Adaptive inexact Newton methods
and their application to multi-phase flows
Martin Vohralík
INRIA Paris-Rocquencourt
in collaboration with Clément Cancès, Daniele A. Di Pietro, Alexandre Ern,
Eric Flauraud, I. Sorin Pop, Mary F. Wheeler, and Soleiman Yousef
Luminy, November 20, 2014
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
1 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
1 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Full adaptivity for multi-phase multi-component flows
Multi-phase multi-component porous media flows
nonlinear (degenerate) systems of PDEs
nonlinear algebraic constraints
⇒ difficult numerical approximation, large troublesome
systems of nonlinear algebraic equations
Goals of this work
derive fully computable a posteriori error upper bounds
distinguish different error components
Full adaptivity
time step choice & mesh adaptivity
stopping criteria for linear and nonlinear solvers
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
2 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Full adaptivity for multi-phase multi-component flows
Multi-phase multi-component porous media flows
nonlinear (degenerate) systems of PDEs
nonlinear algebraic constraints
⇒ difficult numerical approximation, large troublesome
systems of nonlinear algebraic equations
Goals of this work
derive fully computable a posteriori error upper bounds
distinguish different error components
Full adaptivity
time step choice & mesh adaptivity
stopping criteria for linear and nonlinear solvers
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
2 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
2 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Model steady problem, discretization
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
u=0
p > 1, q :=
p
p−1 ,
in Ω,
on ∂Ω
f ∈ Lq (Ω)
example: p-Laplacian with σ(u, ∇u) = |∇u|p−2 ∇u
Numerical approximation
mesh Th , linearization step k , algebraic step i: uhk ,i
uhk ,i ∈ V (Th ) typically piecewise polynomial (discontinuous)
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
3 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Model steady problem, discretization
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
u=0
p > 1, q :=
p
p−1 ,
in Ω,
on ∂Ω
f ∈ Lq (Ω)
example: p-Laplacian with σ(u, ∇u) = |∇u|p−2 ∇u
Numerical approximation
mesh Th , linearization step k , algebraic step i: uhk ,i
uhk ,i ∈ V (Th ) typically piecewise polynomial (discontinuous)
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
3 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
3 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Abstract assumptions
Assumption A (Total flux reconstruction)
There exists tkh ,i ∈ Hq (div, Ω) and ρkh ,i ∈ Lq (Ω) such that
∇·tkh ,i = f − ρkh ,i .
Assumption B (Discretization, linearization, and alg. fluxes)
There exist fluxes dhk ,i , lkh ,i , akh ,i ∈ [Lq (Ω)]d such that
(i) tkh ,i = dkh ,i + lkh ,i + akh ,i ;
(ii) as the linear solver converges, kakh ,i kq → 0;
(iii) as the nonlinear solver converges, klkh ,i kq → 0.
Assumption C (Approximation property)
kσ(uhk ,i , ∇uhk ,i ) + dkh ,i kq is small.
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
4 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Abstract assumptions
Assumption A (Total flux reconstruction)
There exists tkh ,i ∈ Hq (div, Ω) and ρkh ,i ∈ Lq (Ω) such that
∇·tkh ,i = f − ρkh ,i .
Assumption B (Discretization, linearization, and alg. fluxes)
There exist fluxes dhk ,i , lkh ,i , akh ,i ∈ [Lq (Ω)]d such that
(i) tkh ,i = dkh ,i + lkh ,i + akh ,i ;
(ii) as the linear solver converges, kakh ,i kq → 0;
(iii) as the nonlinear solver converges, klkh ,i kq → 0.
Assumption C (Approximation property)
kσ(uhk ,i , ∇uhk ,i ) + dkh ,i kq is small.
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
4 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Abstract assumptions
Assumption A (Total flux reconstruction)
There exists tkh ,i ∈ Hq (div, Ω) and ρkh ,i ∈ Lq (Ω) such that
∇·tkh ,i = f − ρkh ,i .
Assumption B (Discretization, linearization, and alg. fluxes)
There exist fluxes dhk ,i , lkh ,i , akh ,i ∈ [Lq (Ω)]d such that
(i) tkh ,i = dkh ,i + lkh ,i + akh ,i ;
(ii) as the linear solver converges, kakh ,i kq → 0;
(iii) as the nonlinear solver converges, klkh ,i kq → 0.
Assumption C (Approximation property)
kσ(uhk ,i , ∇uhk ,i ) + dkh ,i kq is small.
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
4 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Estimate distinguishing error components
Theorem (Estimate distinguishing different error components)
Let
u ∈ V be the weak solution,
uhk ,i ∈ V (Th ) be arbitrary,
Assumptions A and B hold.
Then there holds
k ,i
k ,i
k ,i
k ,i
Ju (uhk ,i ) ≤ ηdisc
+ ηlin
+ ηalg
+ ηrem
.
Moreover, under Assumption C and under appropriate stopping
criteria,
k ,i
k ,i
k ,i
k ,i
+ ηalg
+ ηrem
≤ C(κT )Ju (uhk ,i ).
ηdisc
+ ηlin
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
5 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Estimate distinguishing error components
Theorem (Estimate distinguishing different error components)
Let
u ∈ V be the weak solution,
uhk ,i ∈ V (Th ) be arbitrary,
Assumptions A and B hold.
Then there holds
k ,i
k ,i
k ,i
k ,i
Ju (uhk ,i ) ≤ ηdisc
+ ηlin
+ ηalg
+ ηrem
.
Moreover, under Assumption C and under appropriate stopping
criteria,
k ,i
k ,i
k ,i
k ,i
+ ηlin
+ ηalg
+ ηrem
≤ C(κT )Ju (uhk ,i ).
ηdisc
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
5 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Estimators
discretization estimator
k ,i
ηdisc,K
:= 2
1
p
)1 !
