A posteriori error control for the heat equation
Martin Vohralík, INRIA Paris-Rocquencourt
Lecture II/IV
IIT Bombay, July 13–17, 2015
I Setting Est. & eff. Applications Numerics R&B
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
2 / 25
I Setting Est. & eff. Applications Numerics R&B
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
2 / 25
I Setting Est. & eff. Applications Numerics R&B
An optimal a posteriori estimate for evolutive problems
Guaranteed upper bound
P
P
n
2
ku − uhτ k2?,Ω×(0,T ) ≤ N
n=1
K ∈T n ηK (uhτ )
no undetermined constant: error control
Local efficiency
ηKn (uhτ ) ≤ Ceff ku − uhτ kTK ×(t n−1 ,t n )
optimal space–time mesh refinement
will only be global in space
Asymptotic exactness
PN P
2
2
n
n=1
K ∈T n ηK (uhτ ) /ku − uhτ k?,Ω×(0,T ) & 1
overestimation factor goes to one with meshes size
Robustness
Ceff indep. of data, domain, final time, meshes, or solution
Small evaluation cost
estimators can be evaluated locally in space and in time
M. Vohralík
II/IV A posteriori error control for the heat equation
3 / 25
I Setting Est. & eff. Applications Numerics R&B
An optimal a posteriori estimate for evolutive problems
Guaranteed upper bound
P
P
n
2
ku − uhτ k2?,Ω×(0,T ) ≤ N
n=1
K ∈T n ηK (uhτ )
no undetermined constant: error control
Local efficiency
ηKn (uhτ ) ≤ Ceff ku − uhτ kTK ×(t n−1 ,t n )
optimal space–time mesh refinement
will only be global in space
Asymptotic exactness
PN P
2
2
n
n=1
K ∈T n ηK (uhτ ) /ku − uhτ k?,Ω×(0,T ) & 1
overestimation factor goes to one with meshes size
Robustness
Ceff indep. of data, domain, final time, meshes, or solution
Small evaluation cost
estimators can be evaluated locally in space and in time
M. Vohralík
II/IV A posteriori error control for the heat equation
3 / 25
I Setting Est. & eff. Applications Numerics R&B
An optimal a posteriori estimate for evolutive problems
Guaranteed upper bound
P
P
n
2
ku − uhτ k2?,Ω×(0,T ) ≤ N
n=1
K ∈T n ηK (uhτ )
no undetermined constant: error control
Local efficiency
ηKn (uhτ ) ≤ Ceff ku − uhτ kTK ×(t n−1 ,t n )
optimal space–time mesh refinement
will only be global in space
Asymptotic exactness
PN P
2
2
n
n=1
K ∈T n ηK (uhτ ) /ku − uhτ k?,Ω×(0,T ) & 1
overestimation factor goes to one with meshes size
Robustness
Ceff indep. of data, domain, final time, meshes, or solution
Small evaluation cost
estimators can be evaluated locally in space and in time
M. Vohralík
II/IV A posteriori error control for the heat equation
3 / 25
I Setting Est. & eff. Applications Numerics R&B
An optimal a posteriori estimate for evolutive problems
Guaranteed upper bound
P
P
n
2
ku − uhτ k2?,Ω×(0,T ) ≤ N
n=1
K ∈T n ηK (uhτ )
no undetermined constant: error control
Local efficiency
ηKn (uhτ ) ≤ Ceff ku − uhτ kTK ×(t n−1 ,t n )
optimal space–time mesh refinement
will only be global in space
Asymptotic exactness
PN P
2
2
n
n=1
K ∈T n ηK (uhτ ) /ku − uhτ k?,Ω×(0,T ) & 1
overestimation factor goes to one with meshes size
Robustness
Ceff indep. of data, domain, final time, meshes, or solution
Small evaluation cost
estimators can be evaluated locally in space and in time
M. Vohralík
II/IV A posteriori error control for the heat equation
3 / 25
I Setting Est. & eff. Applications Numerics R&B
An optimal a posteriori estimate for evolutive problems
Guaranteed upper bound
P
P
n
2
ku − uhτ k2?,Ω×(0,T ) ≤ N
n=1
K ∈T n ηK (uhτ )
no undetermined constant: error control
Local efficiency
ηKn (uhτ ) ≤ Ceff ku − uhτ kTK ×(t n−1 ,t n )
optimal space–time mesh refinement
will only be global in space
Asymptotic exactness
PN P
2
2
n
n=1
K ∈T n ηK (uhτ ) /ku − uhτ k?,Ω×(0,T ) & 1
overestimation factor goes to one with meshes size
Robustness
Ceff indep. of data, domain, final time, meshes, or solution
Small evaluation cost
estimators can be evaluated locally in space and in time
M. Vohralík
II/IV A posteriori error control for the heat equation
3 / 25
I Setting Est. & eff. Applications Numerics R&B
An optimal a posteriori estimate for evolutive problems
Guaranteed upper bound
P
P
n
2
ku − uhτ k2?,Ω×(0,T ) ≤ N
n=1
K ∈T n ηK (uhτ )
no undetermined constant: error control
Local efficiency
ηKn (uhτ ) ≤ Ceff ku − uhτ kTK ×(t n−1 ,t n )
optimal space–time mesh refinement
will only be global in space
Asymptotic exactness
PN P
2
2
n
n=1
K ∈T n ηK (uhτ ) /ku − uhτ k?,Ω×(0,T ) & 1
overestimation factor goes to one with meshes size
Robustness
Ceff indep. of data, domain, final time, meshes, or solution
Small evaluation cost
estimators can be evaluated locally in space and in time
M. Vohralík
II/IV A posteriori error control for the heat equation
3 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
3 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
3 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
The heat equation
The heat equation
∂t u − ∆u = f
u=0
u(·, 0) = u0
a.e. in Q := Ω × (0, T ),
a.e. on ∂Ω × (0, T ),
a.e. in Ω
Assumptions
Ω ⊂ Rd , d ≥ 2, polygonal domain
T > 0, final simulation time
f ∈ L2 (Q), u0 ∈ L2 (Ω)
Spaces
X := L2 (0, T ; H01 (Ω)), X 0 = L2 (0, T , H −1 (Ω))
Y := {y ∈ X ; ∂t y ∈ X 0 }
Weak solution
Find u ∈ Y such that, for a.e. t ∈ (0, T ) and for all v ∈ H01 (Ω),
h∂t u, v iH −1 (Ω),H 1 (Ω) (t) + (∇u, ∇v )(t) = (f , v )(t).
