Full adaptivity for unsteady nonlinear problems
Martin Vohralík, INRIA Paris-Rocquencourt
Lecture IV/IV
IIT Bombay, July 13–17, 2015
I Stefan problem Two-phase flow R&B
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
2 / 35
I Stefan problem Two-phase flow R&B
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
2 / 35
I Stefan problem Two-phase flow R&B
Full adaptivity for unsteady nonlinear problems
Real (porous media) flows
systems of PDEs
nonlinear (degenerate)
unsteady
⇒ difficult numerical approximation, large troublesome
systems of nonlinear algebraic equations
Goals
derive fully computable a posteriori error upper bounds
distinguish different error components
Full adaptivity
time step choice & mesh adaptivity
stopping criteria for regularization and linear and
nonlinear solvers
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
3 / 35
I Stefan problem Two-phase flow R&B
Full adaptivity for unsteady nonlinear problems
Real (porous media) flows
systems of PDEs
nonlinear (degenerate)
unsteady
⇒ difficult numerical approximation, large troublesome
systems of nonlinear algebraic equations
Goals
derive fully computable a posteriori error upper bounds
distinguish different error components
Full adaptivity
time step choice & mesh adaptivity
stopping criteria for regularization and linear and
nonlinear solvers
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
3 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
3 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
The Stefan problem
The Stefan problem
∂t u − ∆β(u) = f
u(·, 0) = u0
β(u) = 0
in Ω × (0, T ),
in Ω,
on ∂Ω × (0, T )
Nomenclature
u enthalpy, β(u) temperature
β: Lβ -Lipschitz continuous, β(s) = 0 in (0, 1), strictly
increasing otherwise
phase change, degenerate parabolic problem
u0 ∈ L2 (Ω), f ∈ L2 (0, T ; L2 (Ω))
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
4 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
The Stefan problem
The Stefan problem
∂t u − ∆β(u) = f
u(·, 0) = u0
β(u) = 0
in Ω × (0, T ),
in Ω,
on ∂Ω × (0, T )
Nomenclature
u enthalpy, β(u) temperature
β: Lβ -Lipschitz continuous, β(s) = 0 in (0, 1), strictly
increasing otherwise
phase change, degenerate parabolic problem
u0 ∈ L2 (Ω), f ∈ L2 (0, T ; L2 (Ω))
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
4 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Numerical practice: regularization
Regularization of β, parameter 1
0.5
−1
−0.5
β(u), β (u)
0.5
1u
1.5
2
−0.5
−1
β (·) smooth and strictly increasing
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
5 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Setting
Functional spaces
Z := H 1 (0, T ; H −1 (Ω))
X := L2 (0, T ; H01 (Ω)),
Weak formulation
u∈Z
with β(u) ∈ X ,
u(·, 0) = u0
in Ω,
h∂t u, ϕi(s) + (∇β(u), ∇ϕ)(s) = (f , ϕ)(s) ∀ϕ ∈ H01 (Ω), s ∈ (0, T )
Approximation (conforming, with linearization & regularization)
,k
,k
,k
uhτ
∈ Z,
∂t uhτ
∈ L2 (0, T ; L2 (Ω)),
β(uhτ
) ∈ X,
,k
uhτ |In is affine in time on In
∀1 ≤ n ≤ N
,k
Residual R(uhτ
) ∈ X 0 and its dual norm, ϕ ∈ X
Z Tn
o
,k
,k
,k
hR(uhτ
), ϕiX 0 ,X :=
h∂t (u −uhτ
), ϕi+(∇(β(u)−β(uhτ
)), ∇ϕ) (s)ds,
0
,k
kR(uhτ )kX 0
M. Vohralík
:=
sup
ϕ∈X , kϕkX =1
,k
hR(uhτ
), ϕiX 0 ,X