(
kσ(uhk ,i , ∇uhk ,i )+dkh ,i kq,K +
q
X
he1−q k[[uhk ,i ]]kqq,e
e∈EK
linearization estimator
k ,i
ηlin,K
:= klkh ,i kq,K
algebraic estimator
k ,i
ηalg,K
:= kakh ,i kq,K
algebraic remainder estimator
k ,i
ηrem,K
:= hΩ kρkh ,i kq,K
)1/q
(
η k· ,i :=
X
,i q
η k·,K
K ∈Th
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
6 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
6 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Applications
Discretization methods
conforming finite elements
nonconforming finite elements
discontinuous Galerkin
various finite volumes
mixed finite elements
Linearizations
fixed point
Newton
Linear solvers
independent of the linear solver
. . . all Assumptions A to C verified
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
7 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Applications
Discretization methods
conforming finite elements
nonconforming finite elements
discontinuous Galerkin
various finite volumes
mixed finite elements
Linearizations
fixed point
Newton
Linear solvers
independent of the linear solver
. . . all Assumptions A to C verified
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
7 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Applications
Discretization methods
conforming finite elements
nonconforming finite elements
discontinuous Galerkin
various finite volumes
mixed finite elements
Linearizations
fixed point
Newton
Linear solvers
independent of the linear solver
. . . all Assumptions A to C verified
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
7 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Applications
Discretization methods
conforming finite elements
nonconforming finite elements
discontinuous Galerkin
various finite volumes
mixed finite elements
Linearizations
fixed point
Newton
Linear solvers
independent of the linear solver
. . . all Assumptions A to C verified
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
7 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Numerical experiment I
Model problem
p-Laplacian
∇·(|∇u|p−2 ∇u) = f
in Ω,
u = uD
on ∂Ω
weak solution (used to impose the Dirichlet BC)
u(x, y ) =
− p−1
p
(x −
1 2
2)
+ (y −
1 2
2)
p
2(p−1)
+
p−1
p
1
2
p
p−1
tested values p = 1.5 and 10
nonconforming finite elements
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
8 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Analytical and approximate solutions
0.04
0.4
0.02
0.04
0.3
0
0
0.2
0.2
−0.02
u
u
0.3
0.4
0.02
−0.02
−0.04
0.1
0.1
0
−0.1
−0.06
0
−0.04
1
1
1
0.5
y
0.5
0 0
x
Case p = 1.5
Martin Vohralík
1
−0.06
0.5
y
−0.1
0.5
0
0
x
Case p = 10
Adaptive inexact Newton methods and multi-phase flows
9 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Error and estimators as a function of CG iterations,
p = 10, 6th level mesh, 6th Newton step.
0
0
10
0
10
10
−2
10
−6
10
Dual error
Dual error
Dual error
−4
10
−1
10
−1
10
−8
10
−10
10
0
−2
100
200
300
400
Algebraic iteration
500
600
Newton
Martin Vohralík
700
10
3
−2
6
9
Algebraic iteration
12
inexact Newton
15
10
0
error up
estimate
disc. est.
lin. est.
alg. est.
alg. rem. est.
5
10
15
20
Algebraic iteration
25
30
35
ad. inexact Newton
Adaptive inexact Newton methods and multi-phase flows
10 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Error and estimators as a function of Newton
iterations, p = 10, 6th level mesh
2
2
10
−2
10
−2
1
−6
10
Dual error
Dual error
10
−4
10
−4
10
−6
10
−8
10
−8
−2
10
−10
0
10
−10
2
4
6
8
10
12
Newton iteration
14
16
Newton
Martin Vohralík
18
20
10
0
10
−1
10
10
error up
estimate
disc. est.
lin. est.
2
10
10
Dual error
10
0
10
10
3
10
0
0
−3
10
20
30
40
50
60
Newton iteration
70
80
90
inexact Newton
100
10
1
2
3
4
5
6
7
8
9
Newton iteration
10
11
12
13
ad. inexact Newton
Adaptive inexact Newton methods and multi-phase flows
11 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Newton and algebraic iterations, p = 10
3
80
4
10
full
inex.
adapt. inex.
Number of algebraic solver iterations
Number of Newton iterations
90
70
60
50
40
30
20
10
full
inex.
adapt. inex.
Total number of algebraic solver iterations
100
2
10
1
10
10
0
1
0
2
3
4
Refinement level
5
6
Newton it. / refinement
Martin Vohralík
10 0
10
full
inex.
adapt. inex.
3
10
2
10
1
1
10
Newton iteration
2
10
alg. it. / Newton step
10
1
2
3
4
Refinement level
5
6
alg. it. / refinement
Adaptive inexact Newton methods and multi-phase flows
12 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Error and estimators as a function of CG iterations,
p = 1.5, 6th level mesh, 1st Newton step.
0
0
10
0
10
10
−2
−1
10
10
−1
10
−6
10
Dual error
−2
Dual error
Dual error
−4
10
10
−3
10
−2
10
−8
−4
10
10
−10
10
0
−3
100
200
300
Algebraic iteration
400
Newton
Martin Vohralík
500
10
3
−5
6
9
Algebraic iteration
12
inexact Newton
15
10
0
error up
estimate
disc. est.
lin. est.
alg. est.
alg. rem. est.
20
40
60
Algebraic iteration
80
100
ad. inexact Newton
Adaptive inexact Newton methods and multi-phase flows
13 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Error and estimators as a function of Newton
iterations, p = 1.5, 6th level mesh
−2
0
10
−4
10
−4
Dual error
10
−8
10
−10
−6
10
−8
10
10
−12
0
10
Dual error
−6
Dual error
10
−2
10
10
−2
10
−10
2
4
6
8
10
12
Newton iteration
14
16
Newton
Martin Vohralík
18
20
10
0
−3
50
100 150 200 250 300 350 400 450 500 550
Newton iteration
inexact Newton
10
1
error up
estimate
disc. est.
lin. est.
2
Newton iteration
3
ad. inexact Newton
Adaptive inexact Newton methods and multi-phase flows
14 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Numerical experiment II
Model problem
p-Laplacian
∇·(|∇u|p−2 ∇u) = f
u = uD
in Ω,
on ∂Ω
weak solution (used to impose the Dirichlet BC)
7
u(r , θ) = r 8 sin(θ 87 )
p = 4, L-shape domain, singularity in the origin
(Carstensen and Klose (2003))
nonconforming finite elements
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
15 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Error distribution on an adaptively refined mesh
−3
−3
x 10
x 10
8
5
7
4.5
4
6
3.5
5
3
4
2.5
2
3
1.5
2
1
1
0.5
Estimated error distribution
Martin Vohralík
Exact error distribution
Adaptive inexact Newton methods and multi-phase flows
16 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Appl. & num. res.
