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Solution Algorithms for the Solution Algorithms for the Incompressible Incompressible NavierNavier--Stokes Stokes
EquationsEquations
Howard ElmanUniversity of Maryland
1
The Incompressible Navier-Stokes Equations
0 div grad)grad( 2
t
=−=+⋅+∇−
ufpuuuu να
α=0 → steady state problemα=1 → evolutionary problem / emphasized here
N
D
snpnu
wu
Ω∂=−∂∂
Ω∂=
on
on
D
N
Ω∂Ω∂Ω
Inflow boundary: ∂Ω+ = x∈∂Ω | u· n > 0
Characteristic boundary:∂Ω0 = x∈∂Ω | u· n = 0
Outflow boundary:∂Ω- = x∈∂Ω | u· n < 0
3,2,1 ,in =⊂Ω ddR
2
The Incompressible Navier-Stokes Equations
For pure Dirichlet problem: ∂ΩD = ∂Ω , pressure solution is defined up to constant. In this case, require
0=⋅+⋅ ∫∫+− Ω∂Ω∂
nwnw
Volume of fluid flowing into Ω equals volume of fluid flowing out
Enclosed flow: everywhere on ∂Ω0=⋅nw
3
The Stokes Equations
0 div
grad- 2
=−=+∇
u
fpu dR⊂Ωin
0) div(
) div(:
)( ),( allfor s.t. ),( ),( Find 21
21
0
=
=−∇∇
Ω∈Ω∈Ω∈Ω∈
∫
∫ ∫∫
Ω
Ω ΩΩ
uq
fvvpvu
LqHvLpHu EE
Weak form:
yyxxyyxx vuvuvuvuvu )()()()()()()()(: Here 22221111 +++=∇∇
4
Approximation by Mixed Finite Elements
Choose bases φj for velocities ψj for pressures
Velocity basis functions have the form
φ
φ 0 and 0
Seek discrete approximationsuh = ∑j ujφj ≈ uph = ∑j pjψj ≈ p
0) div(
) div(:
),( allfor s.t. , , Find 0
=
=−∇∇
∈Ω∈∈∈
∫
∫ ∫∫
Ω
Ω ΩΩ
hh
hhhhh
hhh
hh
hhEh
uq
fvvpvu
MqXvMpXu
5
Result: Matrix Equation
=
gf
pu
BBA T
0
A=[aij], aij=∫Ω ∇φj∇φi, discrete vector Laplacian
B=[Bij], bij=∫Ω ψj divφj , discrete (-)divergence
Coefficient matrix is symmetric indefinite
0 BBA T
One other important matrix: pressure mass matrix Q=[qij], qij=∫Ω ψjψi
Important property: uniformly well-conditioned, κ(Q)=O(1)
6
Mass Matrices
ψj ψi
h
Q=[qij], qij=∫Ω ψjψI =2h/3 j=ih/6 j=i+1, i-1
In one dimension: Q~ hI
For problems in d dimensions, Q~hdI
We will see a velocity mass matrix as well
1D Example
7
The Inf-sup Condition
0|||||||||) div,(|
supinf 0constant >≥∇≠≠ γ
qvvq
vq
Equivalently, given any q, there exists v such that
For continuous spaces:
Velocity space is “rich enough” for this bound to hold
0||||||||) div,( >≥
∇γ
qvvq
8
The Discrete Inf-sup Condition
0|||||||||) div,(|
mmin 0constant >≥∇≠≠ γ
hh
hhvq qv
vqax
hh
These conditions are not guaranteed. If they hold, then
Analogous condition for discrete spaces:
The Schur complement is spectrally equivalent to the pressuremass matrix (and hence is uniformly well-conditioned)
21
2
),(),( Γ≤≤
−
qQqqqBBA T
γ
Matrix version:
0),(),(
|)B,(|mmin
2/12/10vconstantq >≥≠≠ γQqqAvv
vqax
9
Iterative Solution of Discrete Stokes Equations
=
00 f
pu
BBA T
||||)/()(1
)/()(12|||| 0
2/
radbc
adbcr
k
k
+−
≤
]][0-b d-a c
[λ ⊂
Coefficient matrix is symmetric indefinite• MINRES algorithm can be used (short recurrences )• Eigenvalues are contained in two real intervals
• Convergence bound forMINRES :
• Seek preconditioner to make these intervals small
10
Iterative Solution of Discrete Stokes Equations
=
pu
MA
pu
BBA
S
T
00
0 λ
SMA0
0Proposed form for preconditioner:
Effectiveness depends on eigenvalues of
pMBu
pBAu
S
T
λλ
==− )1(
1)-( ,1 λλµµ ==− pMpBBA ST
For MS=Q, pressure mass matrix, inf-sup bound µ (whence λ) is independent of discretization parameter h
λ = (½) (1±(1+4µ)1/2)
11
Consequence
• Convergence rate is independent of mesh
• Implementation: requires Poisson solve A-1vmass matrix solve Q-1q
• Practical choice: use fast methods (multigrid) for first taskdiagonal approximation for second task
• Result: Optimal solver for steady Stokes
QA
00
=
gf
pu
BBA T
0 For MINRES applied to
with preconditioner
12
Stabilization
hhh
hh vqvvq
allfor 0|||||||||) div,(|=
∇
Discrete inf-sup condition (typically) fails to hold if for some qh
Equivalently, for some vector q
vQqqAvv
vq vectorsallfor 0
),(),(
|)B,(|2/12/1
=
⇒ q∈ null(BT), problem is rank-deficient
This is true for “natural” elements, such as equal order velocitiespressures on common grids!
