An adaptive inexact Newton method
Martin Vohralík, INRIA Paris-Rocquencourt
Lecture III/IV
IIT Bombay, July 13–17, 2015
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Outline
1
Introduction
2
A posteriori error estimate
3
Stopping criteria, efficiency, and robustness
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
III/IV Adaptive inexact Newton method
2 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Outline
1
Introduction
2
A posteriori error estimate
3
Stopping criteria, efficiency, and robustness
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
III/IV Adaptive inexact Newton method
2 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 an 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.
M. Vohralík
III/IV Adaptive inexact Newton method
3 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 an 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.
M. Vohralík
III/IV Adaptive inexact Newton method
3 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 an 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.
M. Vohralík
III/IV Adaptive inexact Newton method
3 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 an 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.
M. Vohralík
III/IV Adaptive inexact Newton method
3 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 an 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.
M. Vohralík
III/IV Adaptive inexact Newton method
3 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 an 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.
M. Vohralík
III/IV Adaptive inexact Newton method
3 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 an 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.
M. Vohralík
III/IV Adaptive inexact Newton method
3 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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?
M. Vohralík
III/IV Adaptive inexact Newton method
4 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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?
M. Vohralík
III/IV Adaptive inexact Newton method
4 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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?
M. Vohralík
III/IV Adaptive inexact Newton method
4 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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?
M. Vohralík
III/IV Adaptive inexact Newton method
4 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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?
M. Vohralík
III/IV Adaptive inexact Newton method
4 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Outline
1
Introduction
2
A posteriori error estimate
3
Stopping criteria, efficiency, and robustness
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
III/IV Adaptive inexact Newton method
4 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Model steady problem, discretization
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
u=0
p > 1, q :=
p
p−1 ,
in Ω,
on ∂Ω
f ∈ Lq (Ω)
example: p-Laplacian with σ(u, ∇u) = |∇u|p−2 ∇u
f piecewise polynomial for simplicity
weak solution: u ∈ V := W01,p (Ω) such that
(σ(u, ∇u), ∇v ) = (f , v )
∀v ∈ V
Numerical approximation
simplicial mesh Th , linearization step k , algebraic step i
uhk ,i ∈ V (Th ) := {v ∈ Lp (Ω), v |K ∈ W 1,p (K ) ∀K ∈ Th } 6⊂ V
M. Vohralík
III/IV Adaptive inexact Newton method
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Model steady problem, discretization
Quasi-linear elliptic problem
−∇·σ(u, ∇u) = f
u=0
p > 1, q :=
p
p−1 ,
in Ω,
on ∂Ω
f ∈ Lq (Ω)
example: p-Laplacian with σ(u, ∇u) = |∇u|p−2 ∇u
f piecewise polynomial for simplicity
weak solution: u ∈ V := W01,p (Ω) such that