Energy error and overall performance
0
10
600
Total number of algebraic solver iterations
energy error uniform
energy error adaptive
−1
Energy error
10
−2
10
−3
10
1
10
2
10
3
10
Number of faces
Energy error
Martin Vohralík
4
10
5
10
uniform
adaptive
550
500
450
400
350
300
250
200
150
100
50
0
1
2
3
4
5
6
7
8
9
Refinement level
10
11
12
13
Overall performance
Adaptive inexact Newton methods and multi-phase flows
17 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
17 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Two-phase flow in porous media
Two-phase flow in porous media
∂t (φsα ) + ∇·uα = qα ,
α ∈ {n, w},
−λα (sw )K(∇pα + ρα g∇z) = uα ,
α ∈ {n, w},
sn + sw = 1,
pn − pw = pc (sw )
+ boundary & initial conditions
Mathematical issues
coupled system
unsteady, nonlinear
elliptic–degenerate parabolic type
dominant advection
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
18 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Two-phase flow in porous media
Two-phase flow in porous media
∂t (φsα ) + ∇·uα = qα ,
α ∈ {n, w},
−λα (sw )K(∇pα + ρα g∇z) = uα ,
α ∈ {n, w},
sn + sw = 1,
pn − pw = pc (sw )
+ boundary & initial conditions
Mathematical issues
coupled system
unsteady, nonlinear
elliptic–degenerate parabolic type
dominant advection
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
18 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Global and complementary pressures
Global pressure
sw
Z
p(sw , pw ) := pw +
0
λn (a)
p0 (a)da
λw (a) + λn (a) c
Complementary pressure
Z sw
λw (a)λn (a) 0
q(sw ) := −
pc (a)da
λ
w (a) + λn (a)
0
Comments
necessary for the correct definition of the weak solution
equivalent Darcy velocities expressions
uw (sw , pw ) := − K λw (sw )∇p(sw , pw ) + ∇q(sw ) + λw (sw )ρw g∇z ,
un (sw , pw ) := − K λn (sw )∇p(sw , pw ) − ∇q(sw ) + λn (sw )ρn g∇z
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
19 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Global and complementary pressures
Global pressure
sw
Z
p(sw , pw ) := pw +
0
λn (a)
p0 (a)da
λw (a) + λn (a) c
Complementary pressure
Z sw
λw (a)λn (a) 0
pc (a)da
q(sw ) := −
λ
w (a) + λn (a)
0
Comments
necessary for the correct definition of the weak solution
equivalent Darcy velocities expressions
uw (sw , pw ) := − K λw (sw )∇p(sw , pw ) + ∇q(sw ) + λw (sw )ρw g∇z ,
un (sw , pw ) := − K λn (sw )∇p(sw , pw ) − ∇q(sw ) + λn (sw )ρn g∇z
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
19 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Global and complementary pressures
Global pressure
sw
Z
p(sw , pw ) := pw +
0
λn (a)
p0 (a)da
λw (a) + λn (a) c
Complementary pressure
Z sw
λw (a)λn (a) 0
pc (a)da
q(sw ) := −
λ
w (a) + λn (a)
0
Comments
necessary for the correct definition of the weak solution
equivalent Darcy velocities expressions
uw (sw , pw ) := − K λw (sw )∇p(sw , pw ) + ∇q(sw ) + λw (sw )ρw g∇z ,
un (sw , pw ) := − K λn (sw )∇p(sw , pw ) − ∇q(sw ) + λn (sw )ρn g∇z
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
19 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Weak formulation
Energy space
X := L2 ((0, T ); HD1 (Ω))
Definition (Weak solution (Arbogast 1992, Chen 2001))
Find (sw , pw ) such that, with sn := 1 − sw ,
sw ∈ C([0, T ]; L2 (Ω)), sw (·, 0) = sw0 ,
∂t sw ∈ L2 ((0, T ); (HD1 (Ω))0 ),
p(sw , pw ) ∈ X ,
q(sw ) ∈ X ,
Z T
h∂t (φsα ), ϕi − (uα (sw , pw ), ∇ϕ) − (qα , ϕ) dt = 0
0
∀ϕ ∈ X , α ∈ {n, w}.
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
20 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Weak formulation
Energy space
X := L2 ((0, T ); HD1 (Ω))
Definition (Weak solution (Arbogast 1992, Chen 2001))
Find (sw , pw ) such that, with sn := 1 − sw ,
sw ∈ C([0, T ]; L2 (Ω)), sw (·, 0) = sw0 ,
∂t sw ∈ L2 ((0, T ); (HD1 (Ω))0 ),
p(sw , pw ) ∈ X ,
q(sw ) ∈ X ,
Z T
h∂t (φsα ), ϕi − (uα (sw , pw ), ∇ϕ) − (qα , ϕ) dt = 0
0
∀ϕ ∈ X , α ∈ {n, w}.
