Adaptive inexact Newton methods
and their application to two-phase flows
Martin Vohralík
INRIA Paris-Rocquencourt
in collaboration with Alexandre Ern and Mary F. Wheeler
Caen, November 10, 2014
I Adaptive inexact Newton Two-phase flow C
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
1 / 56
I Adaptive inexact Newton Two-phase flow C
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
1 / 56
I Adaptive inexact Newton Two-phase flow C
Inexact iterative linearization
System of nonlinear algebraic equations
Nonlinear operator A : RN → RN , vector F ∈ RN : find U ∈ RN s.t.
A(U) = F
Algorithm (Inexact iterative linearization)
1
Choose initial vector U 0 . Set k := 1.
2
U k −1 ⇒ matrix Ak −1 and vector F k −1 : find U k s.t.
Ak −1 U k ≈ F k −1 .
3
1
2
Set U k ,0 := U k −1 and i := 1.
Do 1 algebraic solver step ⇒ U k ,i s.t. (R k ,i algebraic res.)
Ak −1 U k ,i = F k −1 − R k ,i .
3
4
Convergence? OK ⇒ U k := U k ,i . KO ⇒ i := i + 1, back
to 3.2.
Convergence? OK ⇒ finish. KO ⇒ k := k + 1, back to 2.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
2 / 56
I Adaptive inexact Newton Two-phase flow C
Inexact iterative linearization
System of nonlinear algebraic equations
Nonlinear operator A : RN → RN , vector F ∈ RN : find U ∈ RN s.t.
A(U) = F
Algorithm (Inexact iterative linearization)
1
Choose initial vector U 0 . Set k := 1.
2
U k −1 ⇒ matrix Ak −1 and vector F k −1 : find U k s.t.
Ak −1 U k ≈ F k −1 .
3
1
2
Set U k ,0 := U k −1 and i := 1.
Do 1 algebraic solver step ⇒ U k ,i s.t. (R k ,i algebraic res.)
Ak −1 U k ,i = F k −1 − R k ,i .
3
4
Convergence? OK ⇒ U k := U k ,i . KO ⇒ i := i + 1, back
to 3.2.
Convergence? OK ⇒ finish. KO ⇒ k := k + 1, back to 2.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
2 / 56
I Adaptive inexact Newton Two-phase flow C
Inexact iterative linearization
System of nonlinear algebraic equations
Nonlinear operator A : RN → RN , vector F ∈ RN : find U ∈ RN s.t.
A(U) = F
Algorithm (Inexact iterative linearization)
1
Choose initial vector U 0 . Set k := 1.
2
U k −1 ⇒ matrix Ak −1 and vector F k −1 : find U k s.t.
Ak −1 U k ≈ F k −1 .
3
1
2
Set U k ,0 := U k −1 and i := 1.
Do 1 algebraic solver step ⇒ U k ,i s.t. (R k ,i algebraic res.)
Ak −1 U k ,i = F k −1 − R k ,i .
3
4
Convergence? OK ⇒ U k := U k ,i . KO ⇒ i := i + 1, back
to 3.2.
Convergence? OK ⇒ finish. KO ⇒ k := k + 1, back to 2.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
2 / 56
I Adaptive inexact Newton Two-phase flow C
Inexact iterative linearization
System of nonlinear algebraic equations
Nonlinear operator A : RN → RN , vector F ∈ RN : find U ∈ RN s.t.
A(U) = F
Algorithm (Inexact iterative linearization)
1
Choose initial vector U 0 . Set k := 1.
2
U k −1 ⇒ matrix Ak −1 and vector F k −1 : find U k s.t.
Ak −1 U k ≈ F k −1 .
3
1
2
Set U k ,0 := U k −1 and i := 1.
Do 1 algebraic solver step ⇒ U k ,i s.t. (R k ,i algebraic res.)
Ak −1 U k ,i = F k −1 − R k ,i .
3
4
Convergence? OK ⇒ U k := U k ,i . KO ⇒ i := i + 1, back
to 3.2.
Convergence? OK ⇒ finish. KO ⇒ k := k + 1, back to 2.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
2 / 56
I Adaptive inexact Newton Two-phase flow C
Inexact iterative linearization
System of nonlinear algebraic equations
Nonlinear operator A : RN → RN , vector F ∈ RN : find U ∈ RN s.t.
A(U) = F
Algorithm (Inexact iterative linearization)
1
Choose initial vector U 0 . Set k := 1.
2
U k −1 ⇒ matrix Ak −1 and vector F k −1 : find U k s.t.
Ak −1 U k ≈ F k −1 .
3
1
2
Set U k ,0 := U k −1 and i := 1.
Do 1 algebraic solver step ⇒ U k ,i s.t. (R k ,i algebraic res.)
Ak −1 U k ,i = F k −1 − R k ,i .
3
4
Convergence? OK ⇒ U k := U k ,i . KO ⇒ i := i + 1, back
to 3.2.
Convergence? OK ⇒ finish. KO ⇒ k := k + 1, back to 2.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
2 / 56
I Adaptive inexact Newton Two-phase flow C
Inexact iterative linearization
System of nonlinear algebraic equations
Nonlinear operator A : RN → RN , vector F ∈ RN : find U ∈ RN s.t.
A(U) = F
Algorithm (Inexact iterative linearization)
1
Choose initial vector U 0 . Set k := 1.
2
U k −1 ⇒ matrix Ak −1 and vector F k −1 : find U k s.t.
Ak −1 U k ≈ F k −1 .
3
1
2
Set U k ,0 := U k −1 and i := 1.
Do 1 algebraic solver step ⇒ U k ,i s.t. (R k ,i algebraic res.)
Ak −1 U k ,i = F k −1 − R k ,i .
3
4
Convergence? OK ⇒ U k := U k ,i . KO ⇒ i := i + 1, back
to 3.2.
Convergence? OK ⇒ finish. KO ⇒ k := k + 1, back to 2.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
2 / 56
I Adaptive inexact Newton Two-phase flow C
Inexact iterative linearization
System of nonlinear algebraic equations
Nonlinear operator A : RN → RN , vector F ∈ RN : find U ∈ RN s.t.
A(U) = F
Algorithm (Inexact iterative linearization)
1
Choose initial vector U 0 . Set k := 1.
2
U k −1 ⇒ matrix Ak −1 and vector F k −1 : find U k s.t.
Ak −1 U k ≈ F k −1 .
3
1
2
Set U k ,0 := U k −1 and i := 1.
Do 1 algebraic solver step ⇒ U k ,i s.t. (R k ,i algebraic res.)
Ak −1 U k ,i = F k −1 − R k ,i .
3
4
Convergence? OK ⇒ U k := U k ,i . KO ⇒ i := i + 1, back
to 3.2.