(σ(u, ∇u), ∇v ) = (f , v )
∀v ∈ V
Numerical approximation
simplicial mesh Th , linearization step k , algebraic step i
uhk ,i ∈ V (Th ) := {v ∈ Lp (Ω), v |K ∈ W 1,p (K ) ∀K ∈ Th } 6⊂ V
M. Vohralík
III/IV Adaptive inexact Newton method
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Intrinsic error measure
Energy error in the Laplace case
k∇(u − uh )k2 =
sup
(∇(u −uh ), ∇ϕ)2 + min k∇(v −uh )k2
v ∈H01 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
|
{z
dual norm of the residual
} |
{z
}
distance of uh to H01 (Ω)
Intrinsic error measure
Ju (uhk ,i ) :=
sup
ϕ∈V ; k∇ϕkp =1
{z
|
+
(σ(u,∇u) − σ(uhk ,i ,∇uhk ,i ), ∇ϕ)
}
dual norm of the residual

X X

K ∈Th e∈EK
|
he1−q k[[u − uhk ,i ]]kqq,e
1
q

{z
distance of uh to V
}
there holds Ju (uhk ,i ) = 0 if and only if u = uhk ,i
M. Vohralík
III/IV Adaptive inexact Newton method
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Intrinsic error measure
Energy error in the Laplace case
k∇(u − uh )k2 =
sup
(∇(u −uh ), ∇ϕ)2 + min k∇(v −uh )k2
v ∈H01 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
|
{z
dual norm of the residual
} |
{z
}
distance of uh to H01 (Ω)
Intrinsic error measure
Ju (uhk ,i ) :=
sup
ϕ∈V ; k∇ϕkp =1
{z
|
+
(σ(u,∇u) − σ(uhk ,i ,∇uhk ,i ), ∇ϕ)
}
dual norm of the residual

X X

K ∈Th e∈EK
|
he1−q k[[u − uhk ,i ]]kqq,e
1
q

{z
distance of uh to V
}
there holds Ju (uhk ,i ) = 0 if and only if u = uhk ,i
M. Vohralík
III/IV Adaptive inexact Newton method
6 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Intrinsic error measure
Energy error in the Laplace case
k∇(u − uh )k2 =
sup
(∇(u −uh ), ∇ϕ)2 + min k∇(v −uh )k2
v ∈H01 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
|
{z
dual norm of the residual
} |
{z
}
distance of uh to H01 (Ω)
Intrinsic error measure
Ju (uhk ,i ) :=
sup
ϕ∈V ; k∇ϕkp =1
{z
|
+
(σ(u,∇u) − σ(uhk ,i ,∇uhk ,i ), ∇ϕ)
}
dual norm of the residual

X X

K ∈Th e∈EK
|
he1−q k[[u − uhk ,i ]]kqq,e
1
q

{z
distance of uh to V
}
there holds Ju (uhk ,i ) = 0 if and only if u = uhk ,i
M. Vohralík
III/IV Adaptive inexact Newton method
6 / 38
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Abstract assumptions
Assumption A (Total flux reconstruction)
There exists tkh ,i ∈ Hq (div, Ω) and ρkh ,i ∈ Lq (Ω) such that
∇·tkh ,i = f −
ρkh ,i
|{z}
.
algebraic
remainder
Assumption B (Discretization, linearization, and alg. fluxes)
There exist fluxes dhk ,i , lkh ,i , akh ,i ∈ [Lq (Ω)]d such that
(i) tkh ,i = dkh ,i + lkh ,i + akh ,i ;
(ii) as the linear solver converges, kakh ,i kq → 0;
(iii) as the nonlinear solver converges, klkh ,i kq → 0.
M. Vohralík
III/IV Adaptive inexact Newton method
7 / 38
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Abstract assumptions
Assumption A (Total flux reconstruction)
There exists tkh ,i ∈ Hq (div, Ω) and ρkh ,i ∈ Lq (Ω) such that
∇·tkh ,i = f −
ρkh ,i
|{z}
.
algebraic
remainder
Assumption B (Discretization, linearization, and alg. fluxes)
There exist fluxes dhk ,i , lkh ,i , akh ,i ∈ [Lq (Ω)]d such that
(i) tkh ,i = dkh ,i + lkh ,i + akh ,i ;
(ii) as the linear solver converges, kakh ,i kq → 0;
(iii) as the nonlinear solver converges, klkh ,i kq → 0.
M. Vohralík
III/IV Adaptive inexact Newton method
7 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Abstract assumptions
Assumption A (Total flux reconstruction)
There exists tkh ,i ∈ Hq (div, Ω) and ρkh ,i ∈ Lq (Ω) such that
∇·tkh ,i = f −
ρkh ,i
|{z}
.
algebraic
remainder
Assumption B (Discretization, linearization, and alg. fluxes)
There exist fluxes dhk ,i , lkh ,i , akh ,i ∈ [Lq (Ω)]d such that
(i) tkh ,i = dkh ,i + lkh ,i + akh ,i ;
(ii) as the linear solver converges, kakh ,i kq → 0;
(iii) as the nonlinear solver converges, klkh ,i kq → 0.