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
20 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
20 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Link energy-type error – dual norm of the residual
Dual norm of the residual on the time interval In
(
(
Z
X
n
Jsw ,pw (sw,hτ , pw,hτ ) :=
sup
h∂t (φsα )−∂t (φsα,hτ ), ϕi
α∈{n,w} ϕ∈X |In , kϕkX |In=1 In
)2 ) 1
2
− (uα (sw , pw ) − uα (sw,hτ , pw,hτ ), ∇ϕ) dt
Theorem (Link energy-type error – dual norm of the residual)
Let (sw , pw ) be the weak solution. Let (sw,hτ , pw,hτ ) be arbitrary
such that p(sw,hτ , pw,hτ ) ∈ X and q(sw,hτ ) ∈ X (and satisfying
the initial and boundary conditions for simplicity). Then
ksw − sw,hτ kL2 ((0,T );H −1 (Ω)) + kq(sw ) − q(sw,hτ )kL2 (Ω×(0,T ))
+ kp(sw , pw ) − p(sw,hτ , pw,hτ )kL2 ((0,T );H 1 (Ω))
0
( N
)1
2
X
≤C
Jsnw ,pw (sw,hτ , pw,hτ )2
n=1
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
21 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Link energy-type error – dual norm of the residual
Dual norm of the residual on the time interval In
(
(
Z
X
n
Jsw ,pw (sw,hτ , pw,hτ ) :=
sup
h∂t (φsα )−∂t (φsα,hτ ), ϕi
α∈{n,w} ϕ∈X |In , kϕkX |In=1 In
)2 ) 1
2
− (uα (sw , pw ) − uα (sw,hτ , pw,hτ ), ∇ϕ) dt
Theorem (Link energy-type error – dual norm of the residual)
Let (sw , pw ) be the weak solution. Let (sw,hτ , pw,hτ ) be arbitrary
such that p(sw,hτ , pw,hτ ) ∈ X and q(sw,hτ ) ∈ X (and satisfying
the initial and boundary conditions for simplicity). Then
ksw − sw,hτ kL2 ((0,T );H −1 (Ω)) + kq(sw ) − q(sw,hτ )kL2 (Ω×(0,T ))
+ kp(sw , pw ) − p(sw,hτ , pw,hτ )kL2 ((0,T );H 1 (Ω))
0
( N
)1
2
X
≤C
Jsnw ,pw (sw,hτ , pw,hτ )2
n=1
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
21 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Link energy-type error – dual norm of the residual
Dual norm of the residual on the time interval In
(
(
Z
X
n
Jsw ,pw (sw,hτ , pw,hτ ) :=
sup
h∂t (φsα )−∂t (φsα,hτ ), ϕi
α∈{n,w} ϕ∈X |In , kϕkX |In=1 In
)2 ) 1
2
− (uα (sw , pw ) − uα (sw,hτ , pw,hτ ), ∇ϕ) dt
Theorem (Link energy-type error – dual norm of the residual)
Let (sw , pw ) be the weak solution. Let (sw,hτ , pw,hτ ) be arbitrary
such that p(sw,hτ , pw,hτ ) ∈ X and q(sw,hτ ) ∈ X (and satisfying
the initial and boundary conditions for simplicity). Then
ksw − sw,hτ kL2 ((0,T );H −1 (Ω)) + kq(sw ) − q(sw,hτ )kL2 (Ω×(0,T ))
+ kp(sw , pw ) − p(sw,hτ , pw,hτ )kL2 ((0,T );H 1 (Ω))
0
( N
)1
2
X
≤C
Jsnw ,pw (sw,hτ , pw,hτ )2
n=1
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
21 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Link energy-type error – dual norm of the residual
Dual norm of the residual on the time interval In
(
(
Z
X
n
Jsw ,pw (sw,hτ , pw,hτ ) :=
sup
h∂t (φsα )−∂t (φsα,hτ ), ϕi
α∈{n,w} ϕ∈X |In , kϕkX |In=1 In
)2 ) 1
2
− (uα (sw , pw ) − uα (sw,hτ , pw,hτ ), ∇ϕ) dt
Theorem (Link energy-type error – dual norm of the residual)
Let (sw , pw ) be the weak solution. Let (sw,hτ , pw,hτ ) be arbitrary
such that p(sw,hτ , pw,hτ ) ∈ X and q(sw,hτ ) ∈ X (and satisfying
the initial and boundary conditions for simplicity). Then
ksw − sw,hτ kL2 ((0,T );H −1 (Ω)) + kq(sw ) − q(sw,hτ )kL2 (Ω×(0,T ))
+ kp(sw , pw ) − p(sw,hτ , pw,hτ )kL2 ((0,T );H 1 (Ω))
0
( N
)1
2
X
≤C
Jsnw ,pw (sw,hτ , pw,hτ )2
n=1
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
21 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Distinguishing the error components
Theorem (Distinguishing the error components)
Consider a vertex-centered finite volume / backward Euler
approximation. Let
n be the time step,
k be the linearization step,
i be the algebraic solver step,
n,k ,i
n,k ,i
with the approximations (sw,hτ
, pw,hτ
). Then
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
.
Jsnw ,pw (sw,hτ
, pw,hτ
) ≤ ηsp
+ ηtm
+ ηlin
+ ηalg
Error components
n,k ,i
ηsp
:
n,k ,i
ηtm :
n,k ,i
ηlin
:
n,k ,i
ηalg :
spatial discretization
temporal discretization
linearization
algebraic solver
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
22 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Distinguishing the error components
Theorem (Distinguishing the error components)
Consider a vertex-centered finite volume / backward Euler
approximation. Let
n be the time step,
k be the linearization step,
i be the algebraic solver step,
n,k ,i
n,k ,i
with the approximations (sw,hτ
, pw,hτ
). Then
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
.