IV/IV Full adaptivity for unsteady nonlinear problems
6 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Setting
Functional spaces
Z := H 1 (0, T ; H −1 (Ω))
X := L2 (0, T ; H01 (Ω)),
Weak formulation
u∈Z
with β(u) ∈ X ,
u(·, 0) = u0
in Ω,
h∂t u, ϕi(s) + (∇β(u), ∇ϕ)(s) = (f , ϕ)(s) ∀ϕ ∈ H01 (Ω), s ∈ (0, T )
Approximation (conforming, with linearization & regularization)
,k
,k
,k
uhτ
∈ Z,
∂t uhτ
∈ L2 (0, T ; L2 (Ω)),
β(uhτ
) ∈ X,
,k
uhτ |In is affine in time on In
∀1 ≤ n ≤ N
,k
Residual R(uhτ
) ∈ X 0 and its dual norm, ϕ ∈ X
Z Tn
o
,k
,k
,k
hR(uhτ
), ϕiX 0 ,X :=
h∂t (u −uhτ
), ϕi+(∇(β(u)−β(uhτ
)), ∇ϕ) (s)ds,
0
,k
kR(uhτ )kX 0
M. Vohralík
:=
sup
,k
hR(uhτ
), ϕiX 0 ,X
ϕ∈X , kϕkX =1
IV/IV Full adaptivity for unsteady nonlinear problems
6 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Setting
Functional spaces
Z := H 1 (0, T ; H −1 (Ω))
X := L2 (0, T ; H01 (Ω)),
Weak formulation
u∈Z
with β(u) ∈ X ,
u(·, 0) = u0
in Ω,
h∂t u, ϕi(s) + (∇β(u), ∇ϕ)(s) = (f , ϕ)(s) ∀ϕ ∈ H01 (Ω), s ∈ (0, T )
Approximation (conforming, with linearization & regularization)
,k
,k
,k
uhτ
∈ Z,
∂t uhτ
∈ L2 (0, T ; L2 (Ω)),
β(uhτ
) ∈ X,
,k
uhτ |In is affine in time on In
∀1 ≤ n ≤ N
,k
Residual R(uhτ
) ∈ X 0 and its dual norm, ϕ ∈ X
Z Tn
o
,k
,k
,k
h∂t (u −uhτ
), ϕi+(∇(β(u)−β(uhτ
)), ∇ϕ) (s)ds,
hR(uhτ
), ϕiX 0 ,X :=
0
,k
kR(uhτ )kX 0
M. Vohralík
:=
sup
,k
hR(uhτ
), ϕiX 0 ,X
ϕ∈X , kϕkX =1
IV/IV Full adaptivity for unsteady nonlinear problems
6 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Setting
Functional spaces
Z := H 1 (0, T ; H −1 (Ω))
X := L2 (0, T ; H01 (Ω)),
Weak formulation
u∈Z
with β(u) ∈ X ,
u(·, 0) = u0
in Ω,
h∂t u, ϕi(s) + (∇β(u), ∇ϕ)(s) = (f , ϕ)(s) ∀ϕ ∈ H01 (Ω), s ∈ (0, T )
Approximation (conforming, with linearization & regularization)
,k
,k
,k
uhτ
∈ Z,
∂t uhτ
∈ L2 (0, T ; L2 (Ω)),
β(uhτ
) ∈ X,
,k
uhτ |In is affine in time on In
∀1 ≤ n ≤ N
,k
Residual R(uhτ
) ∈ X 0 and its dual norm, ϕ ∈ X
Z Tn
o
,k
,k
,k
h∂t (u −uhτ
), ϕi+(∇(β(u)−β(uhτ
)), ∇ϕ) (s)ds,
hR(uhτ
), ϕiX 0 ,X :=
0
,k
kR(uhτ )kX 0
M. Vohralík
:=
sup
,k
hR(uhτ
), ϕiX 0 ,X
ϕ∈X , kϕkX =1
IV/IV Full adaptivity for unsteady nonlinear problems
6 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Time-localization of the dual norm of the residual
Time interval In
Xn :=L2 (In ; H01 (Ω))
Z
,k
,k
kR(uhτ )kXn0 :=
sup
{h∂t (u − uhτ
), ϕi
ϕ∈Xn , kϕkXn =1 In
,k
+ (∇β(u) − ∇β(uhτ
), ∇ϕ)}(s) ds
L2 in time:
,k 2
kR(uhτ
)kX 0 =
M. Vohralík
X
1≤n≤N
,k 2
kR(uhτ
)kXn0
IV/IV Full adaptivity for unsteady nonlinear problems
7 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Time-localization of the dual norm of the residual
Time interval In
Xn :=L2 (In ; H01 (Ω))
Z
,k
,k
kR(uhτ )kXn0 :=
sup
{h∂t (u − uhτ
), ϕi
ϕ∈Xn , kϕkXn =1 In
,k
+ (∇β(u) − ∇β(uhτ
), ∇ϕ)}(s) ds
L2 in time:
,k 2
kR(uhτ
)kX 0 =
M. Vohralík
X
1≤n≤N
,k 2
kR(uhτ
)kXn0
IV/IV Full adaptivity for unsteady nonlinear problems
7 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
7 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Estimate distinguishing different error components
Assumption A (Equilibrated flux reconstruction)
For all n ≥ 1, k ≥ 1, and > 0, there exists σhn,,k ∈ H(div; Ω) s.t.
,k
(∇·σhn,,k , 1)K = (f |In , 1)K − (∂t uhτ
|In , 1)K
∀K ∈ T n .
