p - Semantic Scholar
NASA
Contractor
ICASE
Report
Report No.
182040
90-35
ICASE FOURIER ANALYSIS OF FINITE ELEMENT PRECONDITIONED COLLOCATION SCHEMES
Michel O. Deville Ernest H. Mund
Contract May
No.
NAS1-18605
1990
Institute NASA Hampton, Operated
for Computer Langley
Applications
Research
Virginia
in Science
and Engineering
Center
23665-5225
by the Universities
Space
Research
Association
IW A National Aeronaulics and Space Administralion Langley Research Center Hamplon, Virginia 23665-5225 (NA_A-C_-132060) FFIUPIE:_ ANALYSIS ELEMENT PR_CONOTIIONZD COLLOCATTLI (TEA3 c ) 25 p
N90-Z3128
OF FINITE _'-jSCHEMES CSCL 12A
Uncl,as
Ri
G3/64
0280814
!
FOURIER
ANALYSIS
OF
PRECONDITIONED
FINITE
ELEMENT
COLLOCATION
Michel Universit6 Unit6
SCHEMES
O. Deville t
Catholique
de Louvain
de Mdcanique
Appllqude
Louvain-La-Neuve,
Belgium
and Ernest Universitd Service
H. Mund
Libre
de Bruxelles
de Mdtrologie BruxeUes,
Nucldaire
Belgium and
Universit6 Unitd
Catholique
de Louvain
de Thermodynamique
et Turbomachines
Louvain-La-Neuve,
Belgium
ABSTRACT This
p_per
investigates
conditioned
Fourier
elliptic
hyperbolic
and
pressions part
of the
of the
Fourier previous
model
considers (in the
conclusions
spectrum
collocation
eigenvalues
paper
analysis
the
schemes_
The
problems are the
tranverse
on the
of the iteration and
obtained
first the
with
part
operator of the paper
use
of symbolic
numerical
differential
of the 2-D Stokes
efficiency
of finite
element
pre-
one-dimensional
equation.
Analytical
computation. equations
problem.
element
finite
analyses
advection-diffusion
set of one-dimensional direction)
of some
The
second
resulting
All results
preconditioning
ex-
agree
from with
schemes.
IThls researchwas supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the author waz in residenceat the Institutefor Computer Applications in Science and Engineering (ICASE), NASA
Langley Research Center, Hampton, VA 23665.
1.
INTRODUCTION In the recent
past,
Chebyshev
collocation
schemes
have
been
applied
extensively
to the
numerical integration of the Navier-Stokes equations [1, 3, 4]. For scalar is well known that the condition number of the matrix system of discrete
elliptic problems, it algebraic equations
increases
at hand.
the
rapidly
with N, the number
preconditioning
this numerical
technique
burden.
constitute powerful fluid flow elements
of degrees
seems
The
present
the
authors
preconditioning
numerically,
were sake
preconditioned algebra
of simplicity, Fourier
process
elements,
collocation
of the eigenvectors analyses consists
that
finite
Therefore, to overcome
elements
(FE)
numerical
the
aims
perform
as preconditioners. carried
element
is uniform
and
is considered. cubic
is set Using
4, the Fourier
out in the framework
numerical
using linear A further
problem is done.
and
model
is reduced
The Q2-Q1
analysis
FE
of the eigenvalues
is investigated Stokes
Hcrrnite structure
the previous
discretization.
the The
these
the spatial
is also examined.
In Section
and
up with
a full analysis
model
technique
A similar
of a finite
size
corroborates
hyperbolic
to Fourier
results
quadratic,
solvers.
investigation
equation.
analytical
model
method
algebraic
one may
at
study
mesh
linear,
iteration
and
amenable
experiments
note to the
case,
Lagrangian
The upwinding
flow model
the Q2-Q1 element is the analyses on finite element
2, a one-dimensional
3, a one-dimensional
are candidates
corroborate
In this
solutions,
in an advection-difI_usion
a 1-D incompressible
by
This theoretical
FE preconditioning.
elements
in order
[10]).
ourselves
Richardson
operators
of the Fourier
[6]. In Section
quadratic
The
present
(cfr.
restrict
In Section
is preconditioned
operator.
the
scheme.
reduced.
as approximate
iteration
tool
it was shown that As all the previous
languages
we will
respectively.
preconditioners of the
out
manipulation
is considerably
collocation cubic
carried
use of symbolic the
of the problem adequate
preconditioners for general second-order elliptic equations. In [3], several in velocity-pressure formulation were investigated. From the analysis of operator, problem.
For
only
[5, 6] demonstrated
the eigenspectrum of the iteration best choice for the steady Stokes through
of freedom
to be
and
to
Q1-P0
The results
of Chebyshev
collocation
[3]. 2.