Convergence? OK ⇒ finish. KO ⇒ k := k + 1, back to 2.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
2 / 56
I Adaptive inexact Newton Two-phase flow C
Context and questions
Approximate solution
approximate solution U k ,i does not solve A(U k ,i ) = F
Numerical method
underlying numerical method: the vector U k ,i is associated
with a (piecewise polynomial) approximation uhk ,i
Partial differential equation
underlying PDE, u its weak solution: A(u) = f
Question (Stopping criteria)
What is a good stopping criterion for the linear solver?
What is a good stopping criterion for the nonlinear solver?
Question (Error)
How big is the error ku − uhk ,i k on Newton step k and
algebraic solver step i, how is it distributed?
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
3 / 56
I Adaptive inexact Newton Two-phase flow C
Context and questions
Approximate solution
approximate solution U k ,i does not solve A(U k ,i ) = F
Numerical method
underlying numerical method: the vector U k ,i is associated
with a (piecewise polynomial) approximation uhk ,i
Partial differential equation
underlying PDE, u its weak solution: A(u) = f
Question (Stopping criteria)
What is a good stopping criterion for the linear solver?
What is a good stopping criterion for the nonlinear solver?
Question (Error)
How big is the error ku − uhk ,i k on Newton step k and
algebraic solver step i, how is it distributed?
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
3 / 56
I Adaptive inexact Newton Two-phase flow C
Context and questions
Approximate solution
approximate solution U k ,i does not solve A(U k ,i ) = F
Numerical method
underlying numerical method: the vector U k ,i is associated
with a (piecewise polynomial) approximation uhk ,i
Partial differential equation
underlying PDE, u its weak solution: A(u) = f
Question (Stopping criteria)
What is a good stopping criterion for the linear solver?
What is a good stopping criterion for the nonlinear solver?
Question (Error)
How big is the error ku − uhk ,i k on Newton step k and
algebraic solver step i, how is it distributed?
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
3 / 56
I Adaptive inexact Newton Two-phase flow C
Context and questions
Approximate solution
approximate solution U k ,i does not solve A(U k ,i ) = F
Numerical method
underlying numerical method: the vector U k ,i is associated
with a (piecewise polynomial) approximation uhk ,i
Partial differential equation
underlying PDE, u its weak solution: A(u) = f
Question (Stopping criteria)
What is a good stopping criterion for the linear solver?
What is a good stopping criterion for the nonlinear solver?
Question (Error)
How big is the error ku − uhk ,i k on Newton step k and
algebraic solver step i, how is it distributed?
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
3 / 56
I Adaptive inexact Newton Two-phase flow C
Context and questions
Approximate solution
approximate solution U k ,i does not solve A(U k ,i ) = F
Numerical method
underlying numerical method: the vector U k ,i is associated
with a (piecewise polynomial) approximation uhk ,i
Partial differential equation
underlying PDE, u its weak solution: A(u) = f
Question (Stopping criteria)
What is a good stopping criterion for the linear solver?
What is a good stopping criterion for the nonlinear solver?
Question (Error)
How big is the error ku − uhk ,i k on Newton step k and
algebraic solver step i, how is it distributed?
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
3 / 56
I Adaptive inexact Newton Two-phase flow C
Previous results
Inexact Newton method
Eisenstat and Walker (1990’s) (conception, convergence, a
priori error estimates)
Moret (1989) (discrete a posteriori error estimates)
Adaptive inexact Newton method
Bank and Rose (1982), combination with multigrid
Hackbusch and Reusken (1989), damping and multigrid
Deuflhard (1990’s, 2004 book), adaptivity
Stopping criteria for algebraic solvers
engineering literature, since 1950’s
Becker, Johnson, and Rannacher (1995), multigrid
stopping criterion
Arioli (2000’s), comparison of the algebraic and
discretization errors
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
4 / 56
I Adaptive inexact Newton Two-phase flow C
Previous results
Inexact Newton method
Eisenstat and Walker (1990’s) (conception, convergence, a
priori error estimates)
Moret (1989) (discrete a posteriori error estimates)
Adaptive inexact Newton method
Bank and Rose (1982), combination with multigrid
Hackbusch and Reusken (1989), damping and multigrid
Deuflhard (1990’s, 2004 book), adaptivity
Stopping criteria for algebraic solvers
engineering literature, since 1950’s
Becker, Johnson, and Rannacher (1995), multigrid
stopping criterion
Arioli (2000’s), comparison of the algebraic and
discretization errors
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
4 / 56
I Adaptive inexact Newton Two-phase flow C
Previous results
Inexact Newton method
Eisenstat and Walker (1990’s) (conception, convergence, a
priori error estimates)
Moret (1989) (discrete a posteriori error estimates)
Adaptive inexact Newton method
Bank and Rose (1982), combination with multigrid
Hackbusch and Reusken (1989), damping and multigrid
Deuflhard (1990’s, 2004 book), adaptivity
Stopping criteria for algebraic solvers
engineering literature, since 1950’s
Becker, Johnson, and Rannacher (1995), multigrid
stopping criterion
Arioli (2000’s), comparison of the algebraic and
discretization errors
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
4 / 56
I Adaptive inexact Newton Two-phase flow C
Previous results
A posteriori error estimates for numerical discretizations
of nonlinear problems
Ladevèze (since 1990’s), guaranteed upper bound
Han (1994), general framework
Verfürth (1994), residual estimates
