Reports on Large Eddy Simulation
and related topics including:
CFD
Finite Elements and Fluids
Subgrid-scale Modelling
High Reynolds Number Flow Problems

This is under construction.I am still learning!
Papers

## SOME REPORTS

Numerical Analysis of Large Eddy Simulation

We are studying the mathematical development of large eddy simulation. Large eddy simulation (or LES) tries to predict the motion of the large structures in the turbulent flow of a fluid. LES has been highly developed by the engineering computational fluid dynamics community since its inception in 1970. The research program of our group at Pitt is to provide a mathematical and numerical analytic foundation for the field. In particular, it includes:

modeling-derivation of improved space filtered flow models,
see the report with G.P.Galdi and the one with Iliescu,

asymptotics- improved boundary conditions for such models,

analysis- rigorous analytical study of the modeling error,

numerical analysis- derivation and validation of algorithms for space filtered models,

direct simulation- a new approach to LES of direct simulation of large eddies,

simulation- computational testing and benchmarking of the models and algorithms under study.

A New Variational Multiscale Method-

Approximate Deconvolution Models for Large Eddy Simulation-

Understanding turbulent flow is central to many important problems including environmental and energy related applications (global change, mixing of fuel and oxidizer in engines and drag reduction), aerodynamics (maneuvering flight of jet aircraft) and biophysical applications (blood flow in the heart, especially the left ventricle). Turbulent flow is composed of coherent patches of swirling fluid called eddies. These range in size from large storm systems such as hurricanes to the little swirls of air shed from a butterflyąs wings. Large Eddy Simulation (LES for short) seeks to predict the motion of the largest and most important eddies uncoupled from the small eddies. This uncoupling is important because the large eddies are resolvable on a computational mesh (a collection of chunks of the physical problem) which can be handled by a supercomputer.
Our research involves modeling the large eddies (such as storm fronts, hurricanes and tornadoes in the atmosphere) in turbulent flow, predicting their motion in computational experiments and validating mathematically the large eddy models and algorithms developed. Current approaches to LES seem to be presently confronting some barriers to resolution, accuracy and predictability. It seems likely that many of these barriers can be traced to the mathematical foundation of the models used, the boundary conditions imposed and the algorithms employed for the simulations. The research undertaken is to develop these mathematical foundations as a guide for practical computation. This research promises to make it possible to extend the range of accuracy and reliability of predictions important to applications, such as those described above, where technological progress requires confronting turbulence! Click here to read more about our work-but check the reports!.

• Ervin, V.J., Layton, W.J., and Neda, M.,
Numerical Analysis of a Higher Order Time Relaxation Model of Fluids,
to appear International Journal of Numerical Analysis and Modeling.

We study the numerical errors in finite element discretizations
of a time relaxation model of fluid motion:
NSE(u) + chi u^{*} =f,
and div u = 0.
In this model, introduced by Stolz, Adams and Kleiser, u^{*}Ę is a
generalized fluctuation and chi the time relaxation parameter. The goal
of inclusion of the chi u^{*} is to drive unresolved fluctuations to
zero exponentially.
We study convergence of discretization of the model to the model's
solution as h and delta -> 0. Next we complement this with an
experimental study of the effect the time relaxation term (and a nonlinear
extension of it) has on the large scales of a flow near a transitional
point. We close by showing that the time relaxation term does not alter
shock speeds in the inviscid, compressible case, giving analytical
confirmation of a result of Stolz, Adams and Kleiser.

• W. Layton, A. Labovschii, C. C. Manica, Monika Neda and L. G. Rebholz,
The stabilized, extrapolated trapezoidal-Galerkin finite element method,
Technical report available by request.

• W. Layton and R. Lewandowski

A high accuracy Leray-deconvolution model of turbulence and its limiting behavior
Submitted, 2007.

• W. Layton, C. Manica, M. Neda and L. Rebholz

The joint helicity-energy cascade for homogeneous, isotropic turbulence generated by approximate deconvolution models
Submitted, 2005.

