicant will have the opportunity to develop
his/her own research, in interaction with other members of the
"Service de Metrologie Nucleaire". Research themes may include
numerical linear algebra and algebraic multigrid methods.

For further information and application, please contact

Prof. Yvan Notay
Service de Metrologie Nucleaire
Universite Libre de Bruxelles (CP165/84)
Avenue F.D.Roosevelt, 50
1050 Bruxelles (Belgium)


From: Helge Galdal <>
Date: Mon, 15 May 2006 10:58:30 +0200
Subject: PhD Scholarships at CMA, Oslo

Centre of Mathematics for Applications (CMA) - a Centre of Excellence at the
University of Oslo, Norway, hereby invites applications for 3-4 PhD
scholarships. The positions are available from 1 July 2006, but the start
date may be delayed if necessary. Applications with all necessary enclosures
must be received by 8 June, 2006.

For details, please check the full announcement:


From: Romas Baronas <>
Date: Thu, 18 May 2006 09:25:29 +0300
Subject: Contents, Nonlinear Analysis: Modelling and Control

Nonlinear Analysis: Modelling and Control, ISSN 1392-5113,
Volume 11, Number 2, 2006

A free on-line edition is available at:


On a Nonlinear System of Reaction-Diffusion Equations, Pages 115-121
G.A. Afrouzi, S.H. Rasouli

The SVD-Fundamental Theorem of Linear Algebra, Pages 123-136
A.G. Akritas, G.I. Malaschonok, P.S. Vigklas

Circle and Popov Criterion for Output Feedback Stabilization of Uncertain
Systems, Pages 137-148
A. Benabdallah, M.A. Hammami

Discrete Multistage Optimization and Hierarchical Market, Pages 149-156
V.J. Bistrickas, N. Simeliene

Stability of Nuclear Reactor: Point Model Analysis, Pages 157-169
K. Bucys, D. Svitra

Persistence and Stability of a Food Chain Model with Mixed Selection of
Functional Responses, Pages 171-185
A. Maiti, B. Patra, G.P. Samanta

On Nonlinear Vekua Type Equations, Pages 187-200
S.V. Rogosin

Lie Group Analysis of Natural Convection Heat and Mass Transfer in an Inclined
Surface, Pages 201--212
S. Sivasankaran, M. Bhuvaneswari, P. Kandaswamy, E.K. Ramasami

Nonlinear Analysis: Modelling and Control, an official journal of
the Lithuanian Association of Nonlinear Analysts (LANA),
welcomes contributions from the international community.

For a paper submission, please refer to

Dr. Romas Baronas, Journal Secretary,
Nonlinear Analysis: Modelling and Control


From: Joseph Traub <>
Date: Tue, 16 May 2006 18:27:20 -0400 (EDT)
Subject: Contents, Journal of Complexity, Vol 22, No.3, 2006

Journal of Complexity
Volume 22, Number 3, June 2006

Solving linear programs with finite precision: II.Algorithms
Dennis Chung, Felipe Cucker

Improved upper bounds on the star discrepancy of (t,m,s)-nets and
Peter Kritzer

Weighted quadrature formulas and approximation by zonal function
networks on the sphere

The discretized discrepancy principle under general source conditions
Peter Mathe, Sergei V. Pereverzev

Expansion and linear complexity of the coordinate sequences over
Galois rings
Nigang Sun, Lei Hu

On the complexity of the resolvent representation of some prime
differential ideals
Lisi D'Alfonso, Gabriela Jeronimo, Paul Solerno


Corrigendum to "On the expressiveness and decidability of
o-minimal hybrid systems" [J.Complexity 21, (2005) 447-478]
Thomas Brihaye, Christian Michaux


From: JNM <>
Date: Wed, 17 May 2006 18:14:50 +0400
Subject: Contents, Journal of Numerical Mathematics

Vol.14, No.2, 2006, pp.83-160


Convergence and superconvergence of a nonconforming
finite element method for the Stokes problem
S.P.Mao and S.Chen

Reuse of standard preconditioners for higher-order
time discretizations of parabolic PDEs
K.-A.Mardal and T.K.Nilssen

On adaptive computational methods:
global norms controlling local errors

On a numerical method for resolution
of integral equations

End of NA Digest