13
StabilizationExample:
Q1 velocities, bilinear P0 pressures, piecewise constant ,Red-black pressure mode ∈ null(BT),
corrupts pressure solution
Fix: relax discrete incompressibility in a consistent mannerregularization procedure
Result:
=
− 0f
pu
CBBA T
C=stabilization operator
14
Solution Strategy
Remains the same: preconditioned MINRES with preconditioner
matrix mass pressure ~ ,00 QMM
AS
S
21
2
),(),)(( Γ≤+≤
−
qQqqqCBBA T
γ
Effectiveness derives from
15
Return to Navier-Stokes Equations
0) div(
) div()(:
)( ),( allfor s.t. ),( ),( Find 21
21
0
=
=−∇⋅+∇∇
Ω∈Ω∈Ω∈Ω∈
∫
∫ ∫∫
Ω
Ω ΩΩ
uq
fvvpvuuvu
LqHvLpHu EE
ν
0 div
grad)grad( 2
=−=+⋅+∇−
u
fpuuuν
Weak Formulation
16
,νULR ≡ Reynolds number, characterizes relative contributions
of convection and diffusion
L = characteristic length scale in boundarye.g. length of inflow boundary
= x/L in normalized domain
U = normalization for velocity ue.g. u=Uu*, where ||u*||=1
x
Conventions of Notation
17
Reference Problems
2. Driven cavity flow
u1=u2=0, except u1=1 at top
1. Flow over a backward facingstep
u1=u2=0, exceptu1=1-y2 at inflow ν ∂u1/∂x=p∂u2/∂x=0 at outflow
18
Linearization
++
uu
ppup
δδ update compute , (iterate) estimateGiven
Consider steady problem
0 div
grad)grad( 2
=−=+⋅+∇−
u
fpuuuν
uu
puuufpuuu
div div
) grad)grad(( grad)grad( 22
=−+⋅+∇−−=+⋅+∇−
δνδδδν
Linearization
1. Picard iteration
=
⋅+∇−gf
puu
ˆ
ˆ
0div-grad )grad( 2
δδν
Oseen equations
19
Linearization
uu
puuuf
puuuuu
div div
) grad)grad((
grad)grad)(()grad( 2
2
=−+⋅+∇−−=
+⋅+⋅+∇−
δν
δδδδν
2. Newton’s method : substitute into N.S. equations,
throw out quadratic term
∂∂∂∂−∂∂+⋅+∇−
∂∂+⋅+∇−
yx
yyuuxu
xyuxuu
//
/)()grad(2 )(
/)()()grad(2
22
11
ν
ν
++
uupp
δδ
=
gf
f
puu
ˆ2ˆ1ˆ
2
1
δδ
20
Convergence
Picard iteration: Sufficient conditions for convergence are the same as sufficient conditions for uniquenessof steady solution:(*) ν > (c2 ||f||-1)1/2/c
c1, c2 continuity, coercivity constants
Global convergenceRate: linear
Newton’s method: Convergent only for large enough ν,typically larger than (*) aboveGuaranteed convergent only for close enoughstarting values.Rate: quadratic
21
Iterative Solution of Linearized System
=
gf
pu
BBF T
0
Analogue of Stokes strategy: preconditioner:
SMF0
0
Same analysis → BF-1BTp=µ MSp, µ=λ(λ-1)⇒ seek approximation MS to Schur complement
Under mapping µ a λ=1± (1+4µ)1/2,eigenvalues λ are clustered in two regions,one on each side of imaginary axis
Suppose MS≈ BF-1BT so that eigenvalues of
BF-1BTp = µ MSp
are tightly clustered.
