Reports on Large Eddy Simulation
and related topics including:
CFD
Finite Elements and Fluids
Subgridscale Modelling
High Reynolds Number Flow Problems
This is under construction.I am still learning!
Papers
SOME REPORTS
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:
modelingderivation 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 butterfly¹s 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 workbut 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^{*}Ê 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 trapezoidalGalerkin finite element method,
Technical report available by request.

W. Layton and R. Lewandowski
A high accuracy Leraydeconvolution model of turbulence and its
limiting behavior
Submitted, 2007.
Click here to
download the report!

W. Layton, C. Manica, M. Neda and L. Rebholz
The joint helicityenergy cascade for homogeneous, isotropic turbulence
generated by approximate deconvolution models
Submitted, 2005.
Click here to
download the report!.
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.
Click here to
download the report!.
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.
Click here to
download the report!.
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.
Click here to
download the report!.
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)788807.
Click here to
download the report!.
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
Click here to
download the report!.
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 wellposed turbulence model,
Discrete and Continuous Dynamical
Systems
series B., 6, 2006,111128.
This report continues the theoretical analysis of approximate
deconvolution models.
Click here to
download the report!.
 W LAYTON
Model reduction by constraints, discretization of flow
problems and an induced pressure stabilization,
Numerical Linear Algebra
with Applications VolumeÊ12, IssueÊ56, Date:ÊJune  August 2005,
Pages:Ê547562 .
Click here to
download the report!.
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 NavierStokes equations
Feb. 2005 Accepted: SIAM
JNA
Click here to
download the report!.
This studies an attractive synthesis of DCM for stabilization,
and efficient solution of the nonlinear system with multiscale VMS for
extending accuracy and correcting some small oscillations observed in DCM
near transition regions.
 V John, S Kaya and W Layton,
A twolevel variational multiscale method
for convection diffusion equations,
Comp. Meth. Appl. Mech. Engrg.,
195, 45944603, 2006.
Click here to
download the report!.
 M. Anitescu and W. Layton,
Uncertainties in large eddy simulation and
improved estimation of flow functionals,
Appeared in:
SIAM J Scientific Computing.
Click here to
download the report!.
 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;
Click here to
download the report!.
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,
Click here to
download the report!.
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 increasesan observation possibly
significant
for problems in geophysics.
Click here to
download the report!.
 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, (ÒElectronic 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
Stolz and Adams.
 W. Layton and R. Lewandowski,
On a WellPosed 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 EulerLES model induced by setting the kinematic
viscosity to zero.
*The model's solution w > u (a weak solution of the NSE) as the
averaging radius >0.
*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. Epshteynanother
strong PittPhD 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 DynamicsMultiscale Methods
(H.
Deconinck, editor) , Von Karman Institute for Fluid Dynamics,
RhodeSaintGen\`ese, Belgium, 2002.
These are lecture notes for a 2002 short course at the von Karman
Institute.
Click here to
download the report!.
 W. Layton,
Advanced models for large eddy simulation.
to appear in: Computational Fluid DynamicsMultiscale Methods
(H.
Deconinck, editor) , Von Karman Institute for Fluid Dynamics,
RhodeSaintGen\`ese, Belgium, 2002.
These are lecture notes for a 2002 short course at the von Karman
Institute.
Click here to
download the report!.
 W. Layton,
Variational Multiscale Methods annd Subgrid Scale Eddy Viscosity.
to appear in: Computational Fluid DynamicsMultiscale Methods
(H.
Deconinck, editor) , Von Karman Institute for Fluid Dynamics,
RhodeSaintGen\`ese, Belgium, 2002.
These are lecture notes for a 2002 short course at the von Karman
Institute.
Click here to
download the report!.
 A. Dunca, V. John and W. Layton,
Approximating local averages of
fluid velocities: the equilibrium NavierStokes 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.
Both reports are available at
Click here to
download the report!.
 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
on his
webpage to be downloaded.
Click here
to
go there to download 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, 359364, 2001.
Click here to download a .ps preliminary
version
of this report.
 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 wg*w can be
unbounded, the natural formulation of the model is in a nonlinearly
and unboundedly weighted Sobolev space  placing it outside the
LerayLions
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.
Click here to download this as a
postscript file
.
 W. Layton, H.K. Lee and J. Peterson,
A defectcorrection method for the incompressible NavierStokes
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 meshhigh 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
on his
webpage to be downloaded.
Click here
to
go there to download 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, nonmonotone 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
on his
webpage to be downloaded.
Click here
to
go there to download 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
Smagorinskylike sgs model for turbulent fluctuations approaches zero.
In other words, it begins to consider wellposedness 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.
Click here.
 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
uniforminRe convergence result.
My collaborator has put a compressed postscript file of this report
on his
webpage to be downloaded.
Click here
to
go there to download 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)546559.
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)17.
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),245261.
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 multilevel method with backtracking for the NSE
This appeared already in SIAM JNA (1998) and is available from their epubs
site.
Click here to
go there.
 W. Layton and W. Lenfrink,
A multiLevel mesh independence principle for the Navier Stokes
Equations
SIAM JNA 33(1996) 1730.
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)269287.
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,
Click here
to
go there to download this report.
 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) , 18
This report proposes
new continuum models for space filtered flow.
Click here to download a postscript version
of this report.
 V. Ervin, W Layton and J Maubach,
Adaptive defect correction methods for viscous, incompressible flow
problems,
SIAM JNA,37(2000)11651185.
. 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 calculationthus 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.
Click
here to download it as .ps file.
 W. Layton,
Weak Imposition of "noslip" Boundary Conditions in Finite Element
Methods
Computers and Mathematics with
Applications,38(1999)129142.
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 "noslip" BC as the
meshwidth "h" goes to zero. This report begins the study of these
ideas:
* weak imposition using Lagrange multipliers,
* penalty imposition of noslip,
* replacing "noslip" with slip with friction/resistance.
In this last case the resistance parameter depends on Re and "h" so
that
"noslip " 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
multilevel finite element method,SIAM J Sci.Computing,1998
This appeared already in SIAM JSciComputing (1998) and is available from
their epubs
site.
Click here to
go there.
 W. Layton,
A nonlinear , subgridscale model for incompressible, viscous flow
problems,
SIAM J. Sci. Computing 17, 1996, 347357.
 W. Layton,
Subgridscale Modelling and Finite Element Methods for the NavierStokes
Equations,
Preprint MBI964, OttovonGuericke 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 twolevel discretization
method for the stationary MHD equations, ETNA, 6(1997) , 198210.
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.
Click here to
go there.
 J Boland, G B Ermentrout, C A Hall, W Layton and H Melhem,
Analytical and numerical studies of natural convection problems,
Proc Int Conf on Thy and Appls of Diff Eqns, 1988.
 J Boland and W Layton,
An analysis of the FEM for natural convection problems,
Num Meth for PDE's 2(1990) 115126.
 J Boland and W Layton,
Error analysis of finite element methods in steady natural convection
problems,
Num. Functional Analysis and Opt. 11(1990), 449483.