Theorem (An estimate distinguishing the error components)
Let Assumption A hold. Then, for any n ≥ 1, k ≥ 1, and > 0,
,k
n,,k
n,,k
n,,k
n,,k
kR(uhτ
)kXn0 ≤ ηsp
+ ηtm
+ ηreg
+ ηlin
.
2
X n,,k
n,,k 2
(ηsp
) := τ n
ηE,K + k∇β ,k −1 (uhn,,k ) + σhn,,k kK ,
∈T n
Z KX
n,,k 2
,k
(ηtm
) :=
k∇(β(uhτ
)(t) − β(uhn,,k ))k2K dt,
In K ∈T n
X
n,,k 2
(ηreg
) := τ n
k∇(β(uhn,,k ) − β (uhn,,k ))k2K ,
KX
∈T n
n,,k 2
n
(ηlin ) := τ
k∇(β (uhn,,k ) − β ,k −1 (uhn,,k ))k2K
K ∈T n
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
8 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Estimate distinguishing different error components
Assumption A (Equilibrated flux reconstruction)
For all n ≥ 1, k ≥ 1, and > 0, there exists σhn,,k ∈ H(div; Ω) s.t.
,k
(∇·σhn,,k , 1)K = (f |In , 1)K − (∂t uhτ
|In , 1)K
∀K ∈ T n .
Theorem (An estimate distinguishing the error components)
Let Assumption A hold. Then, for any n ≥ 1, k ≥ 1, and > 0,
,k
n,,k
n,,k
n,,k
n,,k
kR(uhτ
)kXn0 ≤ ηsp
+ ηtm
+ ηreg
+ ηlin
.
2
X n,,k
n,,k 2
(ηsp
) := τ n
ηE,K + k∇β ,k −1 (uhn,,k ) + σhn,,k kK ,
∈T n
Z KX
n,,k 2
,k
(ηtm
) :=
k∇(β(uhτ
)(t) − β(uhn,,k ))k2K dt,
In K ∈T n
X
n,,k 2
(ηreg
) := τ n
k∇(β(uhn,,k ) − β (uhn,,k ))k2K ,
KX
∈T n
n,,k 2
n
(ηlin ) := τ
k∇(β (uhn,,k ) − β ,k −1 (uhn,,k ))k2K
K ∈T n
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
8 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Estimate distinguishing different error components
Assumption A (Equilibrated flux reconstruction)
For all n ≥ 1, k ≥ 1, and > 0, there exists σhn,,k ∈ H(div; Ω) s.t.
,k
(∇·σhn,,k , 1)K = (f |In , 1)K − (∂t uhτ
|In , 1)K
∀K ∈ T n .
Theorem (An estimate distinguishing the error components)
Let Assumption A hold. Then, for any n ≥ 1, k ≥ 1, and > 0,
,k
n,,k
n,,k
n,,k
n,,k
kR(uhτ
)kXn0 ≤ ηsp
+ ηtm
+ ηreg
+ ηlin
.
2
X n,,k
n,,k 2
(ηsp
) := τ n
ηE,K + k∇β ,k −1 (uhn,,k ) + σhn,,k kK ,
∈T n
Z KX
n,,k 2
,k
(ηtm
) :=
k∇(β(uhτ
)(t) − β(uhn,,k ))k2K dt,
In K ∈T n
X
n,,k 2
(ηreg
) := τ n
k∇(β(uhn,,k ) − β (uhn,,k ))k2K ,
KX
∈T n
n,,k 2
n
(ηlin ) := τ
k∇(β (uhn,,k ) − β ,k −1 (uhn,,k ))k2K
K ∈T n
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
8 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
8 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Efficiency assumptions
Assumption B (Piecewise polynomials, data, and meshes)
The approximations and the data f and u0 are piecewise
polynomial. The meshes are shape-regular.
Residual estimators
2
X
n,,k
ηres,1
:= τ n
K ∈T n−1,n
n,,k
ηres,2
2
:= τ n
,k
hK2 kf |In − ∂t uhτ
|In + ∆β ,k −1 (uhn,,k )k2K ,
X
e∈E int,n−1,n
he k[[∇β ,k −1 (uhn,,k )·ne ]]k2e
Assumption C (Approximation property)
For all 1 ≤ n ≤ N, there holds
τn
X
k∇β ,k −1 (uhn,,k )+σhn,,k k2K ≤ C
n,,k
ηres,1
2
2 n,,k
+ ηres,2
.
K ∈T n−1,n
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
9 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Efficiency assumptions
Assumption B (Piecewise polynomials, data, and meshes)
The approximations and the data f and u0 are piecewise
polynomial. The meshes are shape-regular.