22
Variant Form of Preconditioner
=
gf
pu
BBF T
0
Observations: • Symmetry is important for Stokes solver
MINRES: optimal with fixed cost per step⇒ need block diagonal preconditioner
• Can also compute in the right norm
• For Navier-Stokes: don’t have symmetryneed Krylov subspace method for nonsymmetricmatrices (e.g. GMRES)
− S
T
MBF
0
Change block diagonal preconditioner
SMF0
0to block triangular
23
Preliminary Analysis of this Variant
−=
pu
MBF
pu
BBF
S
TT
00µGeneralized eigenvalue problem
Fu+BTp = µ (Fu+BTp) , µ ≠ 1 ⇒ u= -F-1BTp Bu = -µ QSp BF-1BTp=µ MSp
µ ⊂ 1 ∪
TheoremFor preconditioned GMRES iteration, let
p0 be arbitrary and u0 = F-1(f-BTp0) (⇒ r0=(0,w0)) (uk,pk)T be generated with block triangular preconditioner,(uk,pk)D be generated with block diagonal preconditioner.
Then (u2k,p2k)D = (u2k+1,p2k+1)D = (uk,pk)T .
24
Computational Requirements
Each step of a Krylov subspace method requires matrix-vectorproduct y=AM-1 x
For preconditioning operation:
Solve Ms r = -q, then solve F w = v-BTr
The only difference from block diagonal solve:matrix-vector product BTr (negligible)
For second step: convection-diffusion solve: can be approximated, e.g. with multigrid
For first step: something new needed: MS
−=
−
qv
MBF
rw
S
T 1
0 compute
25
Approximation for the Schur Complement (I)Suppose the gradient and convection-diffusion operators approximately commute (w=u(m)):
∇∇⋅+∇−≈∇⋅+∇−∇ up ww )()( 22 ννRequire pressure convection-diffusion operator
Discrete analogue:
ppT
uT
Tuupp
Tu
QFBBQBBF
BQFQFQBQ
111
1111
−−−
−−−−
=⇒
=
pA
In practice: don’t have equality, instead pppS QFAM 1−≡
Nomenclature: Pressure Convection-Diffusion Preconditioner
26
Approximation for the Schur Complement (I)
pppS QFAM 1−≡
Requirements: Poisson solve 1−pA
Mass matrix solve 1−pQ
+ Convection-diffusion solve 1−F
1for −SM
Pressure Convection-Diffusion Preconditioner
27
Approximation for the Schur Complement (II)
pww )grad(gradgrad)grad( 22 ⋅+∇−−⋅+∇− ννTry to make commutator
small. Viewed as a discrete operator, norm is
u
h
Qpu
Tu
Tuu
p
hpp
pFQBQBQFQ
pww
)])(())([(sup
])grad(gradgrad)grad[(sup
1111
22
−−−− −=
⋅+∇−−⋅+∇− νν
Make small by minimizing columns of Fp in least squares sense.
Result: )()( 1111 Tuu
Tupp BFQBQBBQQF −−−−=
)())(()( 11111111 Tu
Tuu
Tupp
Tu
T BBQBFQBQBBQQFBBQBBF −−−−−−−− =≈
28
Approximation for the Schur Complement (II)
11111111 ))(()()( −−−−−−−− ≈ Tu
Tuu
Tu
T BBMBFMBMBBMBBF
Nomenclature: Least-Squares Commutator Preconditioner
• Fully automated: does not require operator defined pressure space
• Takes properties of F into account
• Requires convection-diffusion solve and two Poisson solves
+∇⋅+∇−+∇⋅+∇−=
22
1
212
wwwwww
Fν
νFor Newton nonlinear iteration:
29
For Problems Arising from Stabilized Elements
+−
=
− −− )(0
011 CBBF
BF
IBF
I
CB
BFT
TT
⇒
=
+−
− −
−
− IBF
I
CBBF
BF
CB
BFT
TT
1
1
1
0
)(0
1
0
−
− S
TF
MBM
≈
Shows what is needed for stabilized discretization:
MS≈ BF-1BT + C
The ideas discussed above can be adapted to this case.