Carstensen and Klose (2003), guaranteed estimates
Chaillou and Suri (2006, 2007), distinguishing
discretization and linearization errors
Kim (2007), guaranteed estimates, locally conservative
methods
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
5 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
5 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Quasi-linear elliptic problem
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
in Ω,
u=0
on ∂Ω
quasi-linear diffusion problem
σ(v , ξ) = A(v )ξ
∀(v , ξ) ∈ R × Rd
Leray–Lions problem
σ(v , ξ) = A(ξ)ξ
∀ξ ∈ Rd
p
p > 1, q := p−1
, f ∈ Lq (Ω)
Example
p-Laplacian: Leray–Lions setting with A(ξ) = |ξ|p−2 I
Nonlinear operator A : V := W01,p (Ω) → V 0
hA(u), v iV 0 ,V := (σ(u, ∇u), ∇v )
Weak formulation
Find u ∈ V such that
Martin Vohralík
A(u) = f in V 0
Adaptive inexact Newton methods and two-phase flows
6 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Quasi-linear elliptic problem
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
in Ω,
u=0
on ∂Ω
quasi-linear diffusion problem
σ(v , ξ) = A(v )ξ
∀(v , ξ) ∈ R × Rd
Leray–Lions problem
σ(v , ξ) = A(ξ)ξ
∀ξ ∈ Rd
p
p > 1, q := p−1
, f ∈ Lq (Ω)
Example
p-Laplacian: Leray–Lions setting with A(ξ) = |ξ|p−2 I
Nonlinear operator A : V := W01,p (Ω) → V 0
hA(u), v iV 0 ,V := (σ(u, ∇u), ∇v )
Weak formulation
Find u ∈ V such that
Martin Vohralík
A(u) = f in V 0
Adaptive inexact Newton methods and two-phase flows
6 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Quasi-linear elliptic problem
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
in Ω,
u=0
on ∂Ω
quasi-linear diffusion problem
σ(v , ξ) = A(v )ξ
∀(v , ξ) ∈ R × Rd
Leray–Lions problem
σ(v , ξ) = A(ξ)ξ
∀ξ ∈ Rd
p
p > 1, q := p−1
, f ∈ Lq (Ω)
Example
p-Laplacian: Leray–Lions setting with A(ξ) = |ξ|p−2 I
Nonlinear operator A : V := W01,p (Ω) → V 0
hA(u), v iV 0 ,V := (σ(u, ∇u), ∇v )
Weak formulation
Find u ∈ V such that
Martin Vohralík
A(u) = f in V 0
Adaptive inexact Newton methods and two-phase flows
6 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Quasi-linear elliptic problem
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
in Ω,
u=0
on ∂Ω
quasi-linear diffusion problem
σ(v , ξ) = A(v )ξ
∀(v , ξ) ∈ R × Rd
Leray–Lions problem
σ(v , ξ) = A(ξ)ξ
∀ξ ∈ Rd
p
p > 1, q := p−1
, f ∈ Lq (Ω)
Example
p-Laplacian: Leray–Lions setting with A(ξ) = |ξ|p−2 I
Nonlinear operator A : V := W01,p (Ω) → V 0
hA(u), v iV 0 ,V := (σ(u, ∇u), ∇v )
Weak formulation
Find u ∈ V such that
Martin Vohralík
A(u) = f in V 0
Adaptive inexact Newton methods and two-phase flows
6 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Quasi-linear elliptic problem
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
in Ω,
u=0
on ∂Ω
quasi-linear diffusion problem
σ(v , ξ) = A(v )ξ
∀(v , ξ) ∈ R × Rd
Leray–Lions problem
σ(v , ξ) = A(ξ)ξ
∀ξ ∈ Rd
p
p > 1, q := p−1
, f ∈ Lq (Ω)
Example
p-Laplacian: Leray–Lions setting with A(ξ) = |ξ|p−2 I
Nonlinear operator A : V := W01,p (Ω) → V 0
hA(u), v iV 0 ,V := (σ(u, ∇u), ∇v )
Weak formulation
Find u ∈ V such that
Martin Vohralík
A(u) = f in V 0
Adaptive inexact Newton methods and two-phase flows
6 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Quasi-linear elliptic problem
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
in Ω,
u=0
on ∂Ω
quasi-linear diffusion problem
σ(v , ξ) = A(v )ξ
∀(v , ξ) ∈ R × Rd
Leray–Lions problem
σ(v , ξ) = A(ξ)ξ
∀ξ ∈ Rd
p
p > 1, q := p−1
, f ∈ Lq (Ω)
Example
p-Laplacian: Leray–Lions setting with A(ξ) = |ξ|p−2 I
Nonlinear operator A : V := W01,p (Ω) → V 0
hA(u), v iV 0 ,V := (σ(u, ∇u), ∇v )
Weak formulation
Find u ∈ V such that
Martin Vohralík
A(u) = f in V 0
Adaptive inexact Newton methods and two-phase flows
6 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Quasi-linear elliptic problem
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
in Ω,
u=0
on ∂Ω
quasi-linear diffusion problem
σ(v , ξ) = A(v )ξ
∀(v , ξ) ∈ R × Rd
Leray–Lions problem
σ(v , ξ) = A(ξ)ξ
∀ξ ∈ Rd
p
p > 1, q := p−1
, f ∈ Lq (Ω)
Example
p-Laplacian: Leray–Lions setting with A(ξ) = |ξ|p−2 I
Nonlinear operator A : V := W01,p (Ω) → V 0
hA(u), v iV 0 ,V := (σ(u, ∇u), ∇v )
Weak formulation
Find u ∈ V such that
Martin Vohralík
A(u) = f in V 0
Adaptive inexact Newton methods and two-phase flows
6 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Approximate solution and error measure
Approximate solution
uhk ,i ∈ V (Th ) 6⊂ V , uhk ,i not necessarily in V
V (Th ) := {v ∈ Lp (Ω), v |K ∈ W 1,p (K ) ∀K ∈ Th }
Error measure
Ju (uhk ,i ) :=
(σ(u,∇u)−σ(uhk ,i ,∇uhk ,i ), ∇ϕ)+Ju,NC (uhk ,i )
ϕ∈V ; k∇ϕkp =1

1
X X
q
1−q
k ,i
k ,i q
Ju,NC (uh ) :=
he k[[u − uh ]]kq,e


sup
K ∈Th e∈EK
dual norm of the residual + nonconformity
there holds Ju (uhk ,i ) = 0 if and only if u = uhk ,i
link: strong difference of the fluxes + nonconformity
Ju (uhk ,i ) ≤ Ju (uhk ,i ) := kσ(u, ∇u) − σ(uhk ,i , ∇uhk ,i )kq + Ju,NC (uhk ,i )
up
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
7 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Approximate solution and error measure
Approximate solution
uhk ,i ∈ V (Th ) 6⊂ V , uhk ,i not necessarily in V
V (Th ) := {v ∈ Lp (Ω), v |K ∈ W 1,p (K ) ∀K ∈ Th }
Error measure
Ju (uhk ,i ) :=
(σ(u,∇u)−σ(uhk ,i ,∇uhk ,i ), ∇ϕ)+Ju,NC (uhk ,i )
ϕ∈V ; k∇ϕkp =1

1
X X
q
1−q
k ,i
k ,i q
Ju,NC (uh ) :=
he k[[u − uh ]]kq,e


sup
K ∈Th e∈EK
dual norm of the residual + nonconformity
there holds Ju (uhk ,i ) = 0 if and only if u = uhk ,i
link: strong difference of the fluxes + nonconformity
Ju (uhk ,i ) ≤ Ju (uhk ,i ) := kσ(u, ∇u) − σ(uhk ,i , ∇uhk ,i )kq + Ju,NC (uhk ,i )
up
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
7 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Approximate solution and error measure
Approximate solution
uhk ,i ∈ V (Th ) 6⊂ V , uhk ,i not necessarily in V