Residual estimators
2
X
n,,k
ηres,1
:= τ n
K ∈T n−1,n
n,,k
ηres,2
2
:= τ n
,k
hK2 kf |In − ∂t uhτ
|In + ∆β ,k −1 (uhn,,k )k2K ,
X
e∈E int,n−1,n
he k[[∇β ,k −1 (uhn,,k )·ne ]]k2e
Assumption C (Approximation property)
For all 1 ≤ n ≤ N, there holds
τn
X
k∇β ,k −1 (uhn,,k )+σhn,,k k2K ≤ C
n,,k
ηres,1
2
2 n,,k
+ ηres,2
.
K ∈T n−1,n
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
9 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Efficiency assumptions
Assumption B (Piecewise polynomials, data, and meshes)
The approximations and the data f and u0 are piecewise
polynomial. The meshes are shape-regular.
Residual estimators
2
X
n,,k
ηres,1
:= τ n
K ∈T n−1,n
n,,k
ηres,2
2
:= τ n
,k
hK2 kf |In − ∂t uhτ
|In + ∆β ,k −1 (uhn,,k )k2K ,
X
e∈E int,n−1,n
he k[[∇β ,k −1 (uhn,,k )·ne ]]k2e
Assumption C (Approximation property)
For all 1 ≤ n ≤ N, there holds
τn
X
k∇β ,k −1 (uhn,,k )+σhn,,k k2K ≤ C
n,,k
ηres,1
2
2 n,,k
+ ηres,2
.
K ∈T n−1,n
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
9 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Efficiency assumptions
Theorem (Efficiency)
Let, for all 1 ≤ n ≤ N, the stopping and balancing criteria be
satisfied with the parameters small enough. Let Assumptions B
and C hold. Then
n,,k
n,,k
n,,k
n,,k
ηsp
+ ηtm
+ ηreg
+ ηlin
≤ CkR(uhn,,k )kXn0 .
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
10 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
10 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Relation residual–energy norm
Energy estimate (by the Gronwall lemma)
Lβ
Lβ
ku − uhτ k2X 0 +
k(u − uhτ )(·, T )k2H −1 (Ω) + kβ(u) − β(uhτ )k2QT
2
2
Lβ
≤ (2eT − 1) kR(uhτ )k2X 0 + k(u − uhτ )(·, 0)k2H −1 (Ω)
2
Theorem (Temperature and enthalpy errors, tight Gronwall)
Let uhτ ∈ Z such that β(uhτ ) ∈ X be arbitrary. There holds
Lβ
Lβ
ku − uhτ k2X 0 + k(u − uhτ )(·, T )k2H −1 (Ω) +kβ(u) − β(uhτ )k2QT
2
2
Z T
Z t
+2
kβ(u) − β(uhτ )k2Qt +
kβ(u) − β(uhτ )k2Qs et−s ds dt
0
(0
Lβ
≤
(2eT − 1)k(u − uhτ )(·, 0)k2H −1 (Ω) + kR(uhτ )k2X 0
2
)
Z T
Z t
2
2 t−s
kR(uhτ )kXs0 e ds dt .
+2
kR(uhτ )kX 0 +
0
M. Vohralík
t
0
IV/IV Full adaptivity for unsteady nonlinear problems
11 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Relation residual–energy norm
Energy estimate (by the Gronwall lemma)
Lβ
Lβ
ku − uhτ k2X 0 +
k(u − uhτ )(·, T )k2H −1 (Ω) + kβ(u) − β(uhτ )k2QT
2
2
Lβ
≤ (2eT − 1) kR(uhτ )k2X 0 + k(u − uhτ )(·, 0)k2H −1 (Ω)
2
Theorem (Temperature and enthalpy errors, tight Gronwall)
Let uhτ ∈ Z such that β(uhτ ) ∈ X be arbitrary. There holds
Lβ
Lβ
ku − uhτ k2X 0 + k(u − uhτ )(·, T )k2H −1 (Ω) +kβ(u) − β(uhτ )k2QT
2
2
Z T
Z t
+2
kβ(u) − β(uhτ )k2Qt +
kβ(u) − β(uhτ )k2Qs et−s ds dt
0
(0
Lβ
≤
(2eT − 1)k(u − uhτ )(·, 0)k2H −1 (Ω) + kR(uhτ )k2X 0
2
)
Z T
Z t
2
2 t−s
kR(uhτ )kXs0 e ds dt .