Are helicity statistics of turbulence predictable by LES models that truncate scales?
Can statistics be predicted when velocities cannot?

• W. Layton

Bounds on energy and helicity dissipation rates of approximate deconvolution models of turbulence
Submitted, 2006.

This report proves rigorous bounds on time averaged
HELICITY dissipation rates of weak solutions of
approximate deconvolution mlodels.
Related bounds are proven for the NSE.

• W. Layton

Superconvergence of finite element discretization of time relaxation models of advection
Submitted, 2006.

Precise Fourier analysis of the errror in FEM discretizations of time relaxation
regularizations establishes that
previous, worst case, upper a priori error estimates are,
at least in their most essential features, sharp.

• V. John, W Layton and C. Manica,
Time Averaged convergence of algorithms for flow problems,
Submitted, 2005.

What statistics of turbulence are numerically predictable?
What is the error in their computed values?
Can statistics be predicted when velocities cannot?

• W. Layton and M. Neda

Truncation of scales by time relaxation
JMAA, 325(2007)788-807.

We prove that the time relaxation regularization of turbulent flows
satisfies several important limiting properties.
Modulo a subsequence, as
filterwidth -> 0 model solution -> NSE weak solution,

As
relxation coefficient -> infinity
fluctuation in model's weak soln -> 0

Next we adapt a K41 like theory to
derive the formula for the relaxation parameter which forces
truncation of scales to EXACTLY the preassigned filter radius.

• William J. Layton and Monika Neda,
A similarity theory of approximate deconvolution models of turbulence,
Technical report,submitted, 2006

A similarity theory of AMs of turbulence is developed.
We show ADMs predict the correct statistics of homogeneous, isotropic turbulence through the cutoff wave number.
Beyond that, the truncation of scales depends upon the filter used.

• W. Layton and R. Lewandowski,
On a well-posed turbulence model,
Discrete and Continuous Dynamical Systems series B., 6, 2006,111-128.

This report continues the theoretical analysis of approximate deconvolution models.

• W LAYTON
Model reduction by constraints, discretization of flow problems and an induced pressure stabilization,
Numerical Linear Algebra with Applications VolumeĘ12, IssueĘ5-6, Date:ĘJune - August 2005, Pages:Ę547-562 .

This report studies an idea of model reduction:
Solve NSE(u,p)=f
Subject to: (u,p) belong to a finite dimensional subspace

From this new regularizations for both the momentum and continuity equations are constructed.
Stability and convergence are proven.

• S. Kaya, W. Layton and B. Riviere,
Subgrid stabilized defect correction methods for the Navier-Stokes equations
Feb. 2005 Accepted: SIAM JNA

This studies an attractive synthesis of DCM for stabilization,
and efficient solution of the nonlinear system with multi-scale VMS for
extending accuracy and correcting some small oscillations observed in DCM
near transition regions.

• V John, S Kaya and W Layton,
A two-level variational multiscale method for convection diffusion equations,
Comp. Meth. Appl. Mech. Engrg., 195, 4594-4603, 2006.

• M. Anitescu and W. Layton,
Uncertainties in large eddy simulation and improved estimation of flow functionals,
Appeared in: SIAM J Scientific Computing.

• Vincent J. Ervin, William J. Layton and Monika Neda,
Numerical Analysis of a Higher Order Time Relaxation Model of Fluids,
Accepted: Inter J Numer Anal & Modeling, 2006;

The numerical analysis of a time relaxation regularization is performed.
We show the rregularization is of arbitrarily high order of accuracy,
it does not alter shock speeds or jumps and
discretizations of it preserve these accuracy and stability properties.