Jsnw ,pw (sw,hτ
, pw,hτ
) ≤ ηsp
+ ηtm
+ ηlin
+ ηalg
Error components
n,k ,i
ηsp
:
n,k ,i
ηtm :
n,k ,i
ηlin
:
n,k ,i
ηalg :
spatial discretization
temporal discretization
linearization
algebraic solver
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
22 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
22 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Iteratively coupled vertex-centered finite volumes
Vertex-centered finite volumes
simplicial meshes Thn , dual meshes Dhn
discrete saturations and pressures continuous and
piecewise affine on Thn
Implicit pressure equation on step k
n,k −1
n,k −1 n,k
− λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
·nD
n,k −1
n,k −1
+λr,n (sw,h
)K∇pc (sw,h
)·nD , 1 ∂D\∂Ω = 0 ∀D ∈ Dhint,n
Explicit saturation equation on step k
n,k
sw,D
:=
τn
n,k −1
n,k
n−1
λr,w (sw,h
)K∇pw,h
·nD , 1 ∂D\∂Ω + sw,D
φ|D|
Martin Vohralík
∀D ∈ Dhint,n
Adaptive inexact Newton methods and multi-phase flows
23 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Iteratively coupled vertex-centered finite volumes
Vertex-centered finite volumes
simplicial meshes Thn , dual meshes Dhn
discrete saturations and pressures continuous and
piecewise affine on Thn
Implicit pressure equation on step k
n,k −1
n,k −1 n,k
− λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
·nD
n,k −1
n,k −1
+λr,n (sw,h
)K∇pc (sw,h
)·nD , 1 ∂D\∂Ω = 0 ∀D ∈ Dhint,n
Explicit saturation equation on step k
n,k
sw,D
:=
τn
n,k −1
n,k
n−1
λr,w (sw,h
)K∇pw,h
·nD , 1 ∂D\∂Ω + sw,D
φ|D|
Martin Vohralík
∀D ∈ Dhint,n
Adaptive inexact Newton methods and multi-phase flows
23 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Iteratively coupled vertex-centered finite volumes
Vertex-centered finite volumes
simplicial meshes Thn , dual meshes Dhn
discrete saturations and pressures continuous and
piecewise affine on Thn
Implicit pressure equation on step k
n,k −1
n,k −1 n,k
− λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
·nD
n,k −1
n,k −1
+λr,n (sw,h
)K∇pc (sw,h
)·nD , 1 ∂D\∂Ω = 0 ∀D ∈ Dhint,n
Explicit saturation equation on step k
n,k
sw,D
:=
τn
n,k −1
n,k
n−1
λr,w (sw,h
)K∇pw,h
·nD , 1 ∂D\∂Ω + sw,D
φ|D|
Martin Vohralík
∀D ∈ Dhint,n
Adaptive inexact Newton methods and multi-phase flows
23 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Linearization and algebraic solution
Iterative coupling step k and algebraic step i
n,k −1
n,k ,i
n,k −1 − λr,w (sw,h
) + λr,n (sw,h
·nD
) K∇pw,h
n,k −1
n,k −1
n,k ,i
+λr,n (sw,h )K∇pc (sw,h )·nD , 1 ∂D\∂Ω = −Rt,D
n,k ,i
sw,D
:=
∀D ∈ Dhint,n
τn
n,k −1
n,k ,i
n−1
λr,w (sw,h
)K∇pw,h
·nD , 1 ∂D\∂Ω + sw,D
φ|D|
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
24 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Linearization and algebraic solution
Iterative coupling step k and algebraic step i
n,k −1
n,k ,i
n,k −1 − λr,w (sw,h
) + λr,n (sw,h
·nD
) K∇pw,h
n,k −1
n,k −1
n,k ,i
+λr,n (sw,h )K∇pc (sw,h )·nD , 1 ∂D\∂Ω = −Rt,D
n,k ,i
sw,D
:=
∀D ∈ Dhint,n
τn
n,k −1
n,k ,i
n−1
λr,w (sw,h
)K∇pw,h
·nD , 1 ∂D\∂Ω + sw,D
φ|D|
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
24 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Flux reconstructions
Total phase velocities reconstructions
n,k ,i
n,k ,i
n,k ,i ·nD
λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
n,k ,i
n,k ,i
+ λr,n (sw,h )K∇pc (sw,h )·nD , 1 e ,
n,k ,i
n,k −1
n,k −1 n,k ,i
+ lt,h )·nD , 1)e := − λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
·nD
n,k −1
n,k −1
+ λr,n (sw,h )K∇pc (sw,h )·nD , 1 e ,
,i
(dn,k
t,h ·nD , 1)e := −
,i
((dn,k
t,h
,i
n,k ,i+ν
n,k ,i+ν
n,k ,i
,i
an,k
+ lt,h
− (dt,h
+ ln,k
t,h := dt,h
t,h )
Wetting phase velocities reconstructions
,i
((dn,k
w,h
n,k ,i
,i
n,k ,i
(dn,k
w,h ·nD , 1)e := − λr,w (sw,h )K∇pw,h ·nD , 1 e ,
,i
n,k −1
n,k ,i
·n
,
1
,
+ ln,k
)·n
,
1)
:=
−
λ
(s
)K∇p
e
r,w
D
D
w,h
w,h
w,h
e
,i
an,k
w,h := 0
Nonwetting phase and total velocities reconstructions
n,k ,i
,i
n,k ,i n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
dn,h
:= dn,k
t,h −dw,h , ln,h := lt,h −lw,h , an,h := at,h −aw,h
n,k ,i
,i
n,k ,i
n,k ,i
t·,h
:= dn,k
·,h + l·,h + a·,h
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
25 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Flux reconstructions
Total phase velocities reconstructions
n,k ,i
n,k ,i
n,k ,i ·nD
λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
n,k ,i
n,k ,i
+ λr,n (sw,h )K∇pc (sw,h )·nD , 1 e ,
n,k ,i
n,k −1
n,k −1 n,k ,i
+ lt,h )·nD , 1)e := − λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
·nD
n,k −1
n,k −1
+ λr,n (sw,h )K∇pc (sw,h )·nD , 1 e ,
,i
(dn,k
t,h ·nD , 1)e := −
,i
((dn,k
t,h
,i
n,k ,i+ν
n,k ,i+ν
n,k ,i
,i
an,k
+ lt,h
− (dt,h
+ ln,k
t,h := dt,h
t,h )
Wetting phase velocities reconstructions
,i
((dn,k
w,h
,i
n,k ,i
n,k ,i
(dn,k
w,h ·nD , 1)e := − λr,w (sw,h )K∇pw,h ·nD , 1 e ,
,i
n,k −1
n,k ,i
·n
,
1
,
+ ln,k
)·n
,
1)
:=
−
λ
(s
)K∇p
e
r,w
D
D
w,h
w,h
w,h
e
,i
an,k
w,h := 0
Nonwetting phase and total velocities reconstructions
n,k ,i
,i
n,k ,i n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
dn,h
:= dn,k