+2
kR(uhτ )kX 0 +
0
M. Vohralík
t
0
IV/IV Full adaptivity for unsteady nonlinear problems
11 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
11 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Regularization stopping criterion
Regularization stopping criterion
Error components estimators
n,,k
n,,k
n,,k
ηreg
≤ Γreg ηsp
+ ηtm
space
time
regularization
10−1
10−2
10−3
10−4
10−5
101
M. Vohralík
102
−1
103
104
IV/IV Full adaptivity for unsteady nonlinear problems
12 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Equilibrating time and space errors
Equilibrating time and space errors
space
time
10−0.8
10−1.8
Overall error estimators
Error components estimators
n,,k
n,,k
n,,k
γtm ηsp
≤ ηtm
≤ Γtm ηsp
10−1.9
10−1
10−2
10−2.1
10−1.2
105
104
Total number of space–time unknowns
M. Vohralík
space over-ref.
time over-ref.
equilibrating
105
104
Total number of space–time unknowns
IV/IV Full adaptivity for unsteady nonlinear problems
13 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Error/error estimate
Error and estimate (dual norm of the residual)
100
err. unif.
err. ad.
est. unif.
est. ad.
10−1
105
107
104
106
Total number of space–time unknowns
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
14 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Effectivity indices (dual norm of the residual)
Effectivity index
4
effectivity unif.
effectivity ad.
3
2
105
107
106
104
Total number of space–time unknowns
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
15 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Error and estimate (energy norm)
Error/error estimate
err. unif.
err. ad.
est. unif.
est. ad.
100
10−1
105
107
104
106
Total number of space–time unknowns
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
16 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Effectivity indices (energy norm)
Effectivity index
12
effectivity unif.
effectivity ad.
10
8
105
107
104
106
Total number of space–time unknowns
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
17 / 35
I Stefan problem Two-phase flow R&B
Dual norm estimate Efficiency Energy estimate Numerics
Actual and estimated error distribution
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
18 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
18 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Two-phase flow in porous media
Two-phase flow in porous media
∂t (φsα ) + ∇·uα = qα ,
−λα (sw )K(∇pα + ρα g∇z) = uα ,
sn + sw = 1,
α ∈ {n, w},
α ∈ {n, w},
pn − pw = pc (sw )
+ boundary & initial conditions
Mathematical issues
coupled system
unsteady, nonlinear
elliptic–degenerate parabolic type
dominant advection
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
19 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Two-phase flow in porous media
Two-phase flow in porous media
∂t (φsα ) + ∇·uα = qα ,
−λα (sw )K(∇pα + ρα g∇z) = uα ,
sn + sw = 1,
α ∈ {n, w},
α ∈ {n, w},
pn − pw = pc (sw )
+ boundary & initial conditions
Mathematical issues
coupled system
unsteady, nonlinear
elliptic–degenerate parabolic type
dominant advection
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
19 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Global and complementary pressures
Global pressure
Z
p(sw , pw ) := pw +
0
sw
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
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
20 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Global and complementary pressures
Global pressure
Z
p(sw , pw ) := pw +
0
sw
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
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
20 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Global and complementary pressures
Global pressure
Z
p(sw , pw ) := pw +
0
sw
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
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
20 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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}.
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
21 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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}.
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
21 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
21 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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∈Xn , kϕkXn=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
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
22 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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∈Xn , kϕkXn=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
