a hybrid newton-krylov method for stabilized fem...

A Hybrid Newton-Krylov method for stabilized FEMdiscretization of Navier-Stokes equations

Stefano BerroneDIPARTIMENTO DI MATEMATICA, POLITECNICO DI TORINO,CORSO DUCA DEGLI ABRUZZI 24, 10129, TORINO, ITALY

e-mail: [email protected]: http://calvino.polito.it/˜sberrone

Stefania BellaviaDIPARTIMENTO DI ENERGETICA, UNIVERSITA DI FIRENZE,

VIA LOMBROSO 6/17, 50134, FIRENZE, ITALY

e-mail: [email protected]

22nd IFIP TC7 Conference on System Modeling and OptimizationTurin, Italy, July 18-22, 2005

IFIP 2005 Torino, July 19, 2005

Outline

• Steady state Navier-Stokes Equations and SUPG-FEM discretization.• Inexact Newton-GMRES Methods.• Globally convergent methods: classical backtracking Newton-GMRES Method, Trust Re-

gion methods.• Global convergence enhancement: A combined Linesearch-subspace trust region

method.• Numerical results: Backward Facing Step, Lid Driven Cavity.

1


Steady-state, incompressible Navier-Stokes Equations

−1

Re4 u + (u · ∇) u + ∇p = f in Ω,

∇ · u = 0 in Ω,

u = 0 on ΓD,

1Re

∂ u

∂n− pn = gN on ΓN,

where:

• Re:the Reynolds number;• Ω ⊂ IR2: ∂Ω = ΓD ∪ ΓN , ΓD ∩ ΓN = ∅ and | ΓD | 6= 0;• n: unit outward normal vector to ∂Ω;• f ∈ [L2(Ω)]2;

• gN ∈ [H12(ΓN)]2.

2


Finite Element Discretization

• Ω: polygonal domain;• Thh a regular family of partitions of Ω into triangles T (usual conformity and minimal-

angle conditions [Ciarlet 1978]);

• Vh ⊂V = [H10,D(Ω)]2, Qh ⊂Q =

(L2

0(Ω) if | ΓN | = 0

L2(Ω) if | ΓN | > 0two conforming finite element spaces based on the partition Th;

• ΠT f , ΠE gN approximations of f , gN by projections.

With the standard Galerkin formulation, only some choices of the spaces Vh and Qh sat-isfies the discrete inf-sup (Babuska-Brezzi) condition needed for uniqueness of the pressure[Brezzi-Fortin 1991].

This may be avoided by resorting to the Streamline Upwind/Petrov Galerkin (SUP G)formulation [Franca-Frey-Hughes 1992, Franca-Frey 1992].

3


SUPG-formulationF ind [uh, ph]∈Vh × Qh such that, ∀[vh, qh]∈Vh × Qh :

(Momentum equations)

1Re

(∇ uh, ∇ vh) + ((uh · ∇) uh, vh) − (ph, ∇ · vh) +

+X

T ∈Th

τT

„−

1Re

4 uh +(uh · ∇) uh + ∇ph, (uh · ∇) vh

«T

+

+X

T ∈Th

δT (∇ · uh, ∇ · vh)T =

= (ΠT f, vh) + (ΠE gN , vh)ΓN+

XT ∈Th

τT (ΠT f, (uh · ∇) vh)T ,

(Continuity)

(qh, ∇ · uh) +

+X

T ∈Th

τT

„−

1Re

4 uh +(uh · ∇) uh + ∇ph, ∇qh

«T

=

=X

T ∈Th

τT (ΠT f, ∇qh)T .

For the stabilization parameters τT and δT we essentially follow [Franca-Frey 1992].

4


Newton Methods for Nonlinear Equations

The discretized steady-state Navier-Stokes equations give rise to a system of nonlinearequations:

F (x) = 0

where x = [uh, ph] is the vector of velocity-pressure values in the grid points.