0
M. Vohralík
II/IV A posteriori error control for the heat equation
4 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
The heat equation
The heat equation
∂t u − ∆u = f
u=0
u(·, 0) = u0
a.e. in Q := Ω × (0, T ),
a.e. on ∂Ω × (0, T ),
a.e. in Ω
Assumptions
Ω ⊂ Rd , d ≥ 2, polygonal domain
T > 0, final simulation time
f ∈ L2 (Q), u0 ∈ L2 (Ω)
Spaces
X := L2 (0, T ; H01 (Ω)), X 0 = L2 (0, T , H −1 (Ω))
Y := {y ∈ X ; ∂t y ∈ X 0 }
Weak solution
Find u ∈ Y such that, for a.e. t ∈ (0, T ) and for all v ∈ H01 (Ω),
h∂t u, v iH −1 (Ω),H 1 (Ω) (t) + (∇u, ∇v )(t) = (f , v )(t).
0
M. Vohralík
II/IV A posteriori error control for the heat equation
4 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
The heat equation
The heat equation
∂t u − ∆u = f
u=0
u(·, 0) = u0
a.e. in Q := Ω × (0, T ),
a.e. on ∂Ω × (0, T ),
a.e. in Ω
Assumptions
Ω ⊂ Rd , d ≥ 2, polygonal domain
T > 0, final simulation time
f ∈ L2 (Q), u0 ∈ L2 (Ω)
Spaces
X := L2 (0, T ; H01 (Ω)), X 0 = L2 (0, T , H −1 (Ω))
Y := {y ∈ X ; ∂t y ∈ X 0 }
Weak solution
Find u ∈ Y such that, for a.e. t ∈ (0, T ) and for all v ∈ H01 (Ω),
h∂t u, v iH −1 (Ω),H 1 (Ω) (t) + (∇u, ∇v )(t) = (f , v )(t).
0
M. Vohralík
II/IV A posteriori error control for the heat equation
4 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
The heat equation
The heat equation
∂t u − ∆u = f
u=0
u(·, 0) = u0
a.e. in Q := Ω × (0, T ),
a.e. on ∂Ω × (0, T ),
a.e. in Ω
Assumptions
Ω ⊂ Rd , d ≥ 2, polygonal domain
T > 0, final simulation time
f ∈ L2 (Q), u0 ∈ L2 (Ω)
Spaces
X := L2 (0, T ; H01 (Ω)), X 0 = L2 (0, T , H −1 (Ω))
Y := {y ∈ X ; ∂t y ∈ X 0 }
Weak solution
Find u ∈ Y such that, for a.e. t ∈ (0, T ) and for all v ∈ H01 (Ω),
h∂t u, v iH −1 (Ω),H 1 (Ω) (t) + (∇u, ∇v )(t) = (f , v )(t).
0
M. Vohralík
II/IV A posteriori error control for the heat equation
4 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
4 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Error and residual in the steady case
Laplace equation
−∆u = f
u=0
in Ω,
on ∂Ω
Weak formulation
Find u ∈ H01 (Ω) such that
(∇u, ∇v ) = (f , v )
Residual of uh ∈ H01 (Ω)
∀v ∈ H01 (Ω)
R(uh ) ∈ H −1 (Ω), the misfit of uh in the weak formulation:
hR(uh ), ϕiH −1 (Ω),H 1 (Ω) := (f , ϕ) − (∇uh , ∇ϕ)
0
ϕ ∈ H01 (Ω)
dual norm of the residual
kR(uh )kH −1 (Ω) :=
sup
hR(uh ), ϕiH −1 (Ω),H 1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
0
Energy error is the dual norm of the residual
k∇(u − uh )k =
M. Vohralík
sup
(∇(u − uh ), ∇ϕ) = kR(uh )kH −1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
II/IV A posteriori error control for the heat equation
5 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Error and residual in the steady case
Laplace equation
−∆u = f
u=0
in Ω,
on ∂Ω
Weak formulation
Find u ∈ H01 (Ω) such that
(∇u, ∇v ) = (f , v )
Residual of uh ∈ H01 (Ω)
∀v ∈ H01 (Ω)
R(uh ) ∈ H −1 (Ω), the misfit of uh in the weak formulation:
hR(uh ), ϕiH −1 (Ω),H 1 (Ω) := (f , ϕ) − (∇uh , ∇ϕ)
0
ϕ ∈ H01 (Ω)
dual norm of the residual
kR(uh )kH −1 (Ω) :=
sup
hR(uh ), ϕiH −1 (Ω),H 1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
0
Energy error is the dual norm of the residual
k∇(u − uh )k =
M. Vohralík
sup
(∇(u − uh ), ∇ϕ) = kR(uh )kH −1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
II/IV A posteriori error control for the heat equation