M. Vohralík
III/IV Adaptive inexact Newton method
7 / 38
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Estimate distinguishing error components
Theorem (Estimate distinguishing different error components)
Let
u ∈ V be the weak solution,
uhk ,i ∈ V (Th ) be arbitrary,
Assumptions A and B hold.
Then there holds
k ,i
k ,i
k ,i
k ,i
Ju (uhk ,i ) ≤ ηdisc
+ ηlin
+ ηalg
+ ηrem
.
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Estimators
discretization estimator
k ,i
ηdisc,K
:= 2
1
p
)1 !
(
kσ(uhk ,i , ∇uhk ,i )+dkh ,i kq,K +
q
X
he1−q k[[uhk ,i ]]kqq,e
e∈EK
linearization estimator
k ,i
ηlin,K
:= klkh ,i kq,K
algebraic estimator
k ,i
ηalg,K
:= kakh ,i kq,K
algebraic remainder estimator
k ,i
ηrem,K
:= hΩ kρkh ,i kq,K
)1/q
(
η k· ,i :=
X
,i
η k·,K
q
K ∈Th
M. Vohralík
III/IV Adaptive inexact Newton method
9 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Outline
1
Introduction
2
A posteriori error estimate
3
Stopping criteria, efficiency, and robustness
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
III/IV Adaptive inexact Newton method
9 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
M. Vohralík
III/IV Adaptive inexact Newton method
10 / 38
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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
M. Vohralík
III/IV Adaptive inexact Newton method
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Assumptions for efficiency
Assumption C (Piecewise polynomials, meshes, quadrature)
The approximation uhk ,i is piecewise polynomial. The meshes
Th are shape-regular. The quadrature error is negligible.
Assumption D (Approximation property)
For all K ∈ Th , there holds
(
kσ(uhk ,i , ∇uhk ,i ) + dkh ,i kq,K ≤ C
X
hKq 0 kf +∇·σ(uhk ,i , ∇uhk ,i )kqq,K 0
K 0 ∈TK
+
X
he k[[σ(uhk ,i , ∇uhk ,i )·ne ]]kqq,e
e∈Eint
K
)1
q
+
X
he1−q k[[uhk ,i ]]kqq,e
.
e∈EK
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III/IV Adaptive inexact Newton method
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Assumptions for efficiency
Assumption C (Piecewise polynomials, meshes, quadrature)
The approximation uhk ,i is piecewise polynomial. The meshes
Th are shape-regular. The quadrature error is negligible.
Assumption D (Approximation property)
For all K ∈ Th , there holds
(
kσ(uhk ,i , ∇uhk ,i ) + dkh ,i kq,K ≤ C
X
hKq 0 kf +∇·σ(uhk ,i , ∇uhk ,i )kqq,K 0
K 0 ∈TK
+
X
he k[[σ(uhk ,i , ∇uhk ,i )·ne ]]kqq,e
e∈Eint
K
)1
q
+
X
he1−q k[[uhk ,i ]]kqq,e
.
e∈EK
M. Vohralík
III/IV Adaptive inexact Newton method
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Global efficiency
Theorem (Global efficiency)
Let the Assumptions C and D be satisfied. Let the global
stopping criteria hold. Then,
k ,i
k ,i
k ,i
k ,i
ηdisc
+ ηlin
+ ηalg
+ ηrem
≤ CJu (uhk ,i ),
where C is independent of σ and q.
Theorem (Local efficiency)
Let the Assumptions C and D be satisfied. Let the local
stopping criteria hold. Then
k ,i
k ,i
k ,i
k ,i
ηdisc,K
+ ηlin,K
+ ηalg,K
+ ηrem,K
≤ CJu,TK (uhk ,i )
∀K ∈ Th .
robustness with respect to the nonlinearity thanks to the
choice of Ju as error measure
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III/IV Adaptive inexact Newton method
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Global efficiency
Theorem (Global efficiency)
Let the Assumptions C and D be satisfied. Let the global
stopping criteria hold. Then,
k ,i
k ,i
k ,i
k ,i
ηdisc
+ ηlin
+ ηalg
+ ηrem
≤ CJu (uhk ,i ),
where C is independent of σ and q.