ELLIPTIC Let us first
MODEL consider
the
simple
elliptic
problem:
=/, with periodic boundary
0 _<x _
c_
9.9.
r
Report 5_clce/_,Orn,n,
1. Repo_
2. Government
No,
NASA
Page
Accession
No.
3. Recipient's
Catalog
No.
CR-182040
ICASE 4. Title
Documentation
SIralOn
Report
No.
90-35
and Subtitle
5. Report
FOURIER ANALYSIS OF COLLOCATION SCHEMES
FINITE
ELEMENT
May
PRECONDITIONED
7. Author(s)
Michel Ernest
Date
1990
6. Performing
Organization
Code
8. Performing
Organization
Report
No.
90-35
O. Deville H. Mund
10. Work
Unit
No.
505-90-21-01 9. Performing
Organization
Name
and Addre_
Institute for Computer Applications in Science and Engineering Mail Stop 132C, NASA Langley Research Center Hampton, VA 23665-5225 12. Sponsoring
Agency
National Langley Hampton,
or Grant
13.
Ty_
of ReportandPeriodCover_
Contractor Space
No.
NASI-18605
and Address
Aeronautics and Research Center VA 23665-5225
15. Supplementaw
Langley Richard
Name
11. Contract
Report
Administration 14. Sponsoring
,_gency
Code
Notes
Submitted Scientific
Technical Monitor: W. Barnwell
to SIAM Journal and Statistical
on
Computing Final
Report
16. Abstract
This paper investigates the spectrum of the iteration operator of some finite element preconditioned Fourier collocation schemes. The first part of the paper analyses one-dimensional elliptic and hyperbolic model problems and the advectiondiffusion equation. Analytical expressions of the eigenvalues are obtained with use of symbolic computation. The second part of the paper considers the set of one-dimensional differential tranverse direction) of the conclusions on the numerical
equations 2-D Stokes efficiency
resulting from Fourier analysis (in the problem. All results agree with previous of finite element preconditioning schemes.
18. Distribution
17. Key Words(SuggestedbyAuthor(s))
finite element, collocation method, eigenvalue analysis, spectral methods
Statement
54 - Numerical Analysis 34 - Fluid Mechanics and
Unclassified 19. SecuriW
Cla_if.
(of this report)
Unclassified
NASA
FORM
_.
SecuriW
Cla_if.
UnClassified
(of this page)
Heat
Transfer
- Unlimited 21. No.
24
of pages
22. Price
A0 3
1626 OCT 86
NASA-La_Iey,
1990
Contractor
ICASE
Report
Report No.
182040
90-35
ICASE FOURIER ANALYSIS OF FINITE ELEMENT PRECONDITIONED COLLOCATION SCHEMES
Michel O. Deville Ernest H. Mund
Contract May
No.
NAS1-18605
1990
Institute NASA Hampton, Operated
for Computer Langley
Applications
Research
Virginia
in Science
and Engineering
Center
23665-5225
by the Universities
Space
Research
Association
IW A National Aeronaulics and Space Administralion Langley Research Center Hamplon, Virginia 23665-5225 (NA_A-C_-132060) FFIUPIE:_ ANALYSIS ELEMENT PR_CONOTIIONZD COLLOCATTLI (TEA3 c ) 25 p
N90-Z3128
OF FINITE _'-jSCHEMES CSCL 12A
Uncl,as
Ri
G3/64
0280814
!
FOURIER
ANALYSIS
OF
PRECONDITIONED
FINITE
ELEMENT
COLLOCATION
Michel Universit6 Unit6
SCHEMES
O. Deville t
Catholique
de Louvain
de Mdcanique
Appllqude
Louvain-La-Neuve,
Belgium
and Ernest Universitd Service
H. Mund
Libre
de Bruxelles
de Mdtrologie BruxeUes,
Nucldaire
Belgium and
Universit6 Unitd
Catholique
de Louvain
de Thermodynamique
et Turbomachines
Louvain-La-Neuve,
Belgium
ABSTRACT This
p_per
investigates
conditioned
Fourier
elliptic
hyperbolic
and
pressions part
of the
of the
Fourier previous
model
considers (in the
conclusions
spectrum
collocation
eigenvalues
paper
analysis
the
schemes_
The
problems are the
tranverse
on the
of the iteration and
obtained
first the
with
part
operator of the paper
use
of symbolic
numerical
differential
of the 2-D Stokes
efficiency
of finite
element
pre-
one-dimensional
equation.
Analytical
computation. equations
problem.
element
finite
analyses
advection-diffusion
set of one-dimensional direction)
of some
The
second
resulting
All results
preconditioning
ex-
agree
from with
schemes.
IThls researchwas supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the author waz in residenceat the Institutefor Computer Applications in Science and Engineering (ICASE), NASA
Langley Research Center, Hampton, VA 23665.
1.