5 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Error and residual in the steady case
Laplace equation
−∆u = f
u=0
in Ω,
on ∂Ω
Weak formulation
Find u ∈ H01 (Ω) such that
(∇u, ∇v ) = (f , v )
Residual of uh ∈ H01 (Ω)
∀v ∈ H01 (Ω)
R(uh ) ∈ H −1 (Ω), the misfit of uh in the weak formulation:
hR(uh ), ϕiH −1 (Ω),H 1 (Ω) := (f , ϕ) − (∇uh , ∇ϕ)
0
ϕ ∈ H01 (Ω)
dual norm of the residual
kR(uh )kH −1 (Ω) :=
sup
hR(uh ), ϕiH −1 (Ω),H 1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
0
Energy error is the dual norm of the residual
k∇(u − uh )k =
M. Vohralík
sup
(∇(u − uh ), ∇ϕ) = kR(uh )kH −1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
II/IV A posteriori error control for the heat equation
5 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Error and residual in the steady case
Laplace equation
−∆u = f
u=0
in Ω,
on ∂Ω
Weak formulation
Find u ∈ H01 (Ω) such that
(∇u, ∇v ) = (f , v )
Residual of uh ∈ H01 (Ω)
∀v ∈ H01 (Ω)
R(uh ) ∈ H −1 (Ω), the misfit of uh in the weak formulation:
hR(uh ), ϕiH −1 (Ω),H 1 (Ω) := (f , ϕ) − (∇uh , ∇ϕ)
0
ϕ ∈ H01 (Ω)
dual norm of the residual
kR(uh )kH −1 (Ω) :=
sup
hR(uh ), ϕiH −1 (Ω),H 1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
0
Energy error is the dual norm of the residual
k∇(u − uh )k =
M. Vohralík
sup
(∇(u − uh ), ∇ϕ) = kR(uh )kH −1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
II/IV A posteriori error control for the heat equation
5 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Error and residual in the steady case
Laplace equation
−∆u = f
u=0
in Ω,
on ∂Ω
Weak formulation
Find u ∈ H01 (Ω) such that
(∇u, ∇v ) = (f , v )
Residual of uh ∈ H01 (Ω)
∀v ∈ H01 (Ω)
R(uh ) ∈ H −1 (Ω), the misfit of uh in the weak formulation:
hR(uh ), ϕiH −1 (Ω),H 1 (Ω) := (f , ϕ) − (∇uh , ∇ϕ)
0
ϕ ∈ H01 (Ω)
dual norm of the residual
kR(uh )kH −1 (Ω) :=
sup
hR(uh ), ϕiH −1 (Ω),H 1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
0
Energy error is the dual norm of the residual
k∇(u − uh )k =
M. Vohralík
sup
(∇(u − uh ), ∇ϕ) = kR(uh )kH −1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
II/IV A posteriori error control for the heat equation
5 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Error and residual in the steady case
Laplace equation
−∆u = f
u=0
in Ω,
on ∂Ω
Weak formulation
Find u ∈ H01 (Ω) such that
(∇u, ∇v ) = (f , v )
Residual of uh ∈ H01 (Ω)
∀v ∈ H01 (Ω)
R(uh ) ∈ H −1 (Ω), the misfit of uh in the weak formulation:
hR(uh ), ϕiH −1 (Ω),H 1 (Ω) := (f , ϕ) − (∇uh , ∇ϕ)
0
ϕ ∈ H01 (Ω)
dual norm of the residual
kR(uh )kH −1 (Ω) :=
sup
hR(uh ), ϕiH −1 (Ω),H 1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
0
Energy error is the dual norm of the residual
k∇(u − uh )k =
M. Vohralík
sup
(∇(u − uh ), ∇ϕ) = kR(uh )kH −1 (Ω)
ϕ∈H01 (Ω); k∇ϕk=1
II/IV A posteriori error control for the heat equation
5 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Intrinsic norms for the heat equation
Norms
Z
energy norm: kv k2X :=
T
0
k∇v k2 (t) dt, v ∈ X
augmented norm: (Verfürth (2003)) kv kY := kv kX + k∂t v kX 0 ,
(Z
)1/2
T
k∂t v kX 0 =
k∂t v k2H −1 (Ω) (t) dt
,v ∈Y
0
link: k∂t v kX 0 =
kv kX ∗ :=
sup
Z
T
ϕ∈X , kϕkX =1 0
Z T
sup
ϕ∈X , kϕkX =1 0
h∂t v , ϕiH −1 (Ω),H 1 (Ω) (t)dt
0
{h∂t v , ϕiH −1 (Ω),H 1 (Ω) + (∇v , ∇ϕ)}(t)dt
Theorem (Norm equivalence, cf.
0
Verfürth (2003))
Let v ∈ Y . Then
kv kY ≤ 3kv kX ∗ + 21/2 kv (·, 0)k,
kv kX ∗ ≤ kv kY .