30
Recapitulating: Two ideas under consideration
Preconditioners
=
− 0
f
p
u
CB
BF T
1. Pressure convection-diffusion(Fp): MS = Ap Fp-1Qp
Requirements: Poisson solve, mass matrix solve, Convection-diffusion on pressure spaceDecisions on boundary conditions
2. Lst-sqrs commutator: QS = (BQ-1BT) (BQ-1FQ-1BT)-1 (Bq-1BT) Requirements: Two Poisson solves
− S
TF
MBM
0for
Requirement common to both: approximation of action of F-1
Overarching philosophy: subsidiary operations Poisson solve convection-diffusion solve are manageable
31
ElaboratingOverarching philosophy: subsidiary operations
Poisson solve convection-diffusion solve
To approximate the action of F-1, Ap-1:
apply a few steps of multigrid to
convection-diffusion equation
Poisson equation
are manageable
32
Reference Problems
2. Driven cavity flow
u1=u2=0, except u1=1 at top
1. Flow over a backward facingstep
u1=u2=0, exceptu1=1-y2 at inflow ν ∂u1/∂x=p∂u2/∂x=0 at outflow
33
Performance
StepNewton system
CavityOseen system
34
Performancewith Multigrid
StepNewton system
CavityOseen system
35
Technical Issues
1. For multigrid used to generate MF-1 (approximately solve
Fv=w), need streamline upwinding:
Approximate the action of F-1 with multigrid applied toto the system matrix F+S where S is the streamline-upwindoperator derived from (u. grad)
=
⋅+∇−gf
puu
ˆ
ˆ
0div-grad )grad( 2
δδν
F
36
Technical Issues
∂∂∂∂−∂∂+⋅+∇−
∂∂+⋅+∇−
yx
yyuuxu
xyuxuu
//
/)()grad(2 )(
/)()()grad(2
22
11
ν
ν
F
2. To generate MF-1 (approximately solve Fv=w) for Newton’smethod:
First approximate F with F from Oseen operator.
Then approximate the action of (F)-1 with multigridas above
+⋅+∇⋅+∇−
yx
y
uuu
uu
)()grad( )(
)()grad(
22
2
12
νν
^
^
37
Technical Issues
3. For enclosed flow, one.by deficient -rank are 0
and
BBFB
TT
Null vectors are p 1 and , respectively.
10
properly. worksGMRES c.
defined.- well
is 00
system onedpreconditi The b.
.0)1,( ,consistent is 0
system The a.
that establish toNeed
1−
−
=
=
S
TT
T
MBF
BBF
ggf
pu
BBF
38
Technical Issues
For issue 3c:
Theorem: If Fx=b is a consistent system where F is rank deficient,then the GMRES method will compute a solution withoutbreaking down iffnull(F) null(FT)=0.
Apply to
For issue 3b: need to show that subsidiary operations such as pressure Poisson solve Apq=s are consistent.
Issue that arises here: multigrid for Poisson equation with Neumannboundary conditions.
1
00
−
−
S
TT
MBF
BBF
39
Picard Iteration vs. Newton’s Method
SlowerMore inner iterations
Disadvantages
More robust, fewer inner iterations
Faster convergence
Advantages
PicardNewton
Our experience: (# Newton steps) / (# Picard steps) < ½(avg. # inner steps (Newton) /
avg. # inner steps (Picard) ≈ 2
40
Software
Experimental results Graphics
Incompressible Flow & Iterative Solver Software(IFISS) Package
for both talks this weekend from
41
IFISS
MATLAB Toolbox for discretizing and solving four problemsarising in models of incompressible flow Authors: HE, A. Ramage and D. Silvester
1. The diffusion (Poisson) equation -∆ u=f
2. The convection-diffusion equation
3. The Stokes equations
4. The Navier-Stokes equations
42
Features of IFISS
For each problem:
1. Several choices of finite element discretization(bilinear orbiquadratic)
2. Several domains (square, L-shaped, step)
3. Error estimates
4. Graphical displays
5. Access to data and problem structure
6. Iterative solution by preconditioned Krylov subspace methods
43
Preconditioned Krylov Subspace Methods
For the diffusion equation:
CGMINRES
+ ICCGMultigrid
For the convection-diffusion equation:GMRESBiCGSTAB(l)
+ ILUMultigrid
For the Stokes equations:
MINRES + Block preconditioningwith inner multigrid
For the Navier Stokes equations:
GMRESBiCGSTAB(l)
+Pressure convection-diffusionLeast squares commutatorwith inner multigrid
44
Last Example: Backward Facing Step,R=100
Least squares commutator preconditioner is much more effectivein this caseBiCGSTAB(2) performance is erratic