Theorem (Local efficiency)
Let the Assumptions C and D be satisfied. Let the local
stopping criteria hold. Then
k ,i
k ,i
k ,i
k ,i
ηdisc,K
+ ηlin,K
+ ηalg,K
+ ηrem,K
≤ CJu,TK (uhk ,i )
∀K ∈ Th .
robustness with respect to the nonlinearity thanks to the
choice of Ju as error measure
M. Vohralík
III/IV Adaptive inexact Newton method
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Global efficiency
Theorem (Global efficiency)
Let the Assumptions C and D be satisfied. Let the global
stopping criteria hold. Then,
k ,i
k ,i
k ,i
k ,i
ηdisc
+ ηlin
+ ηalg
+ ηrem
≤ CJu (uhk ,i ),
where C is independent of σ and q.
Theorem (Local efficiency)
Let the Assumptions C and D be satisfied. Let the local
stopping criteria hold. Then
k ,i
k ,i
k ,i
k ,i
ηdisc,K
+ ηlin,K
+ ηalg,K
+ ηrem,K
≤ CJu,TK (uhk ,i )
∀K ∈ Th .
robustness with respect to the nonlinearity thanks to the
choice of Ju as error measure
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Global efficiency
Theorem (Global efficiency)
Let the Assumptions C and D be satisfied. Let the global
stopping criteria hold. Then,
k ,i
k ,i
k ,i
k ,i
ηdisc
+ ηlin
+ ηalg
+ ηrem
≤ CJu (uhk ,i ),
where C is independent of σ and q.
Theorem (Local efficiency)
Let the Assumptions C and D be satisfied. Let the local
stopping criteria hold. Then
k ,i
k ,i
k ,i
k ,i
ηdisc,K
+ ηlin,K
+ ηalg,K
+ ηrem,K
≤ CJu,TK (uhk ,i )
∀K ∈ Th .
robustness with respect to the nonlinearity thanks to the
choice of Ju as error measure
M. Vohralík
III/IV Adaptive inexact Newton method
12 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Outline
1
Introduction
2
A posteriori error estimate
3
Stopping criteria, efficiency, and robustness
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
III/IV Adaptive inexact Newton method
12 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Nonconforming finite elements for the p-Laplacian
Discretization
Find uh ∈ Vh such that
(σ(∇uh ), ∇vh ) = (f , vh )
∀vh ∈ Vh .
σ(∇uh ) = |∇uh |p−2 ∇uh
Vh 6⊂ V the Crouzeix–Raviart space
leads to the system of nonlinear algebraic equations
A(U) = F
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III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Nonconforming finite elements for the p-Laplacian
Discretization
Find uh ∈ Vh such that
(σ(∇uh ), ∇vh ) = (f , vh )
∀vh ∈ Vh .
σ(∇uh ) = |∇uh |p−2 ∇uh
Vh 6⊂ V the Crouzeix–Raviart space
leads to the system of nonlinear algebraic equations
A(U) = F
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Linearization
Linearization
Find uhk ∈ Vh such that
(σ k −1 (∇uhk ), ∇ψe ) = (f , 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
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Linearization
Linearization
Find uhk ∈ Vh such that
(σ k −1 (∇uhk ), ∇ψe ) = (f , 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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III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Algebraic solution
Algebraic solution
Find uhk ,i ∈ Vh such that
(σ k −1 (∇uhk ,i ), ∇ψe ) = (f , 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
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Algebraic solution
Algebraic solution
Find uhk ,i ∈ Vh such that
(σ k −1 (∇uhk ,i ), ∇ψe ) = (f , 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
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
f |K
(x−xK )−
(x−xK )|Ke ,
d
d|De |
where Rek ,i = (f , 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
f |K
(x − xK ) −
(x − xK )|Ke ,
d
d|De |
where R̄ek ,i := (f , ψe ) − (σ(∇uhk ,i ), ∇ψe )
e∈EK
∀e ∈ Ehint .