V (Th ) := {v ∈ Lp (Ω), v |K ∈ W 1,p (K ) ∀K ∈ Th }
Error measure
Ju (uhk ,i ) :=
(σ(u,∇u)−σ(uhk ,i ,∇uhk ,i ), ∇ϕ)+Ju,NC (uhk ,i )
ϕ∈V ; k∇ϕkp =1

1
X X
q
1−q
k ,i
k ,i q
Ju,NC (uh ) :=
he k[[u − uh ]]kq,e


sup
K ∈Th e∈EK
dual norm of the residual + nonconformity
there holds Ju (uhk ,i ) = 0 if and only if u = uhk ,i
link: strong difference of the fluxes + nonconformity
Ju (uhk ,i ) ≤ Ju (uhk ,i ) := kσ(u, ∇u) − σ(uhk ,i , ∇uhk ,i )kq + Ju,NC (uhk ,i )
up
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
7 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Approximate solution and error measure
Approximate solution
uhk ,i ∈ V (Th ) 6⊂ V , uhk ,i not necessarily in V
V (Th ) := {v ∈ Lp (Ω), v |K ∈ W 1,p (K ) ∀K ∈ Th }
Error measure
Ju (uhk ,i ) :=
(σ(u,∇u)−σ(uhk ,i ,∇uhk ,i ), ∇ϕ)+Ju,NC (uhk ,i )
ϕ∈V ; k∇ϕkp =1

1
X X
q
1−q
k ,i
k ,i q
Ju,NC (uh ) :=
he k[[u − uh ]]kq,e


sup
K ∈Th e∈EK
dual norm of the residual + nonconformity
there holds Ju (uhk ,i ) = 0 if and only if u = uhk ,i
link: strong difference of the fluxes + nonconformity
Ju (uhk ,i ) ≤ Ju (uhk ,i ) := kσ(u, ∇u) − σ(uhk ,i , ∇uhk ,i )kq + Ju,NC (uhk ,i )
up
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
7 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Approximate solution and error measure
Approximate solution
uhk ,i ∈ V (Th ) 6⊂ V , uhk ,i not necessarily in V
V (Th ) := {v ∈ Lp (Ω), v |K ∈ W 1,p (K ) ∀K ∈ Th }
Error measure
Ju (uhk ,i ) :=
(σ(u,∇u)−σ(uhk ,i ,∇uhk ,i ), ∇ϕ)+Ju,NC (uhk ,i )
ϕ∈V ; k∇ϕkp =1

1
X X
q
1−q
k ,i
k ,i q
Ju,NC (uh ) :=
he k[[u − uh ]]kq,e


sup
K ∈Th e∈EK
dual norm of the residual + nonconformity
there holds Ju (uhk ,i ) = 0 if and only if u = uhk ,i
link: strong difference of the fluxes + nonconformity
Ju (uhk ,i ) ≤ Ju (uhk ,i ) := kσ(u, ∇u) − σ(uhk ,i , ∇uhk ,i )kq + Ju,NC (uhk ,i )
up
Martin Vohralík
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I Adaptive inexact Newton Two-phase flow C
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Approximate solution and error measure
Approximate solution
uhk ,i ∈ V (Th ) 6⊂ V , uhk ,i not necessarily in V
V (Th ) := {v ∈ Lp (Ω), v |K ∈ W 1,p (K ) ∀K ∈ Th }
Error measure
Ju (uhk ,i ) :=
(σ(u,∇u)−σ(uhk ,i ,∇uhk ,i ), ∇ϕ)+Ju,NC (uhk ,i )
ϕ∈V ; k∇ϕkp =1

1
X X
q
1−q
k ,i
k ,i q
Ju,NC (uh ) :=
he k[[u − uh ]]kq,e


sup
K ∈Th e∈EK
dual norm of the residual + nonconformity
there holds Ju (uhk ,i ) = 0 if and only if u = uhk ,i
link: strong difference of the fluxes + nonconformity
Ju (uhk ,i ) ≤ Ju (uhk ,i ) := kσ(u, ∇u) − σ(uhk ,i , ∇uhk ,i )kq + Ju,NC (uhk ,i )
up
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
7 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
7 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
A posteriori error estimate
Assumption A (Total quasi-equilibrated flux reconstruction)
There exists a flux reconstruction tkh ,i ∈ Hq (div, Ω) and an
algebraic remainder ρkh ,i ∈ Lq (Ω) such that
∇·tkh ,i = fh − ρkh ,i ,
with the data approximation fh s.t. (fh , 1)K = (f , 1)K ∀K ∈ Th .
Theorem (A guaranteed a posteriori error estimate)
Let
u ∈ V be the weak solution,
uhk ,i ∈ V (Th ) be arbitrary,
Assumption A hold.
Then there holds
where η k ,i
Ju (uhk ,i )≤η k ,i ,
is fully computable from uhk ,i , tkh ,i , and ρkh ,i .
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
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I Adaptive inexact Newton Two-phase flow C
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A posteriori error estimate
Assumption A (Total quasi-equilibrated flux reconstruction)
There exists a flux reconstruction tkh ,i ∈ Hq (div, Ω) and an
algebraic remainder ρkh ,i ∈ Lq (Ω) such that
∇·tkh ,i = fh − ρkh ,i ,
with the data approximation fh s.t. (fh , 1)K = (f , 1)K ∀K ∈ Th .
Theorem (A guaranteed a posteriori error estimate)
Let
u ∈ V be the weak solution,
uhk ,i ∈ V (Th ) be arbitrary,
Assumption A hold.
Then there holds
where η k ,i
Ju (uhk ,i )≤η k ,i ,
is fully computable from uhk ,i , tkh ,i , and ρkh ,i .
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
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I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
A posteriori error estimate
Assumption A (Total quasi-equilibrated flux reconstruction)
There exists a flux reconstruction tkh ,i ∈ Hq (div, Ω) and an
algebraic remainder ρkh ,i ∈ Lq (Ω) such that
∇·tkh ,i = fh − ρkh ,i ,
with the data approximation fh s.t. (fh , 1)K = (f , 1)K ∀K ∈ Th .
Theorem (A guaranteed a posteriori error estimate)
Let
u ∈ V be the weak solution,
uhk ,i ∈ V (Th ) be arbitrary,
Assumption A hold.
Then there holds
where η k ,i
Ju (uhk ,i )≤η k ,i ,
is fully computable from uhk ,i , tkh ,i , and ρkh ,i .
Martin Vohralík
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I Adaptive inexact Newton Two-phase flow C
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Distinguishing error components
Assumption B (Discretization, linearization, and algebraic
errors)
There exist fluxes dhk ,i , lkh ,i , akh ,i ∈ [Lq (Ω)]d such that
(i) dkh ,i + lkh ,i + akh ,i = tkh ,i ;
(ii) as the linear solver converges, kakh ,i kq → 0;
(iii) as the nonlinear solver converges, klkh ,i kq → 0.
Comments
dkh ,i : discretization flux reconstruction
lkh ,i : linearization error flux reconstruction
akh ,i : algebraic error flux reconstruction
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
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I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Distinguishing error components
Assumption B (Discretization, linearization, and algebraic
errors)
There exist fluxes dhk ,i , lkh ,i , akh ,i ∈ [Lq (Ω)]d such that
(i) dkh ,i + lkh ,i + akh ,i = tkh ,i ;
(ii) as the linear solver converges, kakh ,i kq → 0;
(iii) as the nonlinear solver converges, klkh ,i kq → 0.
