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Wabro M. Coupled AMG for Navie-Stocks Equation 2003

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InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.1/31
CoupledAlgebraicMultigrid
Methodsforthe
p
Navier-Stokes
Equations
MarkusWabro
JointworkwithWalterZulehner
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
www.numa.uni-linz.ac.at
ThisworkhasbeensupportedbytheAustrianScienceFoundationFondszurF
¨
orderungder
wissenschaftlichenForschung(FWF)underthegrantP14953RobustAlgebraicMultigridMethods
andtheirParallelization
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.2/31
Overview
Overview
Introduction(Navier-Stokes/Oseenequations)
(Algebraic)multigridmethodsforthesolutionofthe
Oseenequations
DecoupledApproach
CoupledApproach
Coarse-levelconstruction
Smoothing
Numericalresults
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.2/31
Overview
Overview
Introduction(Navier-Stokes/Oseenequations)
(Algebraic)multigridmethodsforthesolutionofthe
Oseenequations
DecoupledApproach
CoupledApproach
Coarse-levelconstruction
Smoothing
Numericalresults
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.2/31
Overview
Overview
Introduction(Navier-Stokes/Oseenequations)
(Algebraic)multigridmethodsforthesolutionofthe
Oseenequations
DecoupledApproach
CoupledApproach
Coarse-levelconstruction
Smoothing
Numericalresults
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.2/31
Overview
Overview
Introduction(Navier-Stokes/Oseenequations)
(Algebraic)multigridmethodsforthesolutionofthe
Oseenequations
DecoupledApproach
CoupledApproach
Coarse-levelconstruction
Smoothing
Numericalresults
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.2/31
Overview
Overview
Introduction(Navier-Stokes/Oseenequations)
(Algebraic)multigridmethodsforthesolutionofthe
Oseenequations
DecoupledApproach
CoupledApproach
Coarse-levelconstruction
Smoothing
Numericalresults
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.3/31
Aim
Aim
Complex3Ddomain
,
theincompressibleNavier-Stokesequations
+boundary/initialconditions,
FiniteElementDiscretization
;
0 @
A
(
u
)
B
T
B
C
1 A
0 @
u
p
1 A
=
0 @
f
g
1 A
Wewantanef?cientSolver.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.3/31
Aim
Aim
Complex3Ddomain
,
theincompressibleNavier-Stokesequations
@
t
uu+(ur)u+rp=f;
divu=0;
+boundary/initialconditions,
FiniteElementDiscretization
;
0 @
A
(
u
)
B
T
B
C
1 A
0 @
u
p
1 A
=
0 @
f
g
1 A
Wewantanef?cientSolver.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.3/31
Aim
Aim
Complex3Ddomain
,
theincompressibleNavier-Stokesequations
@
t
u
u+(
u
r
)u+
r
p=f;
div
u=0;
+boundary/initialconditions,
FiniteElementDiscretization
;
0 @
A
(
u
)
B
T
B
C
1 A
0 @
u
p
1 A
=
0 @
f
g
1 A