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
22 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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∈Xn , kϕkXn=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
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
22 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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∈Xn , kϕkXn=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
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
22 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
22 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Distinguishing the error components
Theorem (Distinguishing the error components)
Consider a vertex-centered finite volume / backward Euler
approximation and Newton linearization. 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 :
M. Vohralík
spatial discretization
temporal discretization
linearization
algebraic solver
IV/IV Full adaptivity for unsteady nonlinear problems
23 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Distinguishing the error components
Theorem (Distinguishing the error components)
Consider a vertex-centered finite volume / backward Euler
approximation and Newton linearization. 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 :
M. Vohralík
spatial discretization
temporal discretization
linearization
algebraic solver
IV/IV Full adaptivity for unsteady nonlinear problems
23 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Full adaptivity
Full adaptivity
only a necessary number of algebraic/linearization
solver iterations
adaptive regularization, model adaptation, adaptive
choice of the scheme parameters
“online decisions”: algebraic step / linearization step /
space mesh refinement / time step modification
important computational savings
guaranteed and robust a posteriori error estimates
Not treated for the moment
convergence and optimality
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
24 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Full adaptivity
Full adaptivity
only a necessary number of algebraic/linearization
solver iterations
adaptive regularization, model adaptation, adaptive
choice of the scheme parameters
“online decisions”: algebraic step / linearization step /
space mesh refinement / time step modification
important computational savings
guaranteed and robust a posteriori error estimates
Not treated for the moment
convergence and optimality
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
24 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
24 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Iteratively coupled vertex-centered finite volumes
Vertex-centered finite volumes
simplicial meshes T n , dual meshes Dn
saturations & pressures continuous and pw affine on T n
Tn
Dn
Implicit pressure equation on step k
n,k −1
n,k −1 n,k
·nD
− λw (sw,h
) + λn (sw,h
) K∇pw,h
n,k −1
n,k −1
+λn (sw,h
)K∇pc (sw,h
)·nD , 1 ∂D\∂Ω = 0
∀D ∈ Dint,n
Explicit saturation equation on step k
n,k
sw,D
:=
τn
n,k −1
n,k
n−1
λw (sw,h
)K∇pw,h
·nD , 1 ∂D\∂Ω + sw,D
φ|D|
M. Vohralík
∀D ∈ Dint,n
IV/IV Full adaptivity for unsteady nonlinear problems
25 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Iteratively coupled vertex-centered finite volumes
Vertex-centered finite volumes
simplicial meshes T n , dual meshes Dn
saturations & pressures continuous and pw affine on T n
Tn
Dn
Implicit pressure equation on step k
n,k −1
n,k −1 n,k
·nD
− λw (sw,h
) + λn (sw,h
) K∇pw,h
n,k −1
n,k −1
+λn (sw,h
)K∇pc (sw,h
)·nD , 1 ∂D\∂Ω = 0
∀D ∈ Dint,n
Explicit saturation equation on step k
n,k
sw,D
:=
τn
n,k −1
n,k
n−1
λw (sw,h
)K∇pw,h
·nD , 1 ∂D\∂Ω + sw,D
φ|D|
M. Vohralík
∀D ∈ Dint,n
IV/IV Full adaptivity for unsteady nonlinear problems
25 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Iteratively coupled vertex-centered finite volumes
Vertex-centered finite volumes
simplicial meshes T n , dual meshes Dn
saturations & pressures continuous and pw affine on T n
Tn
Dn
Implicit pressure equation on step k
n,k −1
n,k −1 n,k
·nD
− λw (sw,h
) + λn (sw,h
) K∇pw,h
n,k −1
n,k −1
+λn (sw,h
)K∇pc (sw,h
)·nD , 1 ∂D\∂Ω = 0
∀D ∈ Dint,n
Explicit saturation equation on step k
n,k
sw,D
:=
τn
n,k −1
n,k
n−1
λw (sw,h
)K∇pw,h
·nD , 1 ∂D\∂Ω + sw,D
φ|D|
M. Vohralík
∀D ∈ Dint,n
IV/IV Full adaptivity for unsteady nonlinear problems
25 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Linearization and algebraic solution
Iterative coupling step k and algebraic step i
n,k −1
n,k ,i