t,h −dw,h , ln,h := lt,h −lw,h , an,h := at,h −aw,h
n,k ,i
,i
n,k ,i
n,k ,i
t·,h
:= dn,k
·,h + l·,h + a·,h
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
25 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Flux reconstructions
Total phase velocities reconstructions
n,k ,i
n,k ,i
n,k ,i ·nD
λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
n,k ,i
n,k ,i
+ λr,n (sw,h )K∇pc (sw,h )·nD , 1 e ,
n,k ,i
n,k −1
n,k −1 n,k ,i
+ lt,h )·nD , 1)e := − λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
·nD
n,k −1
n,k −1
+ λr,n (sw,h )K∇pc (sw,h )·nD , 1 e ,
,i
(dn,k
t,h ·nD , 1)e := −
,i
((dn,k
t,h
,i
n,k ,i+ν
n,k ,i+ν
n,k ,i
,i
an,k
+ lt,h
− (dt,h
+ ln,k
t,h := dt,h
t,h )
Wetting phase velocities reconstructions
,i
((dn,k
w,h
,i
n,k ,i
n,k ,i
(dn,k
w,h ·nD , 1)e := − λr,w (sw,h )K∇pw,h ·nD , 1 e ,
,i
n,k −1
n,k ,i
·n
,
1
,
+ ln,k
)·n
,
1)
:=
−
λ
(s
)K∇p
e
r,w
D
D
w,h
w,h
w,h
e
,i
an,k
w,h := 0
Nonwetting phase and total velocities reconstructions
n,k ,i
,i
n,k ,i n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
dn,h
:= dn,k
t,h −dw,h , ln,h := lt,h −lw,h , an,h := at,h −aw,h
n,k ,i
,i
n,k ,i
n,k ,i
t·,h
:= dn,k
·,h + l·,h + a·,h
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
25 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Model problem
Horizontal flow
∂t (φsα ) − ∇·
kr,α (sw )
K∇pα = 0,
µα
sn + sw = 1,
pn − pw = pc (sw )
Brooks–Corey model
relative permeabilities
kr,w (sw ) = se4 ,
kr,n (sw ) = (1 − se )2 (1 − se2 )
capillary pressure
− 21
pc (sw ) = pd se
se :=
Martin Vohralík
sw − srw
1 − srw − srn
Adaptive inexact Newton methods and multi-phase flows
26 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Model problem
Horizontal flow
∂t (φsα ) − ∇·
kr,α (sw )
K∇pα = 0,
µα
sn + sw = 1,
pn − pw = pc (sw )
Brooks–Corey model
relative permeabilities
kr,w (sw ) = se4 ,
kr,n (sw ) = (1 − se )2 (1 − se2 )
capillary pressure
− 21
pc (sw ) = pd se
se :=
Martin Vohralík
sw − srw
1 − srw − srn
Adaptive inexact Newton methods and multi-phase flows
26 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Data from Klieber & Rivière (2006)
Data
Ω = (0, 300)m × (0, 300)m,
φ = 0.2,
T = 4·106 s,
K = 10−11 I m2 ,
µw = 5·10−4 kg m−1 s−1 ,
srw = srn = 0,
µn = 2·10−3 kg m−1 s−1 ,
pd = 5·103 kg m−1 s−2
e 18m × 18m lower left corner block)
Initial condition (K
e,
s0 = 0.2 on K ∈ Th , K 6∈ K
w
sw0
e
= 0.95 on K ∈ Th , K ∈ K
b 18m × 18m upper right corner block)
Boundary conditions (K
no flow Neumann boundary conditions everywhere except
e ∩ ∂Ω and ∂ K
b ∩ ∂Ω
of ∂ K
e – injection well: sw = 0.95, pw = 3.45·106 kg m−1 s−2
K
b – production well: sw = 0.2, pw = 2.41·106 kg m−1 s−2
K
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
27 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Data from Klieber & Rivière (2006)
Data
Ω = (0, 300)m × (0, 300)m,
φ = 0.2,
T = 4·106 s,
K = 10−11 I m2 ,
µw = 5·10−4 kg m−1 s−1 ,
srw = srn = 0,
µn = 2·10−3 kg m−1 s−1 ,
pd = 5·103 kg m−1 s−2
e 18m × 18m lower left corner block)
Initial condition (K
e,
s0 = 0.2 on K ∈ Th , K 6∈ K
w
sw0
e
= 0.95 on K ∈ Th , K ∈ K
b 18m × 18m upper right corner block)
Boundary conditions (K
no flow Neumann boundary conditions everywhere except
e ∩ ∂Ω and ∂ K
b ∩ ∂Ω
of ∂ K
e – injection well: sw = 0.95, pw = 3.45·106 kg m−1 s−2
K
b – production well: sw = 0.2, pw = 2.41·106 kg m−1 s−2
K
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
27 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Data from Klieber & Rivière (2006)
Data
Ω = (0, 300)m × (0, 300)m,
φ = 0.2,
T = 4·106 s,
K = 10−11 I m2 ,
µw = 5·10−4 kg m−1 s−1 ,
srw = srn = 0,
µn = 2·10−3 kg m−1 s−1 ,
pd = 5·103 kg m−1 s−2
e 18m × 18m lower left corner block)
Initial condition (K
e,
s0 = 0.2 on K ∈ Th , K 6∈ K
w
sw0
e
= 0.95 on K ∈ Th , K ∈ K
b 18m × 18m upper right corner block)
Boundary conditions (K
no flow Neumann boundary conditions everywhere except
e ∩ ∂Ω and ∂ K
b ∩ ∂Ω
of ∂ K
e – injection well: sw = 0.95, pw = 3.45·106 kg m−1 s−2
K
b – production well: sw = 0.2, pw = 2.41·106 kg m−1 s−2
K
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
27 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Estimators and stopping criteria
4
4
10
2
10
0
10
10
2
10
0
Estimators
Estimators
10
−2
10
adaptive stopping criterion
−4
10
−6
10
−8
10
0
total
spatial
temporal
linearization
algebraic
20
40
60
adaptive stopping criterion
−2
10
−4
10
−6
10
100 120 140
GMRes iteration
160
180
10
200
Estimators in function of
GMRes iterations
Martin Vohralík
classical stopping criterion
−8
classical stopping criterion
80
total
spatial
temporal
linearization
algebraic
220
1
2
3
4
5
6
7
Iterative coupling iteration
8
9
10
Estimators in function of
iterative coupling iterations
Adaptive inexact Newton methods and multi-phase flows
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I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
GMRes relative residual/iterative coupling iterations
Number of iterative coupling iterations
18
−6
GMRes relative residual
10
−8
10
−10
10
−12
classical
adaptive
10
−14
10
2
2.05
2.1