M. Vohralík
II/IV A posteriori error control for the heat equation
6 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Intrinsic norms for the heat equation
Norms
Z
energy norm: kv k2X :=
T
0
k∇v k2 (t) dt, v ∈ X
augmented norm: (Verfürth (2003)) kv kY := kv kX + k∂t v kX 0 ,
(Z
)1/2
T
k∂t v kX 0 =
k∂t v k2H −1 (Ω) (t) dt
,v ∈Y
0
link: k∂t v kX 0 =
kv kX ∗ :=
sup
Z
T
ϕ∈X , kϕkX =1 0
Z T
sup
ϕ∈X , kϕkX =1 0
h∂t v , ϕiH −1 (Ω),H 1 (Ω) (t)dt
0
{h∂t v , ϕiH −1 (Ω),H 1 (Ω) + (∇v , ∇ϕ)}(t)dt
Theorem (Norm equivalence, cf.
0
Verfürth (2003))
Let v ∈ Y . Then
kv kY ≤ 3kv kX ∗ + 21/2 kv (·, 0)k,
kv kX ∗ ≤ kv kY .
M. Vohralík
II/IV A posteriori error control for the heat equation
6 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Intrinsic norms for the heat equation
Norms
Z
energy norm: kv k2X :=
T
0
k∇v k2 (t) dt, v ∈ X
augmented norm: (Verfürth (2003)) kv kY := kv kX + k∂t v kX 0 ,
(Z
)1/2
T
k∂t v kX 0 =
k∂t v k2H −1 (Ω) (t) dt
,v ∈Y
0
link: k∂t v kX 0 =
kv kX ∗ :=
sup
Z
T
ϕ∈X , kϕkX =1 0
Z T
sup
ϕ∈X , kϕkX =1 0
h∂t v , ϕiH −1 (Ω),H 1 (Ω) (t)dt
0
{h∂t v , ϕiH −1 (Ω),H 1 (Ω) + (∇v , ∇ϕ)}(t)dt
Theorem (Norm equivalence, cf.
0
Verfürth (2003))
Let v ∈ Y . Then
kv kY ≤ 3kv kX ∗ + 21/2 kv (·, 0)k,
kv kX ∗ ≤ kv kY .
M. Vohralík
II/IV A posteriori error control for the heat equation
6 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Norm equivalence I
Proof.
straightforwardly: Z
T
kv kX ∗ = sup
{h∂t v , ϕiH −1 (Ω),H 1 (Ω) + (∇v , ∇ϕ)}(t)dt
0
ϕ∈X , kϕkX =1 0
≤
Z
sup
ϕ∈X , kϕkX =1 0
sup
+
T
h∂t v , ϕiH −1 (Ω),H 1 (Ω) (t)dt
0
Z
ϕ∈X , kϕkX =1 0
T
(∇v , ∇ϕ)(t)dt
= k∂t v kX 0 + kv kX = kv kY
classically:
2
1
2 kv (·, T )k
=
2
1
2 kv (·, 0)k
+
Z
T
h∂t v , v iH −1 (Ω),H 1 (Ω) (t)dt
0
0
thus kv k2X ≤ 21 kv (·, T )k2 + kv k2X =
R
1
2 + T {h∂ v , v i
kv
(·,
0)k
t
H −1 (Ω),H 1 (Ω) + (∇v , ∇v )}(t)dt
2
0
0
M. Vohralík
II/IV A posteriori error control for the heat equation
7 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Norm equivalence I
Proof.
straightforwardly: Z
T
kv kX ∗ = sup
{h∂t v , ϕiH −1 (Ω),H 1 (Ω) + (∇v , ∇ϕ)}(t)dt
0
ϕ∈X , kϕkX =1 0
≤
Z
sup
ϕ∈X , kϕkX =1 0
sup
+
T
h∂t v , ϕiH −1 (Ω),H 1 (Ω) (t)dt
0
Z
ϕ∈X , kϕkX =1 0
T
(∇v , ∇ϕ)(t)dt
= k∂t v kX 0 + kv kX = kv kY
classically:
2
1
2 kv (·, T )k
=
2
1
2 kv (·, 0)k
+
Z
T
h∂t v , v iH −1 (Ω),H 1 (Ω) (t)dt
0
0
thus kv k2X ≤ 21 kv (·, T )k2 + kv k2X =
R
1
2 + T {h∂ v , v i
kv
(·,
0)k
t
H −1 (Ω),H 1 (Ω) + (∇v , ∇v )}(t)dt
2
0
0
M. Vohralík
II/IV A posteriori error control for the heat equation
7 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Norm equivalence II
Proof.
passing to supremum:
kv k2X
≤
sup
Z
ϕ∈X , kϕkX =1 0
T
{h∂t v , ϕiH −1 (Ω),H 1 (Ω)
0
+ (∇v , ∇ϕ)}(t)dtkv kX + 21 kv (·, 0)k2
x 2 ≤ ax + b2 ⇒ x ≤ a + b, a, b ≥ 0:
kv kX ≤ kv kX ∗ + 2−1/2 kv (·, 0)k
for any ϕ ∈ X :
Z T
Z T
h∂t v , ϕiH −1 (Ω),H 1 (Ω) (t)dt = {h∂t v , ϕiH −1 (Ω),H 1 (Ω)
0
0
0
0
+ (∇v , ∇ϕ) − (∇v , ∇ϕ)}(t)dt
thus k∂t v kX 0 ≤ kv kX ∗ + kv kX
finally kv kY = k∂t v kX 0 + kv kX ≤ 3kv kX ∗ + 21/2 kv (·, 0)k
M. Vohralík
II/IV A posteriori error control for the heat equation
8 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Error and residual for the heat equation
Residual and its dual norm for the heat equation
For uhτ ∈ Y , the residual R(uhτ ) ∈ X 0 :
hR(uhτ ), ϕiX 0 ,X :=
Z
0
T
{(f , ϕ) − h∂t uhτ , ϕiH −1 (Ω),H 1 (Ω)
0
− (∇uhτ , ∇ϕ)}(t)dt
kR(uhτ )kX 0 :=
ϕ ∈ X,
hR(uhτ ), ϕiX 0 ,X .
sup
ϕ∈X , kϕkX =1
Corollary (Equivalence of the k·kY error and of the residual)
Let u be the weak solution. Let uhτ ∈ Y be arbitrary. Then
ku − uhτ kY ≤ 3kR(uhτ )kX 0 + 21/2 k(u − uhτ )(·, 0)k,
kR(uhτ )kX 0 ≤ ku − uhτ kY .