Wewantanef?cientSolver.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.3/31
Aim
Aim
Complex3Ddomain
,
theincompressibleNavier-Stokesequations
@
t
u
u+(
u
r
)u+
r
p=f;
div
u=0;
+boundary/initialconditions,
FiniteElementDiscretization
;
0 @
A
(
u
)
B
T
B
C
1 A
0 @
u
p
1 A
=
0 @
f
g
1 A
Wewantanef?cientSolver.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.4/31
ConvectiveTerm
ConvectiveTerm
Linearization:Fixedpointiteration
;
Oseen
equation
u+(wr)u+rp=f
Stabilization(SUPG):
replace
v
by
v+
h
(wr)v
;
((wr)u;v)
+
h
(wru;wrv)
;
+
(amongstothers)
with
h
=O(h)
.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.4/31
ConvectiveTerm
ConvectiveTerm
Linearization:Fixedpointiteration
;
Oseen
equation
u+(wr)u+rp=f
Stabilization(SUPG):
replace
v
by
v+
h
(wr)v
;
((wr)u;v)
+
h
(wru;wrv)
;
+
(amongstothers)
with
h
=O(h)
.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.5/31
UsedMixedFiniteElements
UsedMixedFiniteElements
P
1
-
P
1
-stab
:
C=
P
elts
K
h
2K
(rp;rq)
0;K
modi?edTaylor-Hood
:
P
nc
1
-
P
0
:
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.5/31
UsedMixedFiniteElements
UsedMixedFiniteElements
P
1
-
P
1
-stab
:
C=
P
elts
K
h
2K
(rp;rq)
0;K
modi?edTaylor-Hood
:
P
nc
1
-
P
0
:
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.5/31
UsedMixedFiniteElements
UsedMixedFiniteElements
P
1
-
P
1
-stab
:
C=
P
elts
K
h
2K
(rp;rq)
0;K
modi?edTaylor-Hood
:
P
nc
1
-
P
0
:
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.6/31
Howshouldwesolvethelinearsaddlepointproblem
0 @
A(w)B
T
BC
1 A
0 @
u
p
1 A
=
0 @
f
g
1 A
(indenite,ingeneralunsymmetric)?
Efciently!
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.6/31
Howshouldwesolvethelinearsaddlepointproblem
0 @
A(w)B
T
BC
1 A
0 @
u
p
1 A
=
0 @
f
g
1 A
(indenite,ingeneralunsymmetric)?
Efciently!
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.7/31
MGMforSaddle-Point-Problems
MGMforSaddle-Point-Problems
IterativeDecoupling
Pressurecorrection
methods:SIMPLE,
Uzawa
Precond.Krylovspace
methods(e.g.GMRES,
BiCGstab)
MGforthe
elliptic
systems.
CoupledMG
forthewhole
system.NeedsappropriateMG
components(smoother,
inter-grid-transfer).
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.7/31
MGMforSaddle-Point-Problems
MGMforSaddle-Point-Problems
IterativeDecoupling
Pressurecorrection
methods:SIMPLE,
Uzawa
Precond.Krylovspace
methods(e.g.GMRES,
BiCGstab)
MGforthe
elliptic
systems.
CoupledMG
forthewhole
system.NeedsappropriateMG
components(smoother,
inter-grid-transfer).
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.7/31
MGMforSaddle-Point-Problems
MGMforSaddle-Point-Problems
IterativeDecoupling
Pressurecorrection
methods:SIMPLE,
Uzawa
Precond.Krylovspace
methods(e.g.GMRES,
BiCGstab)
MGforthe
elliptic
systems.
CoupledMG
forthewhole
system.NeedsappropriateMG
components(smoother,
inter-grid-transfer).