• W. Layton,
Calculating functionals of solutions of large sparse systems,
This research white paper is about a simple problem in numerical
linear algebra which has many applications to CFD and other areas:
Suppose we wish to solve the very large, sparse, nonsymmetric system Au=f.
Often, the solution u is only used to calculate linear and nonlinear
functionals of u , F(u), such as lift, drag,...
One approach is to generate u_n by an iterative method then calculate
F(u_n). Clearly, we should seek iterative methods and functional
recovery methods for which functional approximations F_n :
converge much faster than u_n converges.
This report has some initial ideas about this problem. When they are
properly tested , it will be available as a research report.
Although this is a NLA white paper, its significance
to CFD and LES is obvious so I'm listing it here.

• W. Layton and R Lewandowski,
Consistency and feasibility of approximate deconvolution models of turbulence,
This report considers the time averaged consistency error in ADM models
of fully developed turbulence. We prove rigorously that for homogeneous
isotropic turbulence the zeroth order model (developed in our reports below)
has consistency error which
converges to zero uniformly in the Reynolds number.
Next the whole family of ADM's (developed by Adams and Stolz) are considered
We prove consistency error bounds which show that the
consistency error -> 0 rapidly for the averaging radius
well within the inertial range.
The derived bounds have the interesting property that for models
of order >2, the error decrreases as L increases-an observation possibly significant
for problems in geophysics.

• R. Lewandowski and W. Layton,
Un Filtre pour la SGE Etudie a la Lumiere du K41,
Journees AUM AFM, 2004, to appear.
This report gives a new way to study and assess LES models.
In the case of homogeneous , isotropic turbulence many estimates can be sone simply and sharply using dimensional annalysis.
Using this tool, we give upper estimates on model's consistency error of the form:

||consistency error||< C(Re) (Averaging Radius)^a

Consistency thus requires an upper bound oon the averagging radius in terms of Re of the form:

Averaging Radius << C1(Re) , which ->0 as Re grown to infinity.

On the other hand, the restriction that the averaging radius be chosen
well in the inertial range gives a LOWER bound on the averaging radius of the form:

C2(Re) < < Averaging Radius.

Clearly, practicability of LES using the given model requires

C2(Re) << C1(Re)

which yields an analytic method of assessing and comparing models!

• M. Anitescu, W. Layton and F. Pahlevani,
Implicit for local effects, explicit for nonlocal is unconditionally stable,
appeared in: ETNA, (ŇElectronic Transactions on Numerical AnalysisÓ)

• W. Layton and R. Lewandowski,
A simple , accurate and stable scale similarity model for large eddy
simulation: Energy balance and existence off weak solutions,
appeared in Applied Math. Letters,2003.

This report began the mathemtical study of the deconvolution models proposed by

• W. Layton and R. Lewandowski,
On a Well-Posed Turbulence Model,
appeared in DCDS series B
This and the above paper both study a very simple closure model that
occurs as a zeroth order model in most families of LES models.
These two papers contail the key ideas in our study of deconvolution
and other models. In particular, we give a simple and clear delineation
of the model's energy balance. Using it, we prove: *Existence, uniqueness and regularity of strong solutions.
*The same for the Euler-LES model induced by setting the kinematic viscosity to zero.
*The model's solution w -> u (a weak solution of the NSE) as the
*Rigorous bounds on the model's connsistency error.
*Rigorous bounds on the model's error:
|| model's solution - average of NSE solution||.
*First results on non generation of spurious vorticity by the model.
and so on.
The results in this report has recently been extended to the entire family of ADModels by my student A. Dunca (with Y. Epshteyn-another
strong Pitt-PhD student whi is working with B. Riviere).
Email me and I'll send a copy.

• W. Layton,
A Mathematical Introduction to Large Eddy Simulation
to appear in: Computational Fluid Dynamics-Multiscale Methods
(H. Deconinck, editor) , Von Karman Institute for Fluid Dynamics,
Rhode-Saint-Gen\`ese, Belgium, 2002.
These are lecture notes for a 2002 short course at the von Karman Institute.