M. Vohralík
II/IV A posteriori error control for the heat equation
9 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Error and residual for the heat equation
Residual and its dual norm for the heat equation
For uhτ ∈ Y , the residual R(uhτ ) ∈ X 0 :
hR(uhτ ), ϕiX 0 ,X :=
Z
0
T
{(f , ϕ) − h∂t uhτ , ϕiH −1 (Ω),H 1 (Ω)
0
− (∇uhτ , ∇ϕ)}(t)dt
kR(uhτ )kX 0 :=
ϕ ∈ X,
hR(uhτ ), ϕiX 0 ,X .
sup
ϕ∈X , kϕkX =1
Corollary (Equivalence of the k·kY error and of the residual)
Let u be the weak solution. Let uhτ ∈ Y be arbitrary. Then
ku − uhτ kY ≤ 3kR(uhτ )kX 0 + 21/2 k(u − uhτ )(·, 0)k,
kR(uhτ )kX 0 ≤ ku − uhτ kY .
M. Vohralík
II/IV A posteriori error control for the heat equation
9 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
9 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Time-dependent meshes and discrete solutions
Approximate solutions
discrete times {t n }0≤n≤N , t 0 = 0 and t N = T
In := (t n−1 , t n ], τ n := t n − t n−1 , 1 ≤ n ≤ N
a different simplicial mesh T n on all 0 ≤ n ≤ N
uhn ∈ Vhn , 0 ≤ n ≤ N, piecewise polynomial for simplicity
uhn possibly nonconforming, Vhn 6∈ H01 (Ω)
uhτ : Q → R continuous and piecewise affine in time
uhτ (·, t) := (1 − %(t))uhn−1 + %(t)uhn ,
%(t) =
1
(t − t n−1 )
τn
tailored to the backward Euler discretization
M. Vohralík
II/IV A posteriori error control for the heat equation
10 / 25
I Setting Est. & eff. Applications Numerics R&B
Heat equation Error and residual Discrete setting
Time-dependent meshes and discrete solutions
t
u3hτ
t3
T3
τ3
t2
τ2
u2hτ
T2
t1
u1hτ
T1
τ1
t0
Ω
T0
u0hτ
x
Time-dependent meshes and discrete solutions
Fig. 2.1. Time-dependent meshes and discrete solutions
M. Vohralík
II/IV A posteriori error control for the heat equation
11 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
11 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
11 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Potential and flux reconstructions
General form
potential reconstruction shτ is continuous and piecewise
affine in time with shn ∈ H01 (Ω) for all 0 ≤ n ≤ N
flux reconstruction σhτ is piecewise constant in time with
σhn := σhτ |In ∈ H(div, Ω) for all 1 ≤ n ≤ N
T n,n+1R is a common refinement of T n and T n+1
n
ef n :=
In f (·, t)dt/τ is piecewise constant in time
Assumption A (Potential and flux reconstructions)
Reconstructions shn preserve mean values of uhn on T n,n+1
(shn , 1)K = (uhn , 1)K
∀K ∈ T n,n+1 , ∀1 ≤ n ≤ N.
Reconstructions σhn satisfy a local conservation property
(ef n − ∂t uhτ |In − ∇·σhn , 1)K = 0
M. Vohralík
∀K ∈ T n , ∀1 ≤ n ≤ N.
II/IV A posteriori error control for the heat equation
12 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Potential and flux reconstructions
General form
potential reconstruction shτ is continuous and piecewise
affine in time with shn ∈ H01 (Ω) for all 0 ≤ n ≤ N
flux reconstruction σhτ is piecewise constant in time with
σhn := σhτ |In ∈ H(div, Ω) for all 1 ≤ n ≤ N
T n,n+1R is a common refinement of T n and T n+1
n
ef n :=
In f (·, t)dt/τ is piecewise constant in time
Assumption A (Potential and flux reconstructions)
Reconstructions shn preserve mean values of uhn on T n,n+1
(shn , 1)K = (uhn , 1)K
∀K ∈ T n,n+1 , ∀1 ≤ n ≤ N.
Reconstructions σhn satisfy a local conservation property
(ef n − ∂t uhτ |In − ∇·σhn , 1)K = 0
M. Vohralík
∀K ∈ T n , ∀1 ≤ n ≤ N.
II/IV A posteriori error control for the heat equation
12 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Potential and flux reconstructions
General form
potential reconstruction shτ is continuous and piecewise
affine in time with shn ∈ H01 (Ω) for all 0 ≤ n ≤ N
flux reconstruction σhτ is piecewise constant in time with
σhn := σhτ |In ∈ H(div, Ω) for all 1 ≤ n ≤ N
T n,n+1R is a common refinement of T n and T n+1
n
ef n :=
In f (·, t)dt/τ is piecewise constant in time
Assumption A (Potential and flux reconstructions)
Reconstructions shn preserve mean values of uhn on T n,n+1
(shn , 1)K = (uhn , 1)K
∀K ∈ T n,n+1 , ∀1 ≤ n ≤ N.