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.8/31
DecoupledApproachSIMPLE
DecoupledApproachSIMPLE
^
A
~
u=fB
T
p
k
;
^
S~p=B
~
uCp
k
g;
u
k+1
=
~
uD
1
B
T
~p;
p
k+1
=p
k
+~p;
with
D
diagonalof
A
,
^
AA
,
^
SC+BD
1
B
T
,
damping
.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.9/31
DecoupledApproachBB
DecoupledApproachBB
?Black-Box?-preconditioner
h
Elman,Kay,Loghin,
Silvester,Wathen
i
Findagood
preconditioner
(notnecessarilysolver),
combineitwith
Krylovspacemethod
(e.g.BiCGstab)
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.9/31
DecoupledApproachBB
DecoupledApproachBB
?Black-Box?-preconditioner
h
Elman,Kay,Loghin,
Silvester,Wathen
i
Findagood
preconditioner
(notnecessarilysolver),
combineitwith
Krylovspacemethod
(e.g.BiCGstab)
^
K
1
=
0 @
^
A
1
0
0I
1 A
0 @
IB
T
0I
1 A
0 @
I0
0
^
S
1
1 A
;
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.9/31
DecoupledApproachBB
DecoupledApproachBB
?Black-Box?-preconditioner
h
Elman,Kay,Loghin,
Silvester,Wathen
i
Findagood
preconditioner
(notnecessarilysolver),
combineitwith
Krylovspacemethod
(e.g.BiCGstab)
^
K
1
=
0 @
^
A
1
0
0I
1 A
0 @
IB
T
0I
1 A
0 @
I0
0
^
S
1
1 A
;
r[+(wr)]
1
r
1
[+(wr)]
s
1
s
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.9/31
DecoupledApproachBB
DecoupledApproachBB
?Black-Box?-preconditioner
h
Elman,Kay,Loghin,
Silvester,Wathen
i
Findagood
preconditioner
(notnecessarilysolver),
combineitwith
Krylovspacemethod
(e.g.BiCGstab)
^
K
1
=
0 @
^
A
1
0
0I
1 A
0 @
IB
T
0I
1 A
0 @
I0
0
^
Q
1
A
p
^
L
1
1 A
;
r[+(wr)]
1
r
1
[+(wr)]
s
1
s
^
L
pressureLaplacian
;
^
AA;
A
p
pressureconv.-diff.operator
;
^
Q
pressuremassmatrix(lumping)
;
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.10/31
AlgebraicMultigrid
AlgebraicMultigrid
Complexgeometry
discretization
!
manyunknowns.
Nofurtherrenementpossible.
!
AlgebraicMultigrid.Startingfromthenestlevels
generatethecoarselevels(almost)onlyfrommatrix
information.
O.K.forellipticproblems,applicationtothe
decoupledapproachisstraightforward.
WhataboutcoupledAMG?
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.10/31
AlgebraicMultigrid
AlgebraicMultigrid
Complexgeometry
discretization
!
manyunknowns.
Nofurtherrenementpossible.
!
AlgebraicMultigrid.Startingfromthenestlevels
generatethecoarselevels(almost)onlyfrommatrix
information.
O.K.forellipticproblems,applicationtothe
decoupledapproachisstraightforward.
WhataboutcoupledAMG?
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.10/31
AlgebraicMultigrid
AlgebraicMultigrid
Complexgeometry
discretization
!
manyunknowns.
Nofurtherrenementpossible.
!
AlgebraicMultigrid.Startingfromthenestlevels
generatethecoarselevels(almost)onlyfrommatrix
information.
O.K.forellipticproblems,applicationtothe
decoupledapproachisstraightforward.
WhataboutcoupledAMG?
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.10/31
AlgebraicMultigrid
AlgebraicMultigrid
Complexgeometry
discretization
!
manyunknowns.
Nofurtherrenementpossible.
!
AlgebraicMultigrid.Startingfromthenestlevels
generatethecoarselevels(almost)onlyfrommatrix
information.
O.K.forellipticproblems,applicationtothe
decoupledapproachisstraightforward.
WhataboutcoupledAMG?
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.11/31
CoupledApproachStrategy
CoupledApproachStrategy
Separate
pressureandvelocity-component
unknowns.
Use
interaction
of
u
and
p
indicatedbynite
element.
Trytotranslateasmuchaspossiblefromgeometric
MGforsaddlepointproblemsandfromellipticAMG.
BlackboxcoupledAMG?
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.11/31
CoupledApproachStrategy
CoupledApproachStrategy
Separate
pressureandvelocity-component
unknowns.
Use
interaction
of
u
and
p
indicatedbynite
element.
Trytotranslateasmuchaspossiblefromgeometric
MGforsaddlepointproblemsandfromellipticAMG.
BlackboxcoupledAMG?
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.11/31
CoupledApproachStrategy
CoupledApproachStrategy
Separate
pressureandvelocity-component
unknowns.