A classical method for solving this problem is the Newton method:

• Given x0 ∈ IRn, a tolerance tol,• set k = 0,• While ‖F (xk)‖ > tol

– solve F ′(xk)sk = −F (xk),– set xk+1 = xk + sk,– k = k + 1

⇒ Convergence: for this kind of applications finding a good initial guess is almost impossi-ble; difficulties arise also in the context of adaptive methods.

⇒ Efficiency: enhanced by inexact methods.

5


Inexact Newton Methods

In an Inexact Newton Method, the Newton equation is relaxed to an Inexact Newtoncondition

‖F′(xk)sk + F (xk)‖ ≤ ηk‖F (xk)‖.

• ηk ∈ [0, 1) forcing term. The ultimate rate of convergence of the Inexact methodsdepends on the choice of the ηk’s.

• This naturally allows the use of iterative linear methods: conditions like this are preciselythe small linear residual termination conditions for iterative linear solver.

6


Inexact Newton-GMRES Methods

GMRES is employed to compute the Inexact Newton step sk:

• Given x0 ∈ IRn, a tolerance tol,• set k = 0,• While ‖F (xk)‖ > tol

– choose ηk ∈ [0, 1)– perform GMRES iterations on F ′(xk)sk = −F (xk)

until‖F ′(xk)sk + F (xk)‖

‖F (xk)‖≤ ηk,

– set xk+1 = xk + sk,– k = k + 1,

• rk = F ′(xk)sk + F (xk) residual;

•‖F ′(xk)sk + F (xk)‖

‖F (xk)‖≤ ηk classical relative residual sopping condition.

7


Globalization Strategies

(Inexact) Newton methods exhibit local convergence properties. In order to enhance theirrobustness, these methods have to be augmented with a suitable globalization strategy.

Newton-type methods combined with globalization strategies produce a sequence xksuch that at each iteration a sufficient decrease of the merit function

f(x) =12‖F (x)‖2

is ensured, i.e.f(xk+1) < αkf(xk), αk ∈ (0, 1)

aiming to create robust and locally fast algorithms.

8


Linesearch

• Given a search direction sk, xk+1 is given by

xk+1 = xk + λsk

where the reduction factor λ ∈ (0, 1] is such that a sufficient decrease condition

f(xk + λsk) < (1 − tλ)f(xk)

is reached. A value λ satisfying the condition exists if sk is a descent direction for f(x) inxk, i.e. if

∇f(xk)Tsk < 0.

The backtrack strategy is: set λ = 1 then reduce λ of a given factor θ ∈ (0, 1) until therequired sufficient decrease condition on f is reached. The parameter t is chosen in (0, 1),usually t = 1.0E − 4.

9


Inexact Newton method with Backtracking (INB)

Let xk, ηmax ∈ (0, 1), t ∈ (0, 1), 0 < θm < θmax < 1 be given.

1. Choose ηk ∈ [0, ηmax].2. Compute sk such that

‖F (xk) + F′(xk)sk‖ ≤ ηk‖F (xk)‖.

3. Perform the INB backtracking strategy:3.1 Set sk = sk, ηk = ηk.3.2. While ‖F (xk + sk)‖ > (1 − t(1 − ηk))‖F (xk)‖ do:

Choose θ ∈ [θm, θmax].Update sk = θsk and ηk = 1 − θ(1 − ηk).

4. Set xk+1 = xk + sk.

[Eisenstat-Walker 1994].

10


Failures of Newton-GMRES with linesearch

• The Newton and Inexact Newton directions can be nearly orthogonal to the gradient of f

when F ′ is ill conditioned. Moreover this may happen in the Inexact Newton methods if thelinear systems are solved at a too low accuracy:

cos θN = −

sTk ∇f(xk)

‖sk‖‖∇f(xk)‖>

1 − ηk

(1 + ηk)k2(F ′(xk)).

• When the direction is “poor”, in order to obtain a reduction in f a very high number ofbacktracks is required ⇒ this in practice implies a failure of the backtracking strategy.