n,k −1 − λw (sw,h
) + λn (sw,h
·nD
) K∇pw,h
n,k −1
n,k −1
n,k ,i
+λn (sw,h )K∇pc (sw,h )·nD , 1 ∂D\∂Ω = −Rt,D
n,k ,i
sw,D
:=
M. Vohralík
∀D ∈ Dint,n
τn
n,k −1
n,k ,i
n−1
λw (sw,h
)K∇pw,h
·nD , 1 ∂D\∂Ω + sw,D
φ|D|
IV/IV Full adaptivity for unsteady nonlinear problems
26 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Linearization and algebraic solution
Iterative coupling step k and algebraic step i
n,k −1
n,k ,i
n,k −1 − λw (sw,h
) + λn (sw,h
·nD
) K∇pw,h
n,k −1
n,k −1
n,k ,i
+λn (sw,h )K∇pc (sw,h )·nD , 1 ∂D\∂Ω = −Rt,D
n,k ,i
sw,D
:=
M. Vohralík
∀D ∈ Dint,n
τn
n,k −1
n,k ,i
n−1
λw (sw,h
)K∇pw,h
·nD , 1 ∂D\∂Ω + sw,D
φ|D|
IV/IV Full adaptivity for unsteady nonlinear problems
26 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Flux reconstructions
Total velocities reconstructions
n,k ,i
,i
n,k ,i
n,k ,i ·nD
λw (sw,h
) + λn (sw,h
) K∇pw,h
(dn,k
t,h ·nD , 1)e := −
n,k ,i
n,k ,i
+ λn (sw,h )K∇pc (sw,h )·nD , 1 e ,
n,k ,i
n,k ,i
n,k ,i
n,k −1
n,k −1 ·nD
((dt,h + lt,h )·nD , 1)e := − λw (sw,h
) + λn (sw,h
) K∇pw,h
n,k −1
n,k −1
+ λn (sw,h )K∇pc (sw,h )·nD , 1 e ,
,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 )
Phases velocities reconstructions
n,k ,i
,i
n,k ,i
(dn,k
w,h ·nD , 1)e := − λw (sw,h )K∇pw,h ·nD , 1 e ,
,i
n,k ,i
n,k −1
n,k ,i
((dn,k
w,h + lw,h )·nD , 1)e := − λw (sw,h )K∇pw,h ·nD , 1 e ,
,i
an,k
w,h := 0
,i
n,k ,i
n,k ,i n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
dn,k
n,h := dt,h − dw,h , ln,h := lt,h − lw,h , an,h := at,h − aw,h
Equilibrated fluxes
n,k ,i
,i
n,k ,i
n,k ,i
σ·,h
:= dn,k
·,h + l·,h + a·,h
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
27 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Flux reconstructions
Total velocities reconstructions
n,k ,i
,i
n,k ,i
n,k ,i ·nD
λw (sw,h
) + λn (sw,h
) K∇pw,h
(dn,k
t,h ·nD , 1)e := −
n,k ,i
n,k ,i
+ λn (sw,h )K∇pc (sw,h )·nD , 1 e ,
n,k ,i
n,k ,i
n,k ,i
n,k −1
n,k −1 ·nD
((dt,h + lt,h )·nD , 1)e := − λw (sw,h
) + λn (sw,h
) K∇pw,h
n,k −1
n,k −1
+ λn (sw,h )K∇pc (sw,h )·nD , 1 e ,
,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 )
Phases velocities reconstructions
,i
n,k ,i
n,k ,i
(dn,k
w,h ·nD , 1)e := − λw (sw,h )K∇pw,h ·nD , 1 e ,
,i
n,k ,i
n,k −1
n,k ,i
((dn,k
w,h + lw,h )·nD , 1)e := − λw (sw,h )K∇pw,h ·nD , 1 e ,
,i
an,k
w,h := 0
,i
n,k ,i
n,k ,i n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
dn,k
n,h := dt,h − dw,h , ln,h := lt,h − lw,h , an,h := at,h − aw,h
Equilibrated fluxes
n,k ,i
,i
n,k ,i
n,k ,i
σ·,h
:= dn,k
·,h + l·,h + a·,h
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
27 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Flux reconstructions
Total velocities reconstructions
n,k ,i
,i
n,k ,i
n,k ,i ·nD
λw (sw,h
) + λn (sw,h
) K∇pw,h
(dn,k
t,h ·nD , 1)e := −
n,k ,i
n,k ,i
+ λn (sw,h )K∇pc (sw,h )·nD , 1 e ,
n,k ,i
n,k ,i
n,k ,i
n,k −1
n,k −1 ·nD
((dt,h + lt,h )·nD , 1)e := − λw (sw,h
) + λn (sw,h
) K∇pw,h
n,k −1
n,k −1
+ λn (sw,h )K∇pc (sw,h )·nD , 1 e ,
,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 )
Phases velocities reconstructions
,i
n,k ,i
n,k ,i
(dn,k
w,h ·nD , 1)e := − λw (sw,h )K∇pw,h ·nD , 1 e ,
,i
n,k ,i
n,k −1
n,k ,i
((dn,k
w,h + lw,h )·nD , 1)e := − λw (sw,h )K∇pw,h ·nD , 1 e ,
,i
an,k
w,h := 0
,i
n,k ,i
n,k ,i n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
dn,k
n,h := dt,h − dw,h , ln,h := lt,h − lw,h , an,h := at,h − aw,h
Equilibrated fluxes
n,k ,i
,i
n,k ,i
n,k ,i
σ·,h
:= dn,k
·,h + l·,h + a·,h
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
27 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Flux reconstructions
Total velocities reconstructions
n,k ,i
,i
n,k ,i
n,k ,i ·nD
λw (sw,h
) + λn (sw,h
) K∇pw,h
(dn,k
t,h ·nD , 1)e := −
n,k ,i
n,k ,i
+ λn (sw,h )K∇pc (sw,h )·nD , 1 e ,
n,k ,i
n,k ,i
n,k ,i
n,k −1
n,k −1 ·nD
((dt,h + lt,h )·nD , 1)e := − λw (sw,h
) + λn (sw,h
) K∇pw,h
n,k −1
n,k −1
+ λn (sw,h )K∇pc (sw,h )·nD , 1 e ,
,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 )
Phases velocities reconstructions
,i
n,k ,i
n,k ,i
(dn,k
w,h ·nD , 1)e := − λw (sw,h )K∇pw,h ·nD , 1 e ,
,i
n,k ,i
n,k −1
n,k ,i
((dn,k
w,h + lw,h )·nD , 1)e := − λw (sw,h )K∇pw,h ·nD , 1 e ,
,i
an,k
w,h := 0
,i
n,k ,i
n,k ,i n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
n,k ,i
dn,k
n,h := dt,h − dw,h , ln,h := lt,h − lw,h , an,h := at,h − aw,h
Equilibrated fluxes
n,k ,i
,i
n,k ,i
n,k ,i
σ·,h
:= dn,k
·,h + l·,h + a·,h
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
27 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
27 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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 :=