Use
interaction
of
u
and
p
indicatedbynite
element.
Trytotranslateasmuchaspossiblefromgeometric
MGforsaddlepointproblemsandfromellipticAMG.
BlackboxcoupledAMG?
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.11/31
CoupledApproachStrategy
CoupledApproachStrategy
Separate
pressureandvelocity-component
unknowns.
Use
interaction
of
u
and
p
indicatedbynite
element.
Trytotranslateasmuchaspossiblefromgeometric
MGforsaddlepointproblemsandfromellipticAMG.
BlackboxcoupledAMG?
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.11/31
CoupledApproachStrategy
CoupledApproachStrategy
Separate
pressureandvelocity-component
unknowns.
Use
interaction
of
u
and
p
indicatedbynite
element.
Trytotranslateasmuchaspossiblefromgeometric
MGforsaddlepointproblemsandfromellipticAMG.
BlackboxcoupledAMG?
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.11/31
CoupledApproachStrategy
CoupledApproachStrategy
Separate
pressureandvelocity-component
unknowns.
Use
interaction
of
u
and
p
indicatedbynite
element.
Trytotranslateasmuchaspossiblefromgeometric
MGforsaddlepointproblemsandfromellipticAMG.
BlackboxcoupledAMG?
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.12/31
Notation
Notation
Gridtransfer:
Prolongators
P
l
l+1
=
~
I
l
l+1
J
l
l+1
,
~
I
l
l+1
=diag(I
l
l+1
;I
l
l+1
;I
l
l+1
)
,
~
I
l
l+1
:U
l+1
!U
l
,
U
l
:=(R
n
l
)
d
J
l
l+1
:Q
l+1
!Q
l
,
Q
l
:=R
m
l
Restrictors
P
l+1
l
(
=P
l
l+1
T
).
Coarselevelsystems:
K
l
=
A
l
B
T
l
B
l
C
l
=P
l
l1
K
l1
P
l1
l
Withtwo
exceptions
(
h
-dependenceinconv.stab.
andinelementstab.)!
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.12/31
Notation
Notation
Gridtransfer:
Prolongators
P
l
l+1
=
~
I
l
l+1
J
l
l+1
,
~
I
l
l+1
=diag(I
l
l+1
;I
l
l+1
;I
l
l+1
)
,
~
I
l
l+1
:U
l+1
!U
l
,
U
l
:=(R
n
l
)
d
J
l
l+1
:Q
l+1
!Q
l
,
Q
l
:=R
m
l
Restrictors
P
l+1
l
(
=P
l
l+1
T
).
Coarselevelsystems:
K
l
=
A
l
B
T
l
B
l
C
l
=P
l
l1
K
l1
P
l1
l
Withtwo
exceptions
(
h
-dependenceinconv.stab.
andinelementstab.)!
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.13/31
Mod.Taylor-HoodCoarsening
Mod.Taylor-HoodCoarsening
PSfragreplacements
u
u
u
u
p
p
p
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.13/31
Mod.Taylor-HoodCoarsening
Mod.Taylor-HoodCoarsening
PSfragreplacements
u
u
u
u
p
p
p
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
PSfragreplacements
u
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.13/31
Mod.Taylor-HoodCoarsening
Mod.Taylor-HoodCoarsening
PSfragreplacements
u
u
u
u
p
p
p
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
PSfragreplacements
u
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.13/31
Mod.Taylor-HoodCoarsening
Mod.Taylor-HoodCoarsening
PSfragreplacements
u
u
u
u
p
p
p
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
PSfragreplacements
u
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.13/31
Mod.Taylor-HoodCoarsening
Mod.Taylor-HoodCoarsening
PSfragreplacements
u
u
u
u
p
p
p
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
PSfragreplacements
u
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.13/31
Mod.Taylor-HoodCoarsening
Mod.Taylor-HoodCoarsening
PSfragreplacements
u
u
u
u
p
p
p
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
PSfragreplacements
u
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.13/31
Mod.Taylor-HoodCoarsening
Mod.Taylor-HoodCoarsening
PSfragreplacements
u
u
u
u
p
p
p
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
PSfragreplacements
u
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.13/31
Mod.Taylor-HoodCoarsening
Mod.Taylor-HoodCoarsening
PSfragreplacements
u
u
u
u
p
p
p
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
PSfragreplacements
u
p
Coarsening
Renement
ne(
u
)mesh
coarse(
p
)mesh
1stlevel
2ndlevel
.