Comments
dkh ,i : discretization flux reconstruction
lkh ,i : linearization error flux reconstruction
akh ,i : algebraic error flux reconstruction
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
9 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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
k ,i
k ,i
Ju (uhk ,i ) ≤ η k ,i := ηdisc
+ ηlin
+ ηalg
+ ηrem
+ ηquad
+ ηosc
.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
10 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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
k ,i
k ,i
Ju (uhk ,i ) ≤ η k ,i := ηdisc
+ ηlin
+ ηalg
+ ηrem
+ ηquad
+ ηosc
.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
10 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Estimators
discretization estimator
k ,i
ηdisc,K
:= 2
1
p
)1 !
(
kσ kh ,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
quadrature estimator
k ,i
ηquad,K
:= kσ(uhk ,i , ∇uhk ,i ) − σ kh ,i kq,K
data oscillation estimator
k ,i
ηosc,K
:= CP,p hK kf − fh kq,K
(
)1/q
X k ,i q
η k· ,i :=
η ·,K
K ∈Th
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
11 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
11 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Stopping criteria
Global stopping criteria
stop whenever:
k ,i k ,i k ,i k ,i
ηrem
≤ γrem max ηdisc
, ηlin , ηalg ,
k ,i
k ,i
k ,i ηalg
≤ γalg max ηdisc
, ηlin
,
k ,i
k ,i
ηlin
≤ γlin ηdisc
γrem , γalg , γlin ≈ 0.1
Local stopping criteria
stop whenever:
k ,i
k ,i
k ,i
k ,i ηrem,K
≤ γrem,K max ηdisc,K
, ηlin,K
, ηalg,K
∀K ∈ Th ,
k ,i
k ,i
k ,i
∀K ∈ Th ,
ηalg,K
≤ γalg,K max ηdisc,K
, ηlin,K
k ,i
k ,i
ηlin,K
≤ γlin,K ηdisc,K
∀K ∈ Th
γrem,K , γalg,K , γlin,K ≈ 0.1
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
12 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Stopping criteria
Global stopping criteria
stop whenever:
k ,i k ,i k ,i k ,i
ηrem
≤ γrem max ηdisc
, ηlin , ηalg ,
k ,i
k ,i
k ,i ηalg
≤ γalg max ηdisc
, ηlin
,
k ,i
k ,i
ηlin
≤ γlin ηdisc
γrem , γalg , γlin ≈ 0.1
Local stopping criteria
stop whenever:
k ,i
k ,i
k ,i
k ,i ηrem,K
≤ γrem,K max ηdisc,K
, ηlin,K
, ηalg,K
∀K ∈ Th ,
k ,i
k ,i
k ,i
∀K ∈ Th ,
ηalg,K
≤ γalg,K max ηdisc,K
, ηlin,K
k ,i
k ,i
ηlin,K
≤ γlin,K ηdisc,K
∀K ∈ Th
γrem,K , γalg,K , γlin,K ≈ 0.1
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
12 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Assumption for efficiency
Assumption C (Approximation property)
For all K ∈ Th , there holds
k ,i
k ,i
kσ kh ,i + dkh ,i kq,K . η],T
+ ηosc,T
,
K
K
where
k ,i
η],T
K
(
X
:=
hKq 0 kfh + ∇·σ kh ,i kqq,K 0 +
K 0 ∈TK
+
he k[[σ kh ,i ·ne ]]kqq,e
e∈Eint
K
)
X
X
he1−q k[[uhk ,i ]]kqq,e
1
q
.
e∈EK
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
13 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Global efficiency
Theorem (Global efficiency)
Let the mesh Th be shape-regular and let the global stopping
criteria hold. Recall that Ju (uhk ,i ) ≤ η k ,i . Then, under
Assumption C,
k ,i
k ,i
,
η k ,i . Ju (uhk ,i ) + ηquad
+ ηosc
where . means up to a constant independent of σ and q.
robustness with respect to the nonlinearity thanks to the
choice of the dual norm as error measure
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
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I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Global efficiency
Theorem (Global efficiency)
Let the mesh Th be shape-regular and let the global stopping
criteria hold. Recall that Ju (uhk ,i ) ≤ η k ,i . Then, under
Assumption C,
k ,i
k ,i
,
η k ,i . Ju (uhk ,i ) + ηquad
+ ηosc
where . means up to a constant independent of σ and q.
robustness with respect to the nonlinearity thanks to the
choice of the dual norm as error measure
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
14 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Global efficiency
Theorem (Global efficiency)
Let the mesh Th be shape-regular and let the global stopping
criteria hold. Recall that Ju (uhk ,i ) ≤ η k ,i . Then, under
Assumption C,
k ,i
k ,i
,
η k ,i . Ju (uhk ,i ) + ηquad
+ ηosc
where . means up to a constant independent of σ and q.
robustness with respect to the nonlinearity thanks to the
choice of the dual norm as error measure
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
14 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Global efficiency
Theorem (Global efficiency)
Let the mesh Th be shape-regular and let the global stopping
criteria hold. Recall that Ju (uhk ,i ) ≤ η k ,i . Then, under
Assumption C,
k ,i
k ,i
,
η k ,i . Ju (uhk ,i ) + ηquad
+ ηosc
where . means up to a constant independent of σ and q.
robustness with respect to the nonlinearity thanks to the
choice of the dual norm as error measure
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
14 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Local efficiency
Theorem (Local efficiency)
Let the mesh Th be shape-regular and let the local stopping
criteria hold. Then, under Assumption C,
k ,i
k ,i
k ,i
k ,i
ηdisc,K
+ ηlin,K
+ ηalg,K
+ ηrem,K
k ,i
k ,i
+ ηosc,T
. Ju,TK (uhk ,i ) + ηquad,T
K
K
up
for all K ∈ Th .
robustness and local efficiency for an upper bound on the
dual norm
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
15 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Local efficiency
Theorem (Local efficiency)
Let the mesh Th be shape-regular and let the local stopping
criteria hold. Then, under Assumption C,
k ,i
k ,i
k ,i
k ,i
ηdisc,K
+ ηlin,K
+ ηalg,K
+ ηrem,K
k ,i
k ,i
+ ηosc,T
. Ju,TK (uhk ,i ) + ηquad,T
K
K
up
for all K ∈ Th .
robustness and local efficiency for an upper bound on the
dual norm
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
15 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Local efficiency
Theorem (Local efficiency)
Let the mesh Th be shape-regular and let the local stopping
criteria hold. Then, under Assumption C,
k ,i
k ,i
k ,i
k ,i
ηdisc,K
+ ηlin,K
+ ηalg,K
+ ηrem,K
k ,i
k ,i
+ ηosc,T
. Ju,TK (uhk ,i ) + ηquad,T
K
K
up
for all K ∈ Th .
robustness and local efficiency for an upper bound on the
dual norm
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
15 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
15 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Algebraic error flux reconstruction and remainder
Construction of akh ,i and ρkh ,i
On linearization step k and algebraic step i, we have
Ak −1 U k ,i = F k −1 − R k ,i .
Do ν additional steps of the algebraic solver, yielding
Ak −1 U k ,i+ν = F k −1 − R k ,i+ν .
Construct the function ρkh ,i from the algebraic residual
vector R k ,i+ν (lifting into appropriate discrete space).