M. Vohralík
sw − srw
1 − srw − srn
IV/IV Full adaptivity for unsteady nonlinear problems
28 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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 :=
M. Vohralík
sw − srw
1 − srw − srn
IV/IV Full adaptivity for unsteady nonlinear problems
28 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
29 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
29 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
29 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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
M. 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
Estimators in function of
iterative coupling iterations
IV/IV Full adaptivity for unsteady nonlinear problems
30 / 35
10
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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
2.2
6
x 10
GMRes relative residual
M. Vohralík
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
IV/IV Full adaptivity for unsteady nonlinear problems
31 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications 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
M. 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
IV/IV Full adaptivity for unsteady nonlinear problems
32 / 35
I Stefan problem Two-phase flow R&B
Estimate Full adaptivity Applications Numerical results
Space/time/nonlinear solver/linear solver adaptivity
Fully adaptive computation
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
33 / 35
I Stefan problem Two-phase flow R&B
Outline
1
Introduction
2
The Stefan problem
Dual norm a posteriori estimate and adaptivity
Efficiency
Energy error a posteriori estimate
Numerical results
3
Two-phase flow
A posteriori error estimate
Full adaptivity
Applications
Numerical results
4
References and bibliography
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
33 / 35
I Stefan problem Two-phase flow R&B
Previous results
Nonlinear unsteady problems
Eriksson and Johnson (1995), L∞ (0, T ; L2 (Ω)) estimates
exploiting stability of the adjoint problem
Gallimard, Ladevèze, Pelle (1997), const. rel. estimates
Verfürth (1998), framework for energy control, efficiency
Ohlberger (2001), non-energy estimates, hyperbolic limit
Akrivis, Makridakis, and Nochetto (2006), higher-order
temporal discretizations
Degenerate parabolic problems
Nochetto, Schmidt, Verdi (2000), Stefan problem
Dolejší, Ern, Vohralík (2013), Richards problem
(advection-dominated), robustness in a space–time dual
mesh-dependent norm
Two-phase flows
Chen and Ewing (2003), mesh adaptivity
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
34 / 35
I Stefan problem Two-phase flow R&B
Previous results
Nonlinear unsteady problems
Eriksson and Johnson (1995), L∞ (0, T ; L2 (Ω)) estimates
exploiting stability of the adjoint problem
Gallimard, Ladevèze, Pelle (1997), const. rel. estimates
Verfürth (1998), framework for energy control, efficiency
Ohlberger (2001), non-energy estimates, hyperbolic limit
Akrivis, Makridakis, and Nochetto (2006), higher-order
temporal discretizations
Degenerate parabolic problems
Nochetto, Schmidt, Verdi (2000), Stefan problem
Dolejší, Ern, Vohralík (2013), Richards problem
(advection-dominated), robustness in a space–time dual
mesh-dependent norm
Two-phase flows
Chen and Ewing (2003), mesh adaptivity
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
34 / 35
I Stefan problem Two-phase flow R&B
Previous results
Nonlinear unsteady problems
Eriksson and Johnson (1995), L∞ (0, T ; L2 (Ω)) estimates
exploiting stability of the adjoint problem
Gallimard, Ladevèze, Pelle (1997), const. rel. estimates
Verfürth (1998), framework for energy control, efficiency
Ohlberger (2001), non-energy estimates, hyperbolic limit
Akrivis, Makridakis, and Nochetto (2006), higher-order
temporal discretizations
Degenerate parabolic problems
Nochetto, Schmidt, Verdi (2000), Stefan problem
Dolejší, Ern, Vohralík (2013), Richards problem
(advection-dominated), robustness in a space–time dual
mesh-dependent norm
Two-phase flows
Chen and Ewing (2003), mesh adaptivity
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
34 / 35
I Stefan problem Two-phase flow R&B
Bibliography
Bibliography
D I P IETRO D. A., VOHRALÍK M., YOUSEF S., Adaptive
regularization, linearization, and discretization and a
posteriori error control for the two-phase Stefan problem,
Math. Comp. 84 (2015), 153–186.
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.
Thank you for your attention!
M. Vohralík
IV/IV Full adaptivity for unsteady nonlinear problems
35 / 35