Reconstructions σhn satisfy a local conservation property
(ef n − ∂t uhτ |In − ∇·σhn , 1)K = 0
M. Vohralík
∀K ∈ T n , ∀1 ≤ n ≤ N.
II/IV A posteriori error control for the heat equation
12 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Potential and flux reconstructions
General form
potential reconstruction shτ is continuous and piecewise
affine in time with shn ∈ H01 (Ω) for all 0 ≤ n ≤ N
flux reconstruction σhτ is piecewise constant in time with
σhn := σhτ |In ∈ H(div, Ω) for all 1 ≤ n ≤ N
T n,n+1R is a common refinement of T n and T n+1
n
ef n :=
In f (·, t)dt/τ is piecewise constant in time
Assumption A (Potential and flux reconstructions)
Reconstructions shn preserve mean values of uhn on T n,n+1
(shn , 1)K = (uhn , 1)K
∀K ∈ T n,n+1 , ∀1 ≤ n ≤ N.
Reconstructions σhn satisfy a local conservation property
(ef n − ∂t uhτ |In − ∇·σhn , 1)K = 0
M. Vohralík
∀K ∈ T n , ∀1 ≤ n ≤ N.
II/IV A posteriori error control for the heat equation
12 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Potential and flux reconstructions
General form
potential reconstruction shτ is continuous and piecewise
affine in time with shn ∈ H01 (Ω) for all 0 ≤ n ≤ N
flux reconstruction σhτ is piecewise constant in time with
σhn := σhτ |In ∈ H(div, Ω) for all 1 ≤ n ≤ N
T n,n+1R is a common refinement of T n and T n+1
n
ef n :=
In f (·, t)dt/τ is piecewise constant in time
Assumption A (Potential and flux reconstructions)
Reconstructions shn preserve mean values of uhn on T n,n+1
(shn , 1)K = (uhn , 1)K
∀K ∈ T n,n+1 , ∀1 ≤ n ≤ N.
Reconstructions σhn satisfy a local conservation property
(ef n − ∂t uhτ |In − ∇·σhn , 1)K = 0
M. Vohralík
∀K ∈ T n , ∀1 ≤ n ≤ N.
II/IV A posteriori error control for the heat equation
12 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Practical construction of shτ and σhτ
Possible construction of shn
n
shn := Iav
(uhn ) +
αKn :=
X
αKn bK ,
K ∈T n,n+1
1
n
(u n − Iav
(uhn ), 1)K
(bK , 1)K h
n : the averaging interpolate on the mesh T n
Iav
bK standard (time-independent) bubble function supported
on the element K
specificity of the parabolic case
Construction of σhn
adapted from the elliptic case
M. Vohralík
II/IV A posteriori error control for the heat equation
13 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Practical construction of shτ and σhτ
Possible construction of shn
n
shn := Iav
(uhn ) +
αKn :=
X
αKn bK ,
K ∈T n,n+1
1
n
(u n − Iav
(uhn ), 1)K
(bK , 1)K h
n : the averaging interpolate on the mesh T n
Iav
bK standard (time-independent) bubble function supported
on the element K
specificity of the parabolic case
Construction of σhn
adapted from the elliptic case
M. Vohralík
II/IV A posteriori error control for the heat equation
13 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Practical construction of shτ and σhτ
Possible construction of shn
n
shn := Iav
(uhn ) +
αKn :=
X
αKn bK ,
K ∈T n,n+1
1
n
(u n − Iav
(uhn ), 1)K
(bK , 1)K h
n : the averaging interpolate on the mesh T n
Iav
bK standard (time-independent) bubble function supported
on the element K
specificity of the parabolic case
Construction of σhn
adapted from the elliptic case
M. Vohralík
II/IV A posteriori error control for the heat equation
13 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
13 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
A posteriori error estimate
Theorem (A posteriori error estimate)
Let
u be the weak solution
uhτ given by uhn ∈ Vhn , 0 ≤ n ≤ N, be arbitrary
shτ be the mean values-preserving potential reconstruction
and σhτ locally conservative flux reconstruction.
Then
ku − uhτ kY ≤ 3
( N Z
X
+
X
n=1 In K ∈T n
( N Z
X
n
+
(ηE,K
X
n=1 In K ∈T n
+
( N
X
n=1
M. Vohralík
τ
n
X
K ∈T
n
n
ηCR,K
(t))2 dt
n
)2 (t) dt
(ηNC1,K
n
(ηNC2,K
)2
)1/2
)1/2
)1/2
+ ηIC + 3kf − ef kX 0 .