Suppose we can obtain discretization and linearization flux
reconstructions dkh ,i , lkh ,i on each algebraic step. Then set
akh ,i := (dkh ,i+ν + lhk ,i+ν ) − (dkh ,i + lkh ,i ).
Independent of the algebraic solver.
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I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Algebraic error flux reconstruction and remainder
Construction of akh ,i and ρkh ,i
On linearization step k and algebraic step i, we have
Ak −1 U k ,i = F k −1 − R k ,i .
Do ν additional steps of the algebraic solver, yielding
Ak −1 U k ,i+ν = F k −1 − R k ,i+ν .
Construct the function ρkh ,i from the algebraic residual
vector R k ,i+ν (lifting into appropriate discrete space).
Suppose we can obtain discretization and linearization flux
reconstructions dkh ,i , lkh ,i on each algebraic step. Then set
akh ,i := (dkh ,i+ν + lhk ,i+ν ) − (dkh ,i + lkh ,i ).
Independent of the algebraic solver.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
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I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Algebraic error flux reconstruction and remainder
Construction of akh ,i and ρkh ,i
On linearization step k and algebraic step i, we have
Ak −1 U k ,i = F k −1 − R k ,i .
Do ν additional steps of the algebraic solver, yielding
Ak −1 U k ,i+ν = F k −1 − R k ,i+ν .
Construct the function ρkh ,i from the algebraic residual
vector R k ,i+ν (lifting into appropriate discrete space).
Suppose we can obtain discretization and linearization flux
reconstructions dkh ,i , lkh ,i on each algebraic step. Then set
akh ,i := (dkh ,i+ν + lhk ,i+ν ) − (dkh ,i + lkh ,i ).
Independent of the algebraic solver.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
16 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Algebraic error flux reconstruction and remainder
Construction of akh ,i and ρkh ,i
On linearization step k and algebraic step i, we have
Ak −1 U k ,i = F k −1 − R k ,i .
Do ν additional steps of the algebraic solver, yielding
Ak −1 U k ,i+ν = F k −1 − R k ,i+ν .
Construct the function ρkh ,i from the algebraic residual
vector R k ,i+ν (lifting into appropriate discrete space).
Suppose we can obtain discretization and linearization flux
reconstructions dkh ,i , lkh ,i on each algebraic step. Then set
akh ,i := (dkh ,i+ν + lkh ,i+ν ) − (dkh ,i + lkh ,i ).
Independent of the algebraic solver.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
16 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Algebraic error flux reconstruction and remainder
Construction of akh ,i and ρkh ,i
On linearization step k and algebraic step i, we have
Ak −1 U k ,i = F k −1 − R k ,i .
Do ν additional steps of the algebraic solver, yielding
Ak −1 U k ,i+ν = F k −1 − R k ,i+ν .
Construct the function ρkh ,i from the algebraic residual
vector R k ,i+ν (lifting into appropriate discrete space).
Suppose we can obtain discretization and linearization flux
reconstructions dkh ,i , lkh ,i on each algebraic step. Then set
akh ,i := (dkh ,i+ν + lkh ,i+ν ) − (dkh ,i + lkh ,i ).
Independent of the algebraic solver.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
16 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Algebraic error flux reconstruction and remainder
Construction of akh ,i and ρkh ,i
On linearization step k and algebraic step i, we have
Ak −1 U k ,i = F k −1 − R k ,i .
Do ν additional steps of the algebraic solver, yielding
Ak −1 U k ,i+ν = F k −1 − R k ,i+ν .
Construct the function ρkh ,i from the algebraic residual
vector R k ,i+ν (lifting into appropriate discrete space).
Suppose we can obtain discretization and linearization flux
reconstructions dkh ,i , lkh ,i on each algebraic step. Then set
akh ,i := (dkh ,i+ν + lkh ,i+ν ) − (dkh ,i + lkh ,i ).
Independent of the algebraic solver.
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
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I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Nonconforming finite elements for the p-Laplacian
Discretization
Find uh ∈ Vh such that
(σ(∇uh ), ∇vh ) = (fh , vh )
∀vh ∈ Vh .
σ(∇uh ) = |∇uh |p−2 ∇uh
Vh the Crouzeix–Raviart space
fh := Π0 f
leads to the system of nonlinear algebraic equations
A(U) = F
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I Adaptive inexact Newton Two-phase flow C
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Nonconforming finite elements for the p-Laplacian
Discretization
Find uh ∈ Vh such that
(σ(∇uh ), ∇vh ) = (fh , vh )
∀vh ∈ Vh .
σ(∇uh ) = |∇uh |p−2 ∇uh
Vh the Crouzeix–Raviart space
fh := Π0 f
leads to the system of nonlinear algebraic equations
A(U) = F
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Linearization
Linearization
Find uhk ∈ Vh such that
(σ k −1 (∇uhk ), ∇ψe ) = (fh , ψe )
∀e ∈ Ehint .
uh0 ∈ Vh yields the initial vector U 0
fixed-point linearization
σ k −1 (ξ) := |∇uhk −1 |p−2 ξ
Newton linearization
σ k −1 (ξ) := |∇uhk −1 |p−2 ξ + (p − 2)|∇uhk −1 |p−4
(∇uhk −1 ⊗ ∇uhk −1 )(ξ − ∇uhk −1 )
leads to the system of linear algebraic equations
Ak −1 U k = F k −1
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I Adaptive inexact Newton Two-phase flow C
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Linearization
Linearization
Find uhk ∈ Vh such that
(σ k −1 (∇uhk ), ∇ψe ) = (fh , ψe )
∀e ∈ Ehint .
uh0 ∈ Vh yields the initial vector U 0
fixed-point linearization
σ k −1 (ξ) := |∇uhk −1 |p−2 ξ
Newton linearization
σ k −1 (ξ) := |∇uhk −1 |p−2 ξ + (p − 2)|∇uhk −1 |p−4
(∇uhk −1 ⊗ ∇uhk −1 )(ξ − ∇uhk −1 )
leads to the system of linear algebraic equations
Ak −1 U k = F k −1
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I Adaptive inexact Newton Two-phase flow C
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Algebraic solution
Algebraic solution
Find uhk ,i ∈ Vh such that
(σ k −1 (∇uhk ,i ), ∇ψe ) = (fh , ψe ) − Rek ,i
∀e ∈ Ehint .
algebraic residual vector R k ,i = {Rek ,i }e∈E int
h
discrete system
Ak −1 U k = F k −1 − R k ,i
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I Adaptive inexact Newton Two-phase flow C
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Algebraic solution
Algebraic solution
Find uhk ,i ∈ Vh such that
(σ k −1 (∇uhk ,i ), ∇ψe ) = (fh , ψe ) − Rek ,i
∀e ∈ Ehint .
algebraic residual vector R k ,i = {Rek ,i }e∈E int
h
discrete system
Ak −1 U k = F k −1 − R k ,i
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I Adaptive inexact Newton Two-phase flow C
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Flux reconstructions
Definition (Construction of (dkh ,i + lkh ,i ))
For all K ∈ Th ,
(dkh ,i+lkh ,i )|K := −σ k −1(∇uhk ,i )|K +
X Rek ,i
fh |K
(x−xK )−
(x−xK )|Ke ,
d
d|De |
where, Rek ,i = (fh , ψe ) − (σ k −1 (∇uhk ,i ), ∇ψe )
e∈EK
∀e ∈ Ehint .