• This kind of failures has been observed in practical applications, in particular in the solutionof nonlinear systems arising from Navier Stokes equations:

– [Tuminaro-Walker-Shadid 2002]– [Pawlowski-Shadid-Simonis-Walker 2004]

• In this cases the Newton direction should be rejected and new directions should be con-sidered.

11


Trust-region methods

• Let us consider the quadratic function f(x) = 12‖F (x)‖2 for which ∇f = (F ′)T F

has the same direction as ∇‖F ‖.• The quadratic model mk(p) of f(x) = 1

2‖F (x)‖2 in xk

mk(p) =12‖F

′(xk)p + F (xk)‖2.

• Given a trust-region radius ∆k, the trial direction pk is the solution of the trust-regionproblem

minp∈IRn

mk(p) : ‖p‖ ≤ ∆k

12


• As ∆k tends to zero, the step pk tends to become parallel to the gradient of f : trustregion methods have the advantage to generate directions that may be stronger descentdirections than sk.

∆∆

∆

p

sk

k

kk

C

N

Dogleg approximate solution

– ∆k: trust region radius,– pC: Cauchy point,– sN

k : Newton step.

13


Low dimensional subspace trust-region

• When dealing with large scale problems, by the computational point of view it may beconvenient to minimize the quadratic model in low dimensional subspaces Sk ⊂ IRn

minp∈Sk

mk(p) : ‖p‖ ≤ ∆k,

• The solution of the subspace trust region problem is usually not expensive due to thelow dimension of the subspace. The main computational cost lies in the construction of thesubspace Sk.

• Our approach: Sk = spansk, ∇fk. The subspace trust-region problem reduces to atrust-region problem in R2.

• This requires the computation of ∇fk = (F ′(xk))T F (xk), in our context the Jacobianis available.

14


A combined global method

• We adopt the following strategy:– perform MAXBT backtracks in the inexact direction sk;

– if the sufficient descent condition is not satisfied, the direction sk is rejected;

– a trust region radius ∆k is selected

– a dogleg approximate solution of the subspace trust region problem is employed asthe new trial step.

• Additional cost in computing the new direction pk:– computation of ∇fk = F ′(xk)TF (xk)– dogleg strategy in R2 that is clearly performed at a very low computational cost.

15


HINM Algorithm (Hybrid Inexact Newton Method)

Given xk, ηmax ∈ (0, 1), ηk ∈ [0, ηmax], t ∈ (0, 1), β1 ∈ (0, 1), α1, α2, ∈ (0, 1),0 < θm < θmax < 1, MAXBT > 0, ∆min > 0 and ∆k > ∆min:

1. Compute sk such that ‖F (xk) + F ′(xk)sk‖ ≤ ηk‖F (xk)‖.2. Apply the INB backtracking strategy and perform at most MAXBT backtracks.3. If ‖F (xk + sk)‖ > (1 − t(1 − ηk))‖F (xk)‖

3.1 Do3.1.1 Compute the approximate dogleg solution sk to argmin

‖s‖≤∆k,s∈Sk

mk(s).

3.1.2 Set ∆k = minα1 ∆k, α2 ‖sk‖.while ρf

k(sk) ≥ β1.4. Set xk+1 = xk + sk.5. Choose ∆k+1 > ∆min.

Here

ρfk(sk) =

f(xk) − f(xk + sk)mk(0) − mk(sk)

is a measure of the agreement between f(x) and its quadratic model m(s) around xk.

16


Convergence properties

Theorem. Let r > 0 and L = ∪∞k=0 x ∈ IRn | ‖x − xk‖ ≤ r be a neighborhood

of sequence xk generated by HINM Algorithm. Assume that F ′ is Lipschitz continuousin L, with Lipschitz constant 2γL and ‖F ′(x)‖ is bounded above on L. Then, if F ′

k isinvertible for k ≥ 0, the repeat loop in Step 3 terminates. Further, if there exists a limit pointx∗ of xk such that F ′(x∗) is invertible, then

a) limk→∞ ‖F (xk)‖ = ‖F (x∗)‖ = 0.b) limk→∞ xk = x∗;c) sk = sk, for sufficiently large k.