INTRODUCTION In the recent
past,
Chebyshev
collocation
schemes
have
been
applied
extensively
to the
numerical integration of the Navier-Stokes equations [1, 3, 4]. For scalar is well known that the condition number of the matrix system of discrete
elliptic problems, it algebraic equations
increases
at hand.
the
rapidly
with N, the number
preconditioning
this numerical
technique
burden.
constitute powerful fluid flow elements
of degrees
seems
The
present
the
authors
preconditioning
numerically,
were sake
preconditioned algebra
of simplicity, Fourier
process
elements,
collocation
of the eigenvectors analyses consists
that
finite
Therefore, to overcome
elements
(FE)
numerical
the
aims
perform
as preconditioners. carried
element
is uniform
and
is considered. cubic
is set Using
4, the Fourier
out in the framework
numerical
using linear A further
problem is done.
and
model
is reduced
The Q2-Q1
analysis
FE
of the eigenvalues
is investigated Stokes
Hcrrnite structure
the previous
discretization.
the The
these
the spatial
is also examined.
In Section
and
up with
a full analysis
model
technique
A similar
of a finite
size
corroborates
hyperbolic
to Fourier
results
quadratic,
solvers.
investigation
equation.
analytical
model
method
algebraic
one may
at
study
mesh
linear,
iteration
and
amenable
experiments
note to the
case,
Lagrangian
The upwinding
flow model
the Q2-Q1 element is the analyses on finite element
2, a one-dimensional
3, a one-dimensional
are candidates
corroborate
In this
solutions,
in an advection-difI_usion
a 1-D incompressible
by
This theoretical
FE preconditioning.
elements
in order
[10]).
ourselves
Richardson
operators
of the Fourier
[6]. In Section
quadratic
The
present
(cfr.
restrict
In Section
is preconditioned
operator.
the
scheme.
reduced.
as approximate
iteration
tool
it was shown that As all the previous
languages
we will
respectively.
preconditioners of the
out
manipulation
is considerably
collocation cubic
carried
use of symbolic the
of the problem adequate
preconditioners for general second-order elliptic equations. In [3], several in velocity-pressure formulation were investigated. From the analysis of operator, problem.
For
only
[5, 6] demonstrated
the eigenspectrum of the iteration best choice for the steady Stokes through
of freedom
to be
and
to
Q1-P0
The results
of Chebyshev
collocation
[3]. 2.
ELLIPTIC Let us first
MODEL consider
the
simple
elliptic
problem:
=/, with periodic boundary
0 _<x _
c_
9.9.
r
Report 5_clce/_,Orn,n,
1. Repo_
2. Government
No,
NASA
Page
Accession
No.
3. Recipient's
Catalog
No.
CR-182040
ICASE 4. Title
Documentation
SIralOn
Report
No.
90-35
and Subtitle
5. Report
FOURIER ANALYSIS OF COLLOCATION SCHEMES
FINITE
ELEMENT
May
PRECONDITIONED
7. Author(s)
Michel Ernest
Date
1990
6. Performing
Organization
Code
8. Performing
Organization
Report
No.
90-35
O. Deville H. Mund
10. Work
Unit
No.
505-90-21-01 9. Performing
Organization
Name
and Addre_
Institute for Computer Applications in Science and Engineering Mail Stop 132C, NASA Langley Research Center Hampton, VA 23665-5225 12. Sponsoring
Agency
National Langley Hampton,
or Grant
13.
Ty_
of ReportandPeriodCover_
Contractor Space
No.
NASI-18605
and Address
Aeronautics and Research Center VA 23665-5225
15. Supplementaw
Langley Richard
Name
11. Contract
Report
Administration 14. Sponsoring
,_gency
Code
Notes
Submitted Scientific
Technical Monitor: W. Barnwell
to SIAM Journal and Statistical
on
Computing Final
Report
16. Abstract
This paper investigates the spectrum of the iteration operator of some finite element preconditioned Fourier collocation schemes. The first part of the paper analyses one-dimensional elliptic and hyperbolic model problems and the advectiondiffusion equation. Analytical expressions of the eigenvalues are obtained with use of symbolic computation. The second part of the paper considers the set of one-dimensional differential tranverse direction) of the conclusions on the numerical
equations 2-D Stokes efficiency
resulting from Fourier analysis (in the problem. All results agree with previous of finite element preconditioning schemes.
18. Distribution
17. Key Words(SuggestedbyAuthor(s))
finite element, collocation method, eigenvalue analysis, spectral methods
Statement
54 - Numerical Analysis 34 - Fluid Mechanics and
Unclassified 19. SecuriW
Cla_if.
(of this report)
Unclassified
NASA
FORM
_.
SecuriW
Cla_if.
UnClassified
(of this page)
Heat
Transfer
- Unlimited 21. No.
24
of pages
22. Price
A0 3
1626 OCT 86
NASA-La_Iey,
1990