Definition (Construction of dkh ,i )
For all K ∈ Th ,
dkh ,i |K := −σ(∇uhk ,i )|K +
X R̄ek ,i
fh |K
(x − xK ) −
(x − xK )|Ke ,
d
d|De |
where R̄ek ,i := (fh , ψe ) − (σ(∇uhk ,i ), ∇ψe )
e∈EK
∀e ∈ Ehint .
Definition (Construction of σ kh ,i )
k ,i
Set σ kh ,i := σ(∇uhk ,i ). Consequently, ηquad,K
= 0 for all K ∈ Th .
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
20 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Flux reconstructions
Definition (Construction of (dkh ,i + lkh ,i ))
For all K ∈ Th ,
(dkh ,i+lkh ,i )|K := −σ k −1(∇uhk ,i )|K +
X Rek ,i
fh |K
(x−xK )−
(x−xK )|Ke ,
d
d|De |
where, Rek ,i = (fh , ψe ) − (σ k −1 (∇uhk ,i ), ∇ψe )
e∈EK
∀e ∈ Ehint .
Definition (Construction of dkh ,i )
For all K ∈ Th ,
dkh ,i |K := −σ(∇uhk ,i )|K +
X R̄ek ,i
fh |K
(x − xK ) −
(x − xK )|Ke ,
d
d|De |
where R̄ek ,i := (fh , ψe ) − (σ(∇uhk ,i ), ∇ψe )
e∈EK
∀e ∈ Ehint .
Definition (Construction of σ kh ,i )
k ,i
Set σ kh ,i := σ(∇uhk ,i ). Consequently, ηquad,K
= 0 for all K ∈ Th .
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
20 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Flux reconstructions
Definition (Construction of (dkh ,i + lkh ,i ))
For all K ∈ Th ,
(dkh ,i+lkh ,i )|K := −σ k −1(∇uhk ,i )|K +
X Rek ,i
fh |K
(x−xK )−
(x−xK )|Ke ,
d
d|De |
where, Rek ,i = (fh , ψe ) − (σ k −1 (∇uhk ,i ), ∇ψe )
e∈EK
∀e ∈ Ehint .
Definition (Construction of dkh ,i )
For all K ∈ Th ,
dkh ,i |K := −σ(∇uhk ,i )|K +
X R̄ek ,i
fh |K
(x − xK ) −
(x − xK )|Ke ,
d
d|De |
where R̄ek ,i := (fh , ψe ) − (σ(∇uhk ,i ), ∇ψe )
e∈EK
∀e ∈ Ehint .
Definition (Construction of σ kh ,i )
k ,i
Set σ kh ,i := σ(∇uhk ,i ). Consequently, ηquad,K
= 0 for all K ∈ Th .
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
20 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Verification of the assumptions – upper bound
Lemma (Assumptions A and B)
Assumptions A and B hold.
Comments
kakh ,i kq,K → 0 as the linear solver converges by definition.
klkh ,i kq,K → 0 as the nonlinear solver converges by the
construction of lkh ,i .
Both (dkh ,i + lkh ,i ) and dkh ,i belong to RTN0 (Sh ) ⇒
akh ,i ∈ RTN0 (Sh ) and tkh ,i ∈ RTN0 (Sh ).
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
21 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Verification of the assumptions – upper bound
Lemma (Assumptions A and B)
Assumptions A and B hold.
Comments
kakh ,i kq,K → 0 as the linear solver converges by definition.
klkh ,i kq,K → 0 as the nonlinear solver converges by the
construction of lkh ,i .
Both (dkh ,i + lkh ,i ) and dkh ,i belong to RTN0 (Sh ) ⇒
akh ,i ∈ RTN0 (Sh ) and tkh ,i ∈ RTN0 (Sh ).
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
21 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Verification of the assumptions – efficiency
Lemma (Assumption C)
Assumption C holds.
Comments
dkh ,i close to σ(∇uhk ,i )
approximation properties of Raviart–Thomas–Nédélec
spaces
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
22 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Verification of the assumptions – efficiency
Lemma (Assumption C)
Assumption C holds.
Comments
dkh ,i close to σ(∇uhk ,i )
approximation properties of Raviart–Thomas–Nédélec
spaces
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
22 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Summary
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 two-phase flows
23 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Summary
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 two-phase flows
23 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Summary
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 two-phase flows
23 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Summary
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 two-phase flows
23 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
23 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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 two-phase flows
24 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Analytical and approximate solutions
0.04
0.4
0.02
0.04
0.3
0
0
−0.02
−0.02
−0.04
0.2
0.2
u
u
0.3
0.4
0.02
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 two-phase flows
25 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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
Newton
Martin Vohralík
600
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 two-phase flows
26 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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
Newton
Martin Vohralík
16
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 two-phase flows
27 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Error and estimators, p = 10
0
0
10
0
10
10
error up
dif. flux est.
nonc. est.
lin. est.
alg. est.
−2
10
−2
10
−4
−1
10
−6
10
−4
10
Dual error
Dual error
Dual error
10
−6
10
−8
−2
10
10
−8
10
−10
10
−12
10
−10
1
10
2
10
3
10
Number of faces
4
10
Newton
Martin Vohralík
5
10
10
−3
1
10
2
10
3
10
Number of faces
4
10
inexact Newton
5
10
10
1
10
2
10
3
10
Number of faces
4
10
5
10
ad. inexact Newton
Adaptive inexact Newton methods and two-phase flows
28 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Effectivity indices, p = 10
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9 1
10
2
10
3
10
Number of faces
4
10
Newton
Martin Vohralík
5
10
2.8
Upper and lower dual error effectivity indices
1.8
Upper and lower dual error effectivity indices
Upper and lower dual error effectivity indices
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9 1
10
2
10
3
10
Number of faces
4
10
inexact Newton
5
10
effectivity ind. up
effectivity ind. low
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1 1
10
2
10
3
10
Number of faces
4
10
5
10
ad. inexact Newton
Adaptive inexact Newton methods and two-phase flows
29 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Error distribution, p = 10
−3
−3
x 10
x 10
5.5
5
5
4.5
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
Estimated error distribution
Martin Vohralík
Exact error distribution
Adaptive inexact Newton methods and two-phase flows
30 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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 two-phase flows
31 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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
Newton
Martin Vohralík
400
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 two-phase flows
32 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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
Newton
Martin Vohralík
16
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 two-phase flows
33 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Error and estimators, p = 1.5
0
0
10
0
10
10
error up
dif. flux est.
nonc. est.
lin. est.
alg. est.