Definition (Construction of akh ,i )
Set akh ,i := (dkh ,i+ν + lkh ,i+ν ) − (dkh ,i + lkh ,i ) for (adaptively chosen)
ν > 0 additional algebraic solvers steps; R k ,i+ν
ρkh ,i .
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
f |K
(x−xK )−
(x−xK )|Ke ,
d
d|De |
where Rek ,i = (f , 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
f |K
(x − xK ) −
(x − xK )|Ke ,
d
d|De |
where R̄ek ,i := (f , ψe ) − (σ(∇uhk ,i ), ∇ψe )
e∈EK
∀e ∈ Ehint .
Definition (Construction of akh ,i )
Set akh ,i := (dkh ,i+ν + lkh ,i+ν ) − (dkh ,i + lkh ,i ) for (adaptively chosen)
ν > 0 additional algebraic solvers steps; R k ,i+ν
ρkh ,i .
M. Vohralík
III/IV Adaptive inexact Newton method
16 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
f |K
(x−xK )−
(x−xK )|Ke ,
d
d|De |
where Rek ,i = (f , 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
f |K
(x − xK ) −
(x − xK )|Ke ,
d
d|De |
where R̄ek ,i := (f , ψe ) − (σ(∇uhk ,i ), ∇ψe )
e∈EK
∀e ∈ Ehint .
Definition (Construction of akh ,i )
Set akh ,i := (dkh ,i+ν + lkh ,i+ν ) − (dkh ,i + lkh ,i ) for (adaptively chosen)
ν > 0 additional algebraic solvers steps; R k ,i+ν
ρkh ,i .
M. Vohralík
III/IV Adaptive inexact Newton method
16 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Verification of the assumptions
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
Lemma (Assumptions C and D)
Assumptions C and D hold.
Comments
quadrature error is zero
dkh ,i is close to σ(∇uhk ,i ): approximation properties of the
Raviart–Thomas–Nédélec spaces
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Verification of the assumptions
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
Lemma (Assumptions C and D)
Assumptions C and D hold.
Comments
quadrature error is zero
dkh ,i is close to σ(∇uhk ,i ): approximation properties of the
Raviart–Thomas–Nédélec spaces
M. Vohralík
III/IV Adaptive inexact Newton method
17 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Verification of the assumptions
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
Lemma (Assumptions C and D)
Assumptions C and D hold.
Comments
quadrature error is zero
dkh ,i is close to σ(∇uhk ,i ): approximation properties of the
Raviart–Thomas–Nédélec spaces
M. Vohralík
III/IV Adaptive inexact Newton method
17 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Verification of the assumptions
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
Lemma (Assumptions C and D)
Assumptions C and D hold.
Comments
quadrature error is zero
dkh ,i is close to σ(∇uhk ,i ): approximation properties of the
Raviart–Thomas–Nédélec spaces
M. Vohralík
III/IV Adaptive inexact Newton method
17 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 D verified
M. Vohralík
III/IV Adaptive inexact Newton method
18 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 D verified
M. Vohralík
III/IV Adaptive inexact Newton method
18 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 D verified
M. Vohralík
III/IV Adaptive inexact Newton method
18 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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 D verified
M. Vohralík
III/IV Adaptive inexact Newton method
18 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
Outline
1
Introduction
2
A posteriori error estimate
3
Stopping criteria, efficiency, and robustness
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
III/IV Adaptive inexact Newton method
18 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
Crouzeix–Raviart nonconforming finite elements
M. Vohralík
III/IV Adaptive inexact Newton method
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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
M. Vohralík
1
−0.06
0.5
y
−0.1
0.5
0
0
x
Case p = 10
III/IV Adaptive inexact Newton method
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Error and estimators as a function of CG iterations,
p = 10, 6th level mesh, 6th Newton step.