II/IV A posteriori error control for the heat equation
14 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Estimators
Estimators
constitutive relation estimator
n
ηCR,K
(t) := k∇shτ (t) + σhn kK , t ∈ In
evaluates that −∇shn 6∈ H(div, Ω) & temporal error
equilibrium estimator
n
:= hπK kef n − ∂t shτ |In − ∇·σhn kK
ηE,K
equilibrium evaluated for ∂t shτ |In
nonconformity (constraint) estimators
n
ηNC1,K
(t) := k∇(shτ − uhτ )(t)kK , t ∈ In
n
ηNC2,K := hπK k∂t (shτ − uhτ )|In kK
evaluate the fact that uhn 6∈ H01 (Ω)
initial condition estimator
ηIC := 21/2 ksh0 − u 0 k
data oscillation estimator
kf − ef kX 0
M. Vohralík
II/IV A posteriori error control for the heat equation
15 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Estimators
Estimators
constitutive relation estimator
n
ηCR,K
(t) := k∇shτ (t) + σhn kK , t ∈ In
evaluates that −∇shn 6∈ H(div, Ω) & temporal error
equilibrium estimator
n
:= hπK kef n − ∂t shτ |In − ∇·σhn kK
ηE,K
equilibrium evaluated for ∂t shτ |In
nonconformity (constraint) estimators
n
ηNC1,K
(t) := k∇(shτ − uhτ )(t)kK , t ∈ In
n
ηNC2,K := hπK k∂t (shτ − uhτ )|In kK
evaluate the fact that uhn 6∈ H01 (Ω)
initial condition estimator
ηIC := 21/2 ksh0 − u 0 k
data oscillation estimator
kf − ef kX 0
M. Vohralík
II/IV A posteriori error control for the heat equation
15 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Estimators
Estimators
constitutive relation estimator
n
ηCR,K
(t) := k∇shτ (t) + σhn kK , t ∈ In
evaluates that −∇shn 6∈ H(div, Ω) & temporal error
equilibrium estimator
n
:= hπK kef n − ∂t shτ |In − ∇·σhn kK
ηE,K
equilibrium evaluated for ∂t shτ |In
nonconformity (constraint) estimators
n
ηNC1,K
(t) := k∇(shτ − uhτ )(t)kK , t ∈ In
n
ηNC2,K := hπK k∂t (shτ − uhτ )|In kK
evaluate the fact that uhn 6∈ H01 (Ω)
initial condition estimator
ηIC := 21/2 ksh0 − u 0 k
data oscillation estimator
kf − ef kX 0
M. Vohralík
II/IV A posteriori error control for the heat equation
15 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Estimators
Estimators
constitutive relation estimator
n
ηCR,K
(t) := k∇shτ (t) + σhn kK , t ∈ In
evaluates that −∇shn 6∈ H(div, Ω) & temporal error
equilibrium estimator
n
:= hπK kef n − ∂t shτ |In − ∇·σhn kK
ηE,K
equilibrium evaluated for ∂t shτ |In
nonconformity (constraint) estimators
n
ηNC1,K
(t) := k∇(shτ − uhτ )(t)kK , t ∈ In
n
ηNC2,K := hπK k∂t (shτ − uhτ )|In kK
evaluate the fact that uhn 6∈ H01 (Ω)
initial condition estimator
ηIC := 21/2 ksh0 − u 0 k
data oscillation estimator
kf − ef kX 0
M. Vohralík
II/IV A posteriori error control for the heat equation
15 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Estimators
Estimators
constitutive relation estimator
n
ηCR,K
(t) := k∇shτ (t) + σhn kK , t ∈ In
evaluates that −∇shn 6∈ H(div, Ω) & temporal error
equilibrium estimator
n
:= hπK kef n − ∂t shτ |In − ∇·σhn kK
ηE,K
equilibrium evaluated for ∂t shτ |In
nonconformity (constraint) estimators
n
ηNC1,K
(t) := k∇(shτ − uhτ )(t)kK , t ∈ In
n
ηNC2,K := hπK k∂t (shτ − uhτ )|In kK
evaluate the fact that uhn 6∈ H01 (Ω)
initial condition estimator
ηIC := 21/2 ksh0 − u 0 k
data oscillation estimator
kf − ef kX 0
M. Vohralík
II/IV A posteriori error control for the heat equation
15 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
15 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Separating the space and time error components
Separating the space and time error components
R
n
2
n
2
n
2
In (ηCR,K (t)) dt ≤ (ηCR,K ,1 ) + (ηCR,K ,2 ) ,
n
2
n
n
n 2
(ηCR,K
,1 ) := 2τ k∇sh + σh kK
Z
2
n
2
(ηCR,K ,2 ) := 2 k∇shτ (t) − ∇shn k2K dt = τ n k∇(shn − shn−1 )k2K
3
In
n uses η n
temporal estimator ηtm
CR,K ,2
n uses η n
n
n
n
spatial estimator ηsp
CR,K ,1 , ηE,K , ηNC1,K , and ηNC2,K
Corollary (Estimate separating the space and time errors)
( N
)1/2 ( N
)1/2
X
X
n 2
n 2
ku − uhτ kY ≤
(ηsp )
+
(ηtm )
+ηIC +3kf − ef kX 0
n=1
M. Vohralík
n=1
II/IV A posteriori error control for the heat equation
16 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Separating the space and time error components
Separating the space and time error components
R
n
2
n
2
n
2
In (ηCR,K (t)) dt ≤ (ηCR,K ,1 ) + (ηCR,K ,2 ) ,
n
2
n
n
n 2
(ηCR,K
,1 ) := 2τ k∇sh + σh kK
Z
2
n
2
(ηCR,K ,2 ) := 2 k∇shτ (t) − ∇shn k2K dt = τ n k∇(shn − shn−1 )k2K
3
In
n uses η n
temporal estimator ηtm
CR,K ,2
n uses η n
n
n
n
spatial estimator ηsp