Time/iterative coupling step
2.15
GMRes relative residual
Martin Vohralík
2.2
6
x 10
16
classical
adaptive
14
12
10
8
6
4
2
0
0.5
1
1.5
2
Time
2.5
3
3.5
4
6
x 10
Iterative coupling iterations
Adaptive inexact Newton methods and multi-phase flows
29 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
GMRes iterations
5
5
classical
adaptive
Cumulative number of GMRes iterations
Number of GMRes iterations
250
200
150
100
50
0
2
2.05
2.1
Time/iterative coupling step
2.15
Per time and iterative
coupling step
Martin Vohralík
2.2
6
x 10
x 10
classical
adaptive
4
3
2
1
0
0
0.5
1
1.5
2
Time
2.5
3
3.5
4
6
x 10
Cumulated
Adaptive inexact Newton methods and multi-phase flows
30 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Space/time/nonlinear solver/linear solver adaptivity
Fully adaptive computation
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
31 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
31 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Multi-phase compositional flows
Governing partial differential equations
conservation of mass for components
∂t lc + ∇·Φc = qc ,
∀c ∈ C
+ boundary & initial conditions
Constitutive laws
phase pressures – reference pressure – capillary pressure
Pp := P + Pcp (S)
Darcy’s law
vp (Pp , C p ) := −K (∇Pp − ρp (Pp , C p )g)
component fluxes
X
Φc :=
Φp,c ,
Φp,c := νp (Pp , S, C p )Cp,c vp (Pp , C p )
p∈Pc
amount of moles of component
c per unit volume
X
lc := φ
ζp (Pp , C p )Sp Cp,c
p∈Pc
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
32 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Multi-phase compositional flows
Governing partial differential equations
conservation of mass for components
∂t lc + ∇·Φc = qc ,
∀c ∈ C
+ boundary & initial conditions
Constitutive laws
phase pressures – reference pressure – capillary pressure
Pp := P + Pcp (S)
Darcy’s law
vp (Pp , C p ) := −K (∇Pp − ρp (Pp , C p )g)
component fluxes
X
Φc :=
Φp,c ,
Φp,c := νp (Pp , S, C p )Cp,c vp (Pp , C p )
p∈Pc
amount of moles of component
c per unit volume
X
lc := φ
ζp (Pp , C p )Sp Cp,c
p∈Pc
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
32 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Multi-phase compositional flows
Closure algebraic equations
conservation of pore volume:
P
Sp = 1
P
conservation of the quantity of the matter: c∈Cp Cp,c = 1
for all p ∈ P
thermodynamic equilibrium
p∈P
Mathematical issues
coupled system
unsteady, nonlinear
elliptic–parabolic degenerate type
dominant advection
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
33 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Multi-phase compositional flows
Closure algebraic equations
conservation of pore volume:
P
Sp = 1
P
conservation of the quantity of the matter: c∈Cp Cp,c = 1
for all p ∈ P
thermodynamic equilibrium
p∈P
Mathematical issues
coupled system
unsteady, nonlinear
elliptic–parabolic degenerate type
dominant advection
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
33 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
33 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Weak solution
Energy spaces
X := L2 ((0, tF ); H 1 (Ω)),
Y := H 1 ((0, tF ); L2 (Ω))
Definition (Weak solution)
Find (P, (Sp )p∈P , (Cp,c )p∈P,c∈Cp such that
lc ∈ Y
∀c ∈ C,
Pp (P, S) ∈ X
∀p ∈ P,
Φc ∈ [L2 ((0, tF ); L2 (Ω))]d
∀c ∈ C,
Z tF
Z tF
{(∂t lc , ϕ)(t)−(Φc , ∇ϕ)(t)} dt = (qc , ϕ)(t)dt
0
∀ϕ ∈ X , ∀c ∈ C,
0
the initial condition holds,
the algebraic closure equations hold.
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
34 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Weak solution
Energy spaces
X := L2 ((0, tF ); H 1 (Ω)),
Y := H 1 ((0, tF ); L2 (Ω))
Definition (Weak solution)
Find (P, (Sp )p∈P , (Cp,c )p∈P,c∈Cp such that
lc ∈ Y
∀c ∈ C,
Pp (P, S) ∈ X
∀p ∈ P,
Φc ∈ [L2 ((0, tF ); L2 (Ω))]d
∀c ∈ C,
Z tF
Z tF
{(∂t lc , ϕ)(t)−(Φc , ∇ϕ)(t)} dt = (qc , ϕ)(t)dt
0
∀ϕ ∈ X , ∀c ∈ C,
0
the initial condition holds,
the algebraic closure equations hold.
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
34 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Fully implicit cell-centered finite volume scheme
Fully implicit cell-centered finite volumes
Time step n, Newton iteration k , and linear solver iteration i:
n,k ,i
n,k ,i
find XKn,k ,i := (PKn,k ,i , (Sp,K
)p∈P , (Cp,c,K
)p∈P,c∈Cp , K ∈ Thn , s. t.
X n,k ,i
|K | n,k −1 n,k ,i
n−1
n
X
l
+
L
Fc,M,e − |K |qc,K
−
l
+
c,K
h
c,K
c,K
τn
int,n
n,k ,i
= Rc,K
∀c ∈ C, ∀K ∈ Thn ,
e∈EK
where
n,k ,i
Fc,M,e
:=
X
n,k ,i
Fp,c,M,e
,
p∈Pc
n,k ,i
Fp,c,M,e
:= Fp,c,M,e
X ∂Fp,c,M,e n,k −1 ,i
−1 Xh
· XKn,k
−XKn,k
,
Xhn,k −1 +
0
0
n
∂XK 0
n
0
K ∈Th
and
,i
Ln,k
c,K :=
X ∂lc,K
n,k −1 n,k ,i
n,k −1 X
·
X
−
X
.
0
K
K0
∂XKn 0 h
n
0
K ∈Th
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
35 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Fully implicit cell-centered finite volume scheme
Fully implicit cell-centered finite volumes
Time step n, Newton iteration k , and linear solver iteration i:
n,k ,i
n,k ,i
find XKn,k ,i := (PKn,k ,i , (Sp,K
)p∈P , (Cp,c,K
)p∈P,c∈Cp , K ∈ Thn , s. t.