.
.
L-thlevel
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.14/31
Convergence
Convergence
PSfragreplacements
ApproximationSmoothing
PropertyProperty
Stability
StandardAMG
StandardGMG
Techniques
?
e.g.LBB-Condition
inf
q2Q
l
sup
v2U
l
vB
T
l
p
kvk
U
l
kqk
Q
l
unstablebehaviour:
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.14/31
Convergence
Convergence
PSfragreplacements
ApproximationSmoothing
PropertyProperty
Stability
StandardAMG
StandardGMG
Techniques
?
e.g.LBB-Condition
inf
q2Q
l
sup
v2U
l
vB
T
l
p
kvk
U
l
kqk
Q
l
unstablebehaviour:
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.14/31
Convergence
Convergence
PSfragreplacements
ApproximationSmoothing
PropertyProperty
Stability
StandardAMG
StandardGMG
Techniques
?
e.g.LBB-Condition
inf
q2Q
l
sup
v2U
l
vB
T
l
p
kvk
U
l
kqk
Q
l
unstablebehaviour:
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.15/31
Mod.Taylor-HoodStability
Mod.Taylor-HoodStability
noanalyticresults
onlyheuristics(enoughvelocityunknowns)
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.15/31
Mod.Taylor-HoodStability
Mod.Taylor-HoodStability
noanalyticresults
onlyheuristics(enoughvelocityunknowns)
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.16/31
P
1
-
P
1
-stabCoarsening
P
1
-
P
1
-stabCoarsening
velocityandpressure`live'onthesamenodes
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.16/31
P
1
-
P
1
-stabCoarsening
P
1
-
P
1
-stabCoarsening
velocityandpressure`live'onthesamenodes
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.16/31
P
1
-
P
1
-stabCoarsening
P
1
-
P
1
-stabCoarsening
velocityandpressure`live'onthesamenodes
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.17/31
P
1
-
P
1
-stabStability(1)
P
1
-
P
1
-stabStability(1)
Geometriccase:Franca,Stenberg(1991)usingideaof
Verf
¨
urth(1984):
sup
06=v2V
h
(divv;p)
kvk
1
kpk
0
X
elts
K
h
2K
krpk
20;K
!
1
2
8p2Q
h
!