CR,K ,1 , ηE,K , ηNC1,K , and ηNC2,K
Corollary (Estimate separating the space and time errors)
( N
)1/2 ( N
)1/2
X
X
n 2
n 2
ku − uhτ kY ≤
(ηsp )
+
(ηtm )
+ηIC +3kf − ef kX 0
n=1
M. Vohralík
n=1
II/IV A posteriori error control for the heat equation
16 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Separating the space and time error components
Separating the space and time error components
R
n
2
n
2
n
2
In (ηCR,K (t)) dt ≤ (ηCR,K ,1 ) + (ηCR,K ,2 ) ,
n
2
n
n
n 2
(ηCR,K
,1 ) := 2τ k∇sh + σh kK
Z
2
n
2
(ηCR,K ,2 ) := 2 k∇shτ (t) − ∇shn k2K dt = τ n k∇(shn − shn−1 )k2K
3
In
n uses η n
temporal estimator ηtm
CR,K ,2
n uses η n
n
n
n
spatial estimator ηsp
CR,K ,1 , ηE,K , ηNC1,K , and ηNC2,K
Corollary (Estimate separating the space and time errors)
( N
)1/2 ( N
)1/2
X
X
n 2
n 2
ku − uhτ kY ≤
(ηsp )
+
(ηtm )
+ηIC +3kf − ef kX 0
n=1
M. Vohralík
n=1
II/IV A posteriori error control for the heat equation
16 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Separating the space and time error components
Separating the space and time error components
R
n
2
n
2
n
2
In (ηCR,K (t)) dt ≤ (ηCR,K ,1 ) + (ηCR,K ,2 ) ,
n
2
n
n
n 2
(ηCR,K
,1 ) := 2τ k∇sh + σh kK
Z
2
n
2
(ηCR,K ,2 ) := 2 k∇shτ (t) − ∇shn k2K dt = τ n k∇(shn − shn−1 )k2K
3
In
n uses η n
temporal estimator ηtm
CR,K ,2
n uses η n
n
n
n
spatial estimator ηsp
CR,K ,1 , ηE,K , ηNC1,K , and ηNC2,K
Corollary (Estimate separating the space and time errors)
( N
)1/2 ( N
)1/2
X
X
n 2
n 2
ku − uhτ kY ≤
(ηsp )
+
(ηtm )
+ηIC +3kf − ef kX 0
n=1
M. Vohralík
n=1
II/IV A posteriori error control for the heat equation
16 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
16 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Assumptions for efficiency
Assumption B (Piecewise polynomials, data, and meshes)
There holds:
the meshes {T n }0≤n≤N are shape-regular uniformly in n;
the meshes cannot be refined or coarsened too quickly;
for nonconforming methods on time-varying meshes,
(hn )2 . τ n (mild inverse parabolic CFL on time step).
Local lower bound for the classical residual estimators:
Assumption C (Approximation property)
For all 1 ≤ n ≤ N and all K ∈ T n , there holds
X
k∇uhn + σhn k2K .
hK2 0 kef n − ∂t uhτ |In + ∆uhn k2K 0
K 0 ∈TK
+
X
e∈Eint
K
M. Vohralík
he k[[∇uhn ]]·ne k2e +
X
e∈EK
he−1 k[[uhn ]]k2e
II/IV A posteriori error control for the heat equation
17 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Assumptions for efficiency
Assumption B (Piecewise polynomials, data, and meshes)
There holds:
the meshes {T n }0≤n≤N are shape-regular uniformly in n;
the meshes cannot be refined or coarsened too quickly;
for nonconforming methods on time-varying meshes,
(hn )2 . τ n (mild inverse parabolic CFL on time step).
Local lower bound for the classical residual estimators:
Assumption C (Approximation property)
For all 1 ≤ n ≤ N and all K ∈ T n , there holds
X
k∇uhn + σhn k2K .
hK2 0 kef n − ∂t uhτ |In + ∆uhn k2K 0
K 0 ∈TK
+
X
e∈Eint
K
M. Vohralík
he k[[∇uhn ]]·ne k2e +
X
e∈EK
he−1 k[[uhn ]]k2e
II/IV A posteriori error control for the heat equation
17 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Efficiency
Theorem (Efficiency)
Let Assumptions B and C hold. Then, for all 1 ≤ n ≤ N,
n
n
ηsp
+ ηtm
. ku − uhτ kY (In ) + J n (uhτ ) + Efn .
Notation
jump seminorm:
J n (uhτ )2 := τ n
X
e∈E n−1
he−1 k[[uhn−1 ]]k2e + τ n
(Efn ) space-time data oscillation term
X
e∈E n
he−1 k[[uhn ]]k2e
Comments
J n (uhτ ) . ku − uhτ kX (In ) if the jumps in uhτ have zero
mean values
J n can be added to error: (J n (uhτ) = J n (u − uhτ))
efficiency is local in time but only global in space
M. Vohralík
II/IV A posteriori error control for the heat equation
18 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Efficiency
Theorem (Efficiency)
Let Assumptions B and C hold. Then, for all 1 ≤ n ≤ N,
n
n
ηsp
+ ηtm
. ku − uhτ kY (In ) + J n (uhτ ) + Efn .
Notation
jump seminorm:
J n (uhτ )2 := τ n
X
e∈E n−1
he−1 k[[uhn−1 ]]k2e + τ n
(Efn ) space-time data oscillation term
X
e∈E n
he−1 k[[uhn ]]k2e
Comments
J n (uhτ ) . ku − uhτ kX (In ) if the jumps in uhτ have zero
mean values
J n can be added to error: (J n (uhτ) = J n (u − uhτ))
efficiency is local in time but only global in space
M. Vohralík
II/IV A posteriori error control for the heat equation
18 / 25
I Setting Est. & eff. Applications Numerics R&B
Reconstructions