⇒ Eventually the adopted step is the Inexact Newton step. Then the ultimate rate ofconvergence depends on the choice of the forcing terms, as in classical backtracking InexactNewton methods.

17


Numerical Results

Comparison between Newton-GMRES with backtrack (INB) and our Hybrid Inexact New-ton Method (HINM).

• Initial guess: x0 = (0, . . . , 0)T ;• GMRES: restart=200, maximal number of iterations 600;• ILU right preconditioning;• Success is declared when ‖F (xk)‖ < 1.0E − 6;• Failure is declared when after 200 iterations ‖F (xk)‖ > 1.0E − 6 holds;

– INB: when after 10 backtracks the sufficient decrease condition is not satisfied;– HINM: when ∆k < 1.0E − 8.

• MAXBT = 4 for HINM;• The backtrack reduction factor θ is chosen using the two point parabolic model.

18


Forcing terms:

• Large and small constant forcing terms (linear convergence):– ηk = 1.0E−1,– ηk = 1.0E−4.

• Adaptive choice: Select η0 ∈ [0, 1) and chose

ηk =

˛‖F (xk+1)‖ − ‖F (xk) + F ′(xk)sk‖

˛‖F (xk)‖

, k = 1, 2, . . .

with the safeguard: ηk = maxηk, η(1+

√5)/2

k−1 if η(1+

√5)/2

k−1 > 0.1.– Prevent the initial ηk’s from becoming too small far away from a solution.– ηk → 0: superlinear convergence .

19


Numerical Results: Backward Facing Step

-

6

- l

- L

6

?

h 6

?

Hx

y

u1 = 0; u2 = 0

u1 = 0; u2 = 0

Geometry of the backward facing step

upper eddy

lower eddy

(Re = 600)

a

Eddies’ position for the BFS problem

Numerical experiments performed with Re = 200, 400, 500, 600, 650, 700, 800

Number of Failures: Comparison between the HINM method and the INB methodGrid ηk → 1.0E-1 1.0E-4 Ad. Choice2460 HINM 4 2 4

INB 5 4 46781 HINM 2 4 3

INB 4 4 413200 HINM 1 0 2

INB 2 1 3

20


Angle between the chosen sk and −∇f(xk)

21


Numerical Results: Lid Driven Cavity

-

6

x

y

1

1

u1 =0; u2 =0

u1 =1; u2 =0

u1 =0

u2 =0

u1 =0

u2 =0

Geometry of the lid driven cavity Eddies’ position for the cavity problem

Numerical experiments performed with Re = 2000, 3000, 4000, 5000, 6000, 8000, 10000

22


Equation Scaling

For this problem we consider the following scaled nonlinear equation

D−1

F (x) = 0

where the constant diagonal matrix D is computed at the first iteration by

Di,i =X

j

|F ′i,j(x0)|.

23


Comparison between the HINM and the INB, Cavity Number of failuresGrid ηk → 1.0E-1 1.0E-4 Ad. Choice1089 HINM 0 0 4

INB 5 0 6

Performances of the HINM on tests unsolved by the INB, Nodes=1089

Test η Re iterlGMiter

mGMitermax sw ‖F ‖

1 1.0E-1 2000 59 28 43 27 4.127E-72 1.0E-1 4000 27 39 67 2 9.426E-73 1.0E-1 6000 50 60 123 18 6.878E-74 1.0E-1 8000 85 57 150 57 9.928E-75 1.0E-1 10000 139 74 137 108 8.076E-76 choice 1 3000 83 39 59 46 8.340E-77 choice 1 4000 64 39 88 26 7.543E-7

24


Node=1089, Re=4000, ηk = 1.0E − 1

25


Node=1089, Re=6000, ηk = 1.0E − 1

26


Node=1089, Re=10000, ηk = 1.0E − 1

27

a hybrid newton-krylov method for stabilized fem...

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