• W. Layton,
Advanced models for large eddy simulation.
to appear in: Computational Fluid Dynamics-Multiscale Methods
(H. Deconinck, editor) , Von Karman Institute for Fluid Dynamics,
Rhode-Saint-Gen\`ese, Belgium, 2002.
These are lecture notes for a 2002 short course at the von Karman Institute.

• W. Layton,
Variational Multiscale Methods annd Subgrid Scale Eddy Viscosity.
to appear in: Computational Fluid Dynamics-Multiscale Methods
(H. Deconinck, editor) , Von Karman Institute for Fluid Dynamics,
Rhode-Saint-Gen\`ese, Belgium, 2002.
These are lecture notes for a 2002 short course at the von Karman Institute.

• A. Dunca, V. John and W. Layton,
Approximating local averages of fluid velocities: the equilibrium Navier-Stokes equations,
appeared: Applied Numerical Mathematics.

• S. Kaya and W. Layton,
Subgrid eddy viscosity methods are Variational Multiscale Methods,
We show a new method we introduced in a paper below is actually a
suitably generalized VMM. One interest in the new method is that it
allows models of fluctuations in which fluctuations are allowed to move.
Thus, it is a conceptual advance over RFB type implementaions
of the variational multiscale idea of Hughes. Songul Kaya has performed a
complete analysis of the method for the NSE at high Reynolds number.

• V. John, W. Layton and N. Sahin,
New Near Wall Models for the Large Eddy Simulation
of Recirculating Flows,
technical report, 2002. appeared.

• A. Dunca, V. John, W. Layton,
The Commutation Error of the Space Averaged Navier Stokes Equations on a Bounded Domain,
appeared in: J. Mathematical Fluid Mechanics.

In deriving LES models of turbulent flows, the first step is usually to
filter the NSE by convolving it with the chosen filter function.
Next, convolution and differentiation are interchanged then closure
addressed, etc. However(!), on a bounded domain, convolution
and differentiation DO NOT COMMUTE. Obviously, the hope is that the
commutation error ->0 as the averaging radius ->0 fast enough to be
negligable.
In this report, we calculate the commutation error. We prove that:
The commutation error->0 in Lp if and only if the normal stress of
the turbulent velocity is identically zero everywhere on the
boundary.
This means that any numerical method discretizing the strong form of the
LES equations, such as a finite difference method, makes an O(1) error!
We next show that the commutation error does ->0 in an appropriate
weak sense. This means that variationally based methods are acceptable.
These include FEM's , spectral methods and spectral element methods.
Finally, we study the effect the commutation error term has
on the global kinetic energy balance in several LES models.

My collaborator has put a compressed postscript file of this report

• A. Dunca, V. John, W. Layton and N. Sahin ,
Numerical Analysis of Large Eddy Simulation,
DNS/LES Progress and Challenges
(eds: C.Liu, L.Sakell.T.Beuntner)Greyden Press, 359-364, 2001.

• W. Layton and R. Lewandowski,
Analysis of an eddy viscosity model for large eddy simulation of turbulent flows,
appeared in: J. Math.Fluid Mech., 2001.

This report begins the analysis of a new, eddy viscosity type
LES model. The model contains an extra, eddy viscosity term:

div[ C delta | w - g*w | D( w ) ] , where:

delta := averaging radius,
w := the models approximation of the local velocity average: g*(fluid velocity)
D := deformation tensor.

The analysis of this model is challenging. Because |w-g*w| can be
unbounded, the natural formulation of the model is in a nonlinearly
and unboundedly weighted Sobolev space - placing it outside the Leray-Lions
theory of weak solutions. Nevertheless, we formulate the correct solution concept
based on the model's natural energy balance.
We prove existence of such a solution.

We also estimate the 'optimal' value of the
moel's constant "C" for homogeneous, isotropic turbulence.
Lastly, we show explore a connection with differential filters.

• W. Layton, F. Schieweck and I. Yotov,
Coupling fluid motion with porous media flow,
technical report, appeared in: SINUM.