−2
10
−2
10
−1
10
−4
10
−6
10
Dual error
Dual error
Dual error
−4
10
−6
10
−2
10
−8
10
−3
10
−8
10
−10
10
−12
10
−10
1
10
2
10
3
10
Number of faces
4
10
Newton
Martin Vohralík
5
10
10
−4
1
10
2
10
3
10
Number of faces
4
10
inexact Newton
5
10
10
1
10
2
10
3
10
Number of faces
4
10
5
10
ad. inexact Newton
Adaptive inexact Newton methods and two-phase flows
34 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Effectivity indices, p = 1.5
1.2
1.15
1.1
1.05
1
0.95 1
10
2
10
3
10
Number of faces
4
10
Newton
Martin Vohralík
5
10
1.5
Upper and lower dual error effectivity indices
1.25
Upper and lower dual error effectivity indices
Upper and lower dual error effectivity indices
1.25
1.2
1.15
1.1
1.05
1
0.95 1
10
2
10
3
10
Number of faces
4
10
inexact Newton
5
10
effectivity ind. up
effectivity ind. low
1.45
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1 1
10
2
10
3
10
Number of faces
4
10
5
10
ad. inexact Newton
Adaptive inexact Newton methods and two-phase flows
35 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Newton and algebraic iterations, p = 1.5
3
3
Number of algebraic solver iterations
Number of Newton iterations
2
10
1
10
0
10
1
4
10
full
inex.
adapt. inex.
10
full
inex.
adapt. inex.
Total number of algebraic solver iterations
10
2
10
1
10
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
10
0
1
2
3
10
Newton iteration
3
4
Refinement level
alg. it. / Newton step
alg. it. / refinement
10
10
10
1
2
Adaptive inexact Newton methods and two-phase flows
5
6
36 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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 two-phase flows
37 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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 two-phase flows
38 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
Estimated and actual errors and the effectivity index
0
3
error up un.
dif. flux est. un.
nonc. est. un.
lin. est. un.
alg. est. un.
osc. est. un.
error up ad.
dif. flux est. ad.
nonc. est. ad.
lin. est. ad.
alg. est. ad.
osc. est. ad.
−1
10
−2
Dual error
10
−3
10
−4
10
−5
10
−6
10
−7
10
1
10
2
10
3
10
Number of faces
4
10
effectivity ind. up un.
effectivity ind. low un.
effectivity ind. up ad.
effectivity ind. low ad.
2.5
2
1.5
1
0.5 1
10
5
10
Estimated and actual errors
Martin Vohralík
Upper and lower dual error effectivity indices
10
2
10
3
10
Number of faces
4
10
5
10
Effectivity index
Adaptive inexact Newton methods and two-phase flows
39 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Stoping criteria & efficiency Applications Numerical results
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 two-phase flows
40 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Application and numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
40 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
41 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
41 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Application and 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
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
42 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Application and 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
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
42 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Application and 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
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
42 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
43 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
43 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Application and numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
43 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
44 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
44 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
44 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
44 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
45 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
45 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Application and numerical results
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
45 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
46 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
46 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
46 / 56
I Adaptive inexact Newton Two-phase flow 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
:=
Martin Vohralík
∀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|
Adaptive inexact Newton methods and two-phase flows
47 / 56
I Adaptive inexact Newton Two-phase flow 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
:=
Martin Vohralík
∀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|
Adaptive inexact Newton methods and two-phase flows
47 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Application and numerical results
Velocities reconstructions
Total phase velocities reconstructions
,i
n,k ,i
n,k ,i n,k ,i
(dn,k
λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
·nD
t,h ·nD , 1)e := −
n,k ,i
n,k ,i
+ λr,n (sw,h )K∇pc (sw,h )·nD , 1 e ,
n,k ,i
n,k ,i
n,k −1
n,k −1 n,k ,i
((dt,h + 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
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
n,k ,i
n,k ,i
(dn,k
w,h ·nD , 1)e := − λr,w (sw,h )K∇pw,h ·nD , 1 e ,
n,k ,i
,i
n,k ,i
n,k −1
((dn,k
w,h + lw,h )·nD , 1)e := − λr,w (sw,h )K∇pw,h ·nD , 1 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 two-phase flows
48 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Application and numerical results
Velocities reconstructions
Total phase velocities reconstructions
,i
n,k ,i
n,k ,i n,k ,i
(dn,k
λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
·nD
t,h ·nD , 1)e := −
n,k ,i
n,k ,i
+ λr,n (sw,h )K∇pc (sw,h )·nD , 1 e ,
n,k ,i
n,k ,i
n,k −1
n,k −1 n,k ,i
((dt,h + 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
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
n,k ,i
n,k ,i
(dn,k
w,h ·nD , 1)e := − λr,w (sw,h )K∇pw,h ·nD , 1 e ,
n,k ,i
,i
n,k ,i
n,k −1
((dn,k
w,h + lw,h )·nD , 1)e := − λr,w (sw,h )K∇pw,h ·nD , 1 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 two-phase flows
48 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Application and numerical results
Velocities reconstructions
Total phase velocities reconstructions
,i
n,k ,i
n,k ,i n,k ,i
(dn,k
λr,w (sw,h
) + λr,n (sw,h
) K∇pw,h
·nD
t,h ·nD , 1)e := −
n,k ,i
n,k ,i
+ λr,n (sw,h )K∇pc (sw,h )·nD , 1 e ,
n,k ,i
n,k ,i
n,k −1
n,k −1 n,k ,i
((dt,h + 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
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
n,k ,i
n,k ,i
(dn,k
w,h ·nD , 1)e := − λr,w (sw,h )K∇pw,h ·nD , 1 e ,
n,k ,i
,i
n,k ,i
n,k −1
((dn,k
w,h + lw,h )·nD , 1)e := − λr,w (sw,h )K∇pw,h ·nD , 1 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 two-phase flows
48 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
49 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
49 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
50 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
50 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
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I Adaptive inexact Newton Two-phase flow 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 two-phase flows
51 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
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I Adaptive inexact Newton Two-phase flow 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 two-phase flows
53 / 56
I Adaptive inexact Newton Two-phase flow C
Estimate Application and numerical results
Space/time/nonlinear solver/linear solver adaptivity
Fully adaptive computation
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
54 / 56
I Adaptive inexact Newton Two-phase flow C
Outline
1
Introduction
2
Adaptive inexact Newton method
A guaranteed a posteriori error estimate
Stopping criteria and efficiency
Application
Numerical results
3
Application to two-phase flow in porous media
A guaranteed a posteriori error estimate
Application and numerical results
4
Conclusions and future directions
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
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I Adaptive inexact Newton Two-phase flow 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 two-phase flows
55 / 56
I Adaptive inexact Newton Two-phase flow 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 two-phase flows
55 / 56
I Adaptive inexact Newton Two-phase flow 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.
Merci de votre attention !
Martin Vohralík
Adaptive inexact Newton methods and two-phase flows
56 / 56