Stability
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.18/31
P
1
-
P
1
-stabStability(2)
P
1
-
P
1
-stabStability(2)
AMGcase:Assume
A
D
l
isofessentiallypositivetypei.e.forall
v
l
2U
l
X
k;j
(a
kj
)(v
k
v
j
)
2
X
k;j
(a
kj
)(v
k
v
j
)
2
:
Forall
v
l
2U
l
wecannda
l+1
l
v
l
2U
l+1
suchthat
kv
l
~
I
l
l+1
l+1
l
v
l
k
2D
l
kv
l
k
2A
D
l
:
Thenforalllevels
l
andall
(u;p)2U
l
Q
l
sup
06=v2U
l
06=q2Q
l
B
l
(u;p;v;q)
kvk
A
D
l
+kqk
M
l
l
kuk
A
D
l
+kpk
M
l
;
with
B
l
(u;p;v;q)=u
T
A
D
l
v+p
T
B
l
v+u
T
B
T
l
qp
T
C
l
q
.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.18/31
P
1
-
P
1
-stabStability(2)
P
1
-
P
1
-stabStability(2)
AMGcase:Assume
A
D
l
isofessentiallypositivetypei.e.forall
v
l
2U
l
X
k;j
(a
kj
)(v
k
v
j
)
2
X
k;j
(a
kj
)(v
k
v
j
)
2
:
Forall
v
l
2U
l
wecannda
l+1
l
v
l
2U
l+1
suchthat
kv
l
~
I
l
l+1
l+1
l
v
l
k
2D
l
kv
l
k
2A
D
l
:
Thenforalllevels
l
andall
(u;p)2U
l
Q
l
sup
06=v2U
l
06=q2Q
l
B
l
(u;p;v;q)
kvk
A
D
l
+kqk
M
l
l
kuk
A
D
l
+kpk
M
l
;
with
B
l
(u;p;v;q)=u
T
A
D
l
v+p
T
B
l
v+u
T
B
T
l
qp
T
C
l
q
.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.19/31
P
nc
1
-
P
0
Coarsening(1)
P
nc
1
-
P
0
Coarsening(1)
Use
AMGe
h
Brezina,Cleary,Falgout,Henson,
Jones,Manteufel,McCormick,Ruge
i
with
element
agglomeration
Jones,
Vassilevski
,
adaptedtooursituation.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.19/31
P
nc
1
-
P
0
Coarsening(1)
P
nc
1
-
P
0
Coarsening(1)
Use
AMGe
h
Brezina,Cleary,Falgout,Henson,
Jones,Manteufel,McCormick,Ruge
i
with
element
agglomeration
Jones,
Vassilevski
,
adaptedtooursituation.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.19/31
P
nc
1
-
P
0
Coarsening(1)
P
nc
1
-
P
0
Coarsening(1)
Use
AMGe
h
Brezina,Cleary,Falgout,Henson,
Jones,Manteufel,McCormick,Ruge
i
with
element
agglomeration
Jones,
Vassilevski
,
adaptedtooursituation.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.20/31
P
nc
1
-
P
0
Coarsening(2)
P
nc
1
-
P
0
Coarsening(2)
agglomeration
identifyunknowns
buildprolongation
pressure
:identityprolongation
velocity
:identityonedges,energyminimizationin
theinterior
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.20/31
P
nc
1
-
P
0
Coarsening(2)
P
nc
1
-
P
0
Coarsening(2)
agglomeration
identifyunknowns
buildprolongation
pressure
:identityprolongation
velocity
:identityonedges,energyminimizationin
theinterior
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.20/31
P
nc
1
-
P
0
Coarsening(2)
P
nc
1
-
P
0
Coarsening(2)
agglomeration
identifyunknowns
buildprolongation
pressure
:identityprolongation
velocity
:identityonedges,energyminimizationin
theinterior
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.20/31
P
nc
1
-
P
0
Coarsening(2)
P
nc
1
-
P
0
Coarsening(2)
agglomeration
identifyunknowns
buildprolongation
pressure
:identityprolongation
velocity
:identityonedges,energyminimizationin
theinterior
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.20/31
P
nc
1
-
P
0
Coarsening(2)
P
nc
1
-
P
0
Coarsening(2)
agglomeration
identifyunknowns
buildprolongation
pressure
:identityprolongation
velocity
:identityonedges,energyminimizationin
theinterior
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.21/31
P
nc
1
-
P
0
Stability
P
nc
1
-
P
0
Stability
Lemma:Assumeexistenceoflin.Op.
ll1
:U
l1
!U
l
with
b(
ll1
v
l1
;q
l
)=b(v
l1
;J
l1
l
q
l
)
8q
l
2Q
l
;v
l1
2U
l1
(1)
and
k
ll1
v
l1
k
A
Ckv
l1
k
A
8v
l1
2U
l1
:
(2)
Theninf-supin
U
l1
Q
l1
impliesinf-supin
U
l
Q
l
.