• W. Layton, H.K. Lee and J. Peterson,
A defect-correction method for the incompressible Navier-Stokes equations,
Applied Math. and Computing, 2001.
This report gives L2 error analysis and tests of a defect correction
method tailored to high Re flow problems. Careful numerical tests,
illustrating the method's possibilities and verifying the theoretical
predictons are given as well.

• M Kaya and W Layton,
On "Verifiability" of models of the motion of large eddies in turbulent flows,
in: Differential and Integral Eqns..

There are many models used in LES and an LES model can be thought of
as an approximation of the true Reynolds stress tensor , R(u), by a
tensor depending only upon the local average of u, g*u:

with R(u) := g*(uu) -g*u g*u find an approximation S ~ R
S=S(g*u) ~ R(u)

The accuracy S~R can be assessed using velocity data by computing
||| S(g*u) - R(u) ||| for some norm |||.||| . This , in effect,
computes the modeling residual and it does not imply that (as desired)
the solution of the LES model equations is close to g*u!

(In fact in the report below we iidentify one model seeming to violate this.)
This report derives structural conditions upon the model (i.e., S )
, under which smallness of the modeling residual implies the
model's predicted large eddy motion is close to the true motion.
Four examples of LES models are evaluated with respect to the derived conditions.

• T Iliescu, V John, W Layton, G Matthies and L Tobiska,
An Assessment of Models in Large Eddy Simulation.
in: IJCFD.

We compare carefully and assess 4 LES models and also a DNS. To try , in
so far as possible, to control for the underlying uncertainties and
instabilities in turbulent flows we base our comparison on global
behavior which is more reliably computable than pointwise velocities.
These points of comparison include:

(i)The global kinetic energy in the large eddies.
(ii)The energy dissipation rates of the large eddies.
(iii)The deviation of the LES predictions from a moderate RE
DNS of the NSE.
(iv)At higher RE, we look at aspects of a 'high RE - fine mesh' simulation
which agree across all the tested models. Accepting these features,
We compare these features to a 'coarse mesh-high RE ' simulation
of the same problem.

By this careful approach, we reach some interesting conclusions about
the models tested. In particular, one popular model is found to be
wanting.

My collaborator has put a compressed postscript file of this report

• T Iliescu, V John and W Layton,
Convergence of finite element approximations of large eddy motion,

in: Numer.Methods for PDE's.

We conder a complex model commonly used in LES ( sometimes
called the gradient model) including
models of the: cross terms, Leonard terms, subgrid scale terms,
and a wall law. We prove convergence of the FEM for this model as h->0
for fixed Re and averaging radius , delta. This convergence is not uniform
in Re (as was proven for a simpler model in our other report,below).
We give experiments illustrating the convergence of the method.
Mathematical details arise because of the higher order, higher
degree, non-monotone nonlinearities of the cross terms. Nevertheless,
convergence is proven and the open question now becomes 'robustness'
meaning uniformity in model parameters, such as Re.

My collaborator has put a compressed postscript file of this report

• L. Berselli,G P Galdi, T Iliescu and W Layton,
Mathematical Analysis for a new large eddy simulation model,
technical report, August,2001.
to appear in: Math. Methods and Models in the Applied Sciences,
This report considers the new model proposed in the report by Galdi and L.
(described below). We prove existence of weak solutions to the model.
The report begins considering the case when the exponent "mu" in the
Smagorinsky-like sgs model for turbulent fluctuations approaches zero.
In other words, it begins to consider well-posedness of the model without
using this extra regularization.

This report took a long time to complete, but it's done at last.

Download a compressed postscript file of this report
from the Argonne website.