0
0
10
0
10
10
−2
10
−6
10
Dual error
Dual error
Dual error
−4
10
−1
10
−1
10
−8
10
−10
10
0
−2
100
200
300
400
Algebraic iteration
500
600
700
Newton
10
3
−2
6
9
Algebraic iteration
12
15
inexact Newton
M. Vohralík
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
III/IV Adaptive inexact Newton method
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Error and estimators as a function of Newton
iterations, p = 10, 6th level mesh
2
2
10
−2
10
−2
1
−6
10
Dual error
Dual error
10
−4
10
−4
10
−6
10
−8
10
−8
−2
10
−10
0
10
−10
2
4
6
8
10
12
Newton iteration
14
16
18
20
Newton
10
0
−3
10
20
30
40
50
60
Newton iteration
70
80
90
inexact Newton
M. Vohralík
0
10
−1
10
10
error up
estimate
disc. est.
lin. est.
2
10
10
Dual error
10
0
10
10
3
10
0
100
10
1
2
3
4
5
6
7
8
9
Newton iteration
10
11
12
13
ad. inexact Newton
III/IV Adaptive inexact Newton method
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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
5
10
Newton
10
−3
1
10
2
10
3
10
Number of faces
4
5
10
10
inexact Newton
M. Vohralík
10
1
10
2
10
3
10
Number of faces
4
10
5
10
ad. inexact Newton
III/IV Adaptive inexact Newton method
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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
5
10
Newton
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
5
10
10
inexact Newton
M. Vohralík
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
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
M. Vohralík
Exact error distribution
III/IV Adaptive inexact Newton method
25 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
M. 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
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Error and estimators as a function of CG iterations,
p = 1.5, 6th level mesh, 1st Newton step.
0
0
10
0
10
10
−2
−1
10
10
−1
10
−6
10
Dual error
−2
Dual error
Dual error
−4
10
10
−3
10
−2
10
−8
−4
10
10
−10
10
0
−3
100
200
300
Algebraic iteration
400
500
Newton
10
3
−5
6
9
Algebraic iteration
12
15
inexact Newton
M. Vohralík
10
0
error up
estimate
disc. est.
lin. est.
alg. est.
alg. rem. est.
20
40
60
Algebraic iteration
80
100
ad. inexact Newton
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Error and estimators as a function of Newton
iterations, p = 1.5, 6th level mesh
−2
0
10
−4
10
−4
Dual error
10
−8
10
−10
−6
10
−8
10
10
−12
0
10
Dual error
−6
Dual error
10
−2
10
10
−2
10
−10
2
4
6
8
10
12
Newton iteration
14
16
18
20
Newton
10
0
−3
50
100 150 200 250 300 350 400 450 500 550
Newton iteration
inexact Newton
M. Vohralík
10
1
error up
estimate
disc. est.
lin. est.
2
Newton iteration
3
ad. inexact Newton
III/IV Adaptive inexact Newton method
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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
5
10
Newton
10
−4
1
10
2
10
3
10
Number of faces
4
5
10
10
inexact Newton
M. Vohralík
10
1
10
2
10
3
10
Number of faces
4
10
5
10
ad. inexact Newton
III/IV Adaptive inexact Newton method
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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
5
10
Newton
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
5
10
10
inexact Newton
M. Vohralík
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
III/IV Adaptive inexact Newton method
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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
M. 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
III/IV Adaptive inexact Newton method
5
6
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Numerical experiment II
Model problem
p-Laplacian
∇·(|∇u|p−2 ∇u) = f
in Ω,
u = uD
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))
Crouzeix–Raviart nonconforming finite elements
M. Vohralík
III/IV Adaptive inexact Newton method
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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
M. Vohralík
Exact error distribution
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
5
10
Estimated and actual errors
M. Vohralík
Upper and lower dual error effectivity indices
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
2
10
3
10
Number of faces
4
10
5
10
Effectivity index
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
M. 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
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Outline
1
Introduction
2
A posteriori error estimate
3
Stopping criteria, efficiency, and robustness
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
M. Vohralík
III/IV Adaptive inexact Newton method
36 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
M. Vohralík
III/IV Adaptive inexact Newton method
36 / 38
I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
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
M. Vohralík
III/IV Adaptive inexact Newton method
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I Estimate Stopping criteria & efficiency Applications Numerics R&B
Bibliography
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.
Thank you for your attention!
M. Vohralík
III/IV Adaptive inexact Newton method
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