(1),(2)canbeshownusingpurely
geometric
arguments.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.21/31
P
nc
1
-
P
0
Stability
P
nc
1
-
P
0
Stability
Lemma:Assumeexistenceoflin.Op.
ll1
:U
l1
!U
l
with
b(
ll1
v
l1
;q
l
)=b(v
l1
;J
l1
l
q
l
)
8q
l
2Q
l
;v
l1
2U
l1
(1)
and
k
ll1
v
l1
k
A
Ckv
l1
k
A
8v
l1
2U
l1
:
(2)
Theninf-supin
U
l1
Q
l1
impliesinf-supin
U
l
Q
l
.
(1),(2)canbeshownusingpurely
geometric
arguments.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.22/31
Smoothing(1)
Smoothing(1)
UsesmoothersfromGMGforsaddlepointsystems,e.g.
Vanka-Smoother
:
Solvesmallerlocalproblems.(1pressureunknown+
connectedvelocities)
Combinesolutionsvia(multiplicative)Schwarz
method.
Zulehner/Sch
¨
oberl:theoreticalbasisforGMG
(additivevariant).
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.23/31
Smoothing(2)
Smoothing(2)
Braess-Sarazin-Smoother
:(symm.inexactUzawa)
^
A(
^
u
j+1
u
j
)=fAu
j
B
T
p
j
;
^
S(p
j+1
p
j
)=B
^
u
j+1
Cp
j
g;
^
A(u
j+1
^
u
j+1
)=B
T
(p
j+1
p
j
);
where
^
AA
(
Gauss-Seidel)
^
SC+B
^
A
1
B
T
(
AMGfor
C+B
~
A
1
B
T
~
A
1
:::!
-Jacobi).
Problem:Mustkeepmatricesforinner(
^
S
)AMG
!
needsmorememory.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.23/31
Smoothing(2)
Smoothing(2)
Braess-Sarazin-Smoother
:(symm.inexactUzawa)
^
A(
^
u
j+1
u
j
)=fAu
j
B
T
p
j
;
^
S(p
j+1
p
j
)=B
^
u
j+1
Cp
j
g;
^
A(u
j+1
^
u
j+1
)=B
T
(p
j+1
p
j
);
where
^
AA
(
Gauss-Seidel)
^
SC+B
^
A
1
B
T
(
AMGfor
C+B
~
A
1
B
T
~
A
1
:::!
-Jacobi).
Problem:Mustkeepmatricesforinner(
^
S
)AMG
!
needsmorememory.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.23/31
Smoothing(2)
Smoothing(2)
Braess-Sarazin-Smoother
:(symm.inexactUzawa)
^
A(
^
u
j+1
u
j
)=fAu
j
B
T
p
j
;
^
S(p
j+1
p
j
)=B
^
u
j+1
Cp
j
g;
^
A(u
j+1
^
u
j+1
)=B
T
(p
j+1
p
j
);
where
^
AA
(
Gauss-Seidel)
^
SC+B
^
A
1
B
T
(
AMGfor
C+B
~
A
1
B
T
~
A
1
:::!
-Jacobi).
Problem:Mustkeepmatricesforinner(
^
S
)AMG
!