• W. Layton,
Bounds on energy dissipation rates of the large eddies in turbulent shear flows,
technical report,June, 2000, submitted to Math. and Computer Modeling.
Shear flows are very important in experimental turbulence studies.
Thus, they have also attacted analytical attention, in spite of the
fact that they are less mathematically 'convenient' than models
with homogeneous Dirichlet boundary conditions. In 3d there has recently
been considerable work bounding the average energy dissipation rate in
the NSE for shear flows. On the other hand, turbulent shear flows
are not solved by DNS but rather by techniques such as LES. Thus,
it's interesting to understand the average energy dissipation rates
in various large eddy models. This can be used for example to assess
and compare models mathematically. This report begins the study of
energy dissipation rates in 'large eddy fluids'.

• V John and W Layton,
Analysis of numerical errors in Large Eddy Simulation,
in SIAM J. on Numerical Analysis, January 2002.
This report considers the errors in numerical approximations of a
commonly used model in LES for turbulent flow. We prove that approximate
solutions of the model converge, **uniformly in the Reynolds number **,
to solutions to the LES model for fixed averaging radius , delta.
This is often claimed as a chief advantage of LES. Thus, we give
apparently the first rigorous analytical support for this claim.
We also give numerical experiments which are fully consistent with our
uniform-in-Re convergence result.

My collaborator has put a compressed postscript file of this report

• W Layton,
A connection between subgrid scale eddy viscosity and mixed methods,

to appear in: Applied Mathematics and Computing, 2001.

-email me if you'd like a copy- April,2000.

This report considers the subgrid artificial viscosity idea , recently
introduced by Guermond (in which the diffusion acts ONLY on the finest
resolved mesh scales). We show that this nonconforming artificial viscosity
method is actually a conforming , mixed approximation to ( u , grad u )
which is then restricted to u. This report establishes another realization
of Guermond's basic idea, gives an optimal approximation property in
( u , grad u ) , provides an idea how to extend the method and analysis to
the case of the Navier Stokes equations, gives an error estimate for
the restriction of (uh , grad uh ) to uh , and shows a surprising
link beween the "EVSS" method (widely used for viscoelastic flow
calculations) and subgrid eddy viscosity.

• W Layton,
Analysis of a scale similarity model of
the motion of large eddies in turbulent flow,

Journal Mathematical Analysis and Applications 264(2001)546-559.

This report studies a scale similarity model for large eddy motion.
If we call the solution of the model w then w approximates
the local spacial averages of the true solution u of the NSE:
w approximates g*u.
This paper considers the modeling error || w - g*u ||.
We prove
(i)Consistency in the limit:
w-> u as the eddy scale delta -> 0.
(ii) A bound on the modeling error by the residual of the true solution:
||| w - g*u ||| < ||| Reynolds Stress(u) - Approx.R.S.(u) |||
This means the modeling error can really be assessed by a moderate Re DNS
or by analysis of flow data from a physical experiment.

• W Layton,
Approximating the larger eddies in fluid motion V: Kinetic
energy balance of scale similarity models,

Math. and Computer Modeling,31(2000)1-7.

This report first reviews the development of the scale similarity
model. The kinetic energy balance of these is very unclear.
However,having the correct balance of kinetic energy is essential in a good model.
Within an approach to modelling the effects of unresolved eddies on the large
eddies many different models are possible.
This report shows that it is possible to use the principle of scale
similarity to develop new scale similarity models which satisfy the
correct kinetic energy balance relation (known as'Leray's inequality').

• N. Sahin,
New perspectives on Boundary conditions for large eddy simulation,
Interested in wall laws for large eddy simulation? Check out the report of
my student Niyazi Sahin. Email him ( nisst6+@pitt.edu) or me and
we'll send you one!

• T Iliescu and W Layton,
Approximating the larger eddies in fluid motion III:
the Boussinesq model for turbulent fluctuations,

Analele Stiintifice ale Universitatii "Al.I.Cuza" ,
Series Mathematics,(published since 1900) TomulXLIV(1998),245-261.
This volume is dedicated to the 70th birthday of Professor C Corduneanu.
If you email, I'll post you a copy.

