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