needsmorememory.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.24/31
NumericalResults
P
1
iso
P
2
-
P
1
NumericalResults
P
1
iso
P
2
-
P
1
1e-6
1e-5
1e-4
1e-3
1e-2
1e-1
1
0
100
200
300
400
residual
CPU-time [s]
BiCGStab,BBPre
AMG-SIMPLE
Coupled,MSM
2Dvalve,
10
5
vars,
1e-11
1e-10
1e-9
1e-8
1e-7
1e-6
1e-5
1e-4
0
1000
2000
3000
4000
5000
6000
7000
residual
CPU-time[s]
AMG-SIMPLE
Coupled,Braess,1-shift
Coupled,Braess,2-shift
3Dvalves,
710
5
vars,
(geometriesprovidedbyAVLListGmbH)
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.25/31
NumericalResults
P
1
-
P
1
-stab
NumericalResults
P
1
-
P
1
-stab
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0
4000
8000
12000
16000
residual
CPU-time [s]
Coupled
AMG-SIMPLE
710
5
vars,
mesh:tets,hexas,pyra-
mids,prisms;
=0:0005
(geometriesprovidedbyAVLListGmbH)
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.26/31
NumericalResults
h
-Dependency
NumericalResults
h
-Dependency
Stokes,drivencavity(unstructuredgrid)
1e-07
1e-06
1e-05
0.0001
0.001
0.01
5
6
7
8
9
refinement level
BiCGStab,BBPre
Coupled,Braess
Coupled,MSM
AMG-SIMPLE
PSfragreplacements
T
0:1
level
unknowns
5
10,571
6
52,446
7
166,691
8
665,155
9
2,657,411
T
0:1
:=
CPUtime[min.]forthered.
ofthenormofres.by0.1
numberofunknowns
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.27/31
NumericalResults
-Dependency
NumericalResults
-Dependency
2Dvalve,Taylor-Hood,103,351unknowns
1e-06
1e-05
0.0001
1
10
100
1000
10000
BiCGStab,BBPre
Coupled,Braess
Coupled,MSM
AMG-SIMPLE
PSfragreplacements
T
0:1
1=
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.28/31
NumericalResults
P
nc
1
-
P
0
NumericalResults
P
nc
1
-
P
0
Standardaggl.
ne
!
coarse
agglomeratecoarser
elements
O(n)
Metis-renement
usemetisto
ndaninitialpartition
(coarsestlevel),
`rene'recoursively.
O(nlogn)
!
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.28/31
NumericalResults
P
nc
1
-
P
0
NumericalResults
P
nc
1
-
P
0
Standardaggl.
Metis-renement
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.28/31
NumericalResults
P
nc
1
-
P
0
NumericalResults
P
nc
1
-
P
0
Standardaggl.
Metis-renement
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.28/31
NumericalResults
P
nc
1
-
P
0
NumericalResults
P
nc
1
-
P
0
Standardaggl.
Metis-renement
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.28/31
NumericalResults
P
nc
1
-
P
0
NumericalResults
P
nc
1
-
P
0
Standardaggl.
Metis-renement
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.28/31
NumericalResults
P
nc
1
-
P
0
NumericalResults
P
nc
1
-
P
0
Standardaggl.
Metis-renement
1e-6
2e-6
4e-6
8e-6
5
6
7
8
refinement level
Standard
Metis
PSfragreplacements
T
0:1
(level8:1,180,930unknowns)
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.29/31
Summary(1)(nearlythelastslide)
Summary(1)(nearlythelastslide)
WehavedevelopedcoupledAMGmethodsfor
saddlepointproblemsoriginatingintheNavier
Stokesequations,for
themod.Taylor-Hoodelement
the
P
1
-
P
1
stabelement(stable!)
the
P
nc
1
-
P
0
element(AMGe,stable!)
Wehavecomparedthemtomethodsusinga
decoupledapproach.
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.30/31
Summary(2)(lastslidebutone)
Summary(2)(lastslidebutone)
BUT
WehaveabsolutelynoBlack-Boxmethod!
Wehavenorigorousproveformostaspects(e.g.
convergence)!
InstituteofComputationalMathematics
JohannesKeplerUniversityLinz
q
Introduction
q
DecoupledMG
q
CoupledAMG
q
Num.Results
AdvancesinNumericalAlgorithms,Sept.1013,2003
p.31/31
TheLastSlide
TheLastSlide
Thankyouforyourattention!
Автор
Redmegaman
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equation, Газовая динамика, математические методы, for, гидродинамика, wabro, 2003, amg, coupled, stocks, navie
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