In this report, we develop an approximation to the kinetic energy in the turbulent fluctuations [the 'turbulent
kinetic energy']. Based on this approximation , we give three new subgrid scale models. The report also
develops an existence theory for the corresponding continum limit [NSE+SGSmodel].

• W.Layton and L Tobiska
A multi-level method with backtracking for the NSE

This appeared already in SIAM JNA (1998) and is available from their epubs site.

• W. Layton and W. Lenfrink,
A multi-Level mesh independence principle for the Navier Stokes Equations
SIAM JNA 33(1996) 17-30.

This article might also be availabe at the above epubs site.

• V John and W. Layton,
Approximating local averages of fluid velocities I: the Stokes problem,
Computing 66(2001)269-287.

We show that it is possible to directly
approximate accuately the large eddies without modeling and with no error
in the boundary conditions. Energy accumulating in the smallest resolved
scales is removed via postprocessing and the remainder is an accurate
approximation of the large eddies.

A .pdf file of this is here,

• G.P. Galdi and W. Layton,
Approximation of the larger eddies in fluid motion II:
A model for space filtered flow.

Mathematical Models and Methods in the Applied Sciences, 10 (2000) , 1-8

This report proposes new continuum models for space filtered flow.

• V. Ervin, W Layton and J Maubach,
Adaptive defect correction methods for viscous, incompressible flow problems,

SIAM JNA,37(2000)1165-1185.

. It considers adaptive methods for high Reynolds number flow problems.
It is available at SIAM's epubs site

In this report, we consider "stationary turbulence" via
a defect correction discretization. One advantage of this
approach is that the SGS model used can be incorporated into
the residual calculation-thus not increasing the difficulty
of resolving the nonlinear system. We show this works in an
adaptive context and prove a posteriori error estimates for the whole
approach.

• W. Layton,
Weak Imposition of "no-slip" Boundary Conditions in Finite Element Methods

Computers and Mathematics with Applications,38(1999)129-142.

In other simulations of underresolved flows at high Reynolds numbers
we noticed spurious eddies being generated near walls where the flow
was strongly convergent. The tangential stresses were also very large
near those locations. The natural idea arose to weaken or underweight
the no slip condition in FEM implementations so as to reduce spuriosity
and still recover the correct "no-slip" BC as the
meshwidth "h" goes to zero. This report begins the study of these ideas:

* weak imposition using Lagrange multipliers,
* penalty imposition of no-slip,
* replacing "no-slip" with slip with friction/resistance.

In this last case the resistance parameter depends on Re and "h" so that
"no-slip " is recovered as h->0. It is interesting to note that
this last idea is also appealing physically since it agrees with
experimental observations of flows at very high Re.
Intuitively, nature does not admit infinite tangential forces. After they
excede a certain level the fluid particles must slide along
the boundary in the direction of those forces!

• W.Layton, J.Peterson and H.K.Lee,
Numerical solution of the stationary Navier Stokes Equations by a
multi-level finite element method,SIAM J Sci.Computing,1998

This appeared already in SIAM JSciComputing (1998) and is available from their epubs site.

• W. Layton,
A nonlinear , subgridscale model for incompressible, viscous flow problems,
SIAM J. Sci. Computing 17, 1996, 347-357.

• W. Layton,
Subgridscale Modelling and Finite Element Methods for the Navier-Stokes Equations,

Preprint MBI-96-4, Otto-von-Guericke University
Magdeburg , Germany, 1996.

This is a report based on lecture notes from a short course on
the topic in Magdeburg. I'll be happy to mail you a copy (until they run out)
if you email me your postal address (wjl+@pitt.edu).

• W Layton, A J Meir and P G Schmidt, A two-level discretization
method for the stationary MHD equations, ETNA, 6(1997) , 198-210.

This is available at the ETNA web page.

• W.Layton, J.Maubach and P.Rabier,
Robustness of an elementwise parallel finite element method for convection dominated,
convection diffusion equations

SIAM J Scientific Computing, 1998.

This appeared already in SIAM (1998) and is available from their epubs site.