X n,k ,i
|K | n,k −1 n,k ,i
n−1
n
X
l
+
L
Fc,M,e − |K |qc,K
−
l
+
c,K
h
c,K
c,K
τn
int,n
n,k ,i
= Rc,K
∀c ∈ C, ∀K ∈ Thn ,
e∈EK
where
n,k ,i
Fc,M,e
:=
X
n,k ,i
Fp,c,M,e
,
p∈Pc
n,k ,i
Fp,c,M,e
:= Fp,c,M,e
X ∂Fp,c,M,e n,k −1 ,i
−1 Xh
· XKn,k
−XKn,k
,
Xhn,k −1 +
0
0
n
∂XK 0
n
0
K ∈Th
and
,i
Ln,k
c,K :=
X ∂lc,K
n,k −1 n,k ,i
n,k −1 X
·
X
−
X
.
0
K
K0
∂XKn 0 h
n
0
K ∈Th
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
35 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Estimate distinguishing different error components
Theorem (Estimate distinguishing different error components)
Consider
time step n,
linearization step k ,
iterative algebraic solver step i,
and the corresponding approximations. Then
n,k ,i
n,k ,i
n,k ,i
n,k ,i
(dual error + nonconformity)In ≤ ηsp
+ ηtm
+ ηlin
+ ηalg
.
Error components
n,k ,i
ηsp
: spatial discretization
n,k ,i
ηtm
: temporal discretization
n,k ,i
ηlin : linearization
n,k ,i
ηalg
: algebraic solver
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
36 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Estimate distinguishing different error components
Theorem (Estimate distinguishing different error components)
Consider
time step n,
linearization step k ,
iterative algebraic solver step i,
and the corresponding approximations. Then
n,k ,i
n,k ,i
n,k ,i
n,k ,i
(dual error + nonconformity)In ≤ ηsp
+ ηtm
+ ηlin
+ ηalg
.
Error components
n,k ,i
ηsp
: spatial discretization
n,k ,i
ηtm
: temporal discretization
n,k ,i
ηlin : linearization
n,k ,i
ηalg
: algebraic solver
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
36 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
36 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Test case and numerical setting
Test case
two-spot setting
two phases and three components
homogeneous/heterogeneous permeability distribution
Discretization and resolution
fully implicit cell-centered finite volumes
Newton linearization
GMRes with ILU0 preconditioning algebraic solver
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
37 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Test case and numerical setting
Test case
two-spot setting
two phases and three components
homogeneous/heterogeneous permeability distribution
Discretization and resolution
fully implicit cell-centered finite volumes
Newton linearization
GMRes with ILU0 preconditioning algebraic solver
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
37 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
Estimators and stopping criteria
space
total
time
linearization
algebraic
Error component estimator
Error component estimator
104
102
10−1
adaptive stopping criterion
10−4
10−7
classical stopping criterion
10−10
0
5
10 15 20 25 30
PETSC-GMRes iteration
35
Estimators w.r.t. GMRes iterations
Martin Vohralík
102
adaptive stopping criterion
100
10−2
10−4
classical stopping criterion
10−6
1
2
5
3
4
Newton iteration
6
Estimators w.r.t. Newton iterations
Adaptive inexact Newton methods and multi-phase flows
38 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
classical
adaptive
Newton iterations
8
Cumulated number of Newton iterations
Newton iterations
classical
adaptive
1,500
6
1758 iterations
1,000
4
2
0
2
4
Time
Per time step
Martin Vohralík
6
8
·106
500
853 iterations
0
0
0.5
1
1.5
Time
2
·108
Cumulated
Adaptive inexact Newton methods and multi-phase flows
39 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Estimate Application and numerical results
GMRes iterations
30
classical
adaptive
20
10
0
0
2
4
6
Time/Newton step
Per time and Newton step
Martin Vohralík
8
·106
Cumulated number of GMRes iterations
GMRes iterations
·104
classical
adaptive
4
54067 iterations
2
5110 iterations
0
0
0.5
1
1.5
Time
2
·108
Cumulated
Adaptive inexact Newton methods and multi-phase flows
40 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Outline
1
Introduction
2
Adaptive inexact Newton method
A posteriori error estimate
Application and numerical results
3
Two-phase flow
A posteriori error estimate
Application and numerical results
4
Multi-phase multi-component flow
A posteriori error estimate
Application and numerical results
5
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
40 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Conclusions and future directions
Entire adaptivity
only a necessary number of algebraic/linearization
solver iterations
“online decisions”: algebraic step / linearization step /
space mesh refinement / time step modification
important computational savings
guaranteed and robust a posteriori error estimates
Future directions
other coupled nonlinear systems
convergence and optimality
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
41 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Conclusions and future directions
Entire adaptivity
only a necessary number of algebraic/linearization
solver iterations
“online decisions”: algebraic step / linearization step /
space mesh refinement / time step modification
important computational savings
guaranteed and robust a posteriori error estimates
Future directions
other coupled nonlinear systems
convergence and optimality
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
41 / 42
I Ad. inex. Newton Two-phase Multi-phase-comp. C
Bibliography
E RN A., VOHRALÍK M., Adaptive inexact Newton methods
with a posteriori stopping criteria for nonlinear diffusion
PDEs, SIAM J. Sci. Comput. 35 (2013), A1761–A1791.
VOHRALÍK M., W HEELER M. F., A posteriori error
estimates, stopping criteria, and adaptivity for two-phase
flows, Comput. Geosci. 17 (2013), 789–812.
C ANCÈS C., P OP I. S., VOHRALÍK M., An a posteriori error
estimate for vertex-centered finite volume discretizations of
immiscible incompressible two-phase flow, Math. Comp.
83 (2014), 153–188.
D I P IETRO D. A., F LAURAUD E., VOHRALÍK M., AND
YOUSEF S., A posteriori error estimates, stopping criteria,
and adaptivity for multiphase compositional Darcy flows in
porous media, J. Comput. Phys. (2014),
DOI 10.1016/j.jcp.2014.06.061.
Thank you for your attention!
Martin Vohralík
Adaptive inexact Newton methods and multi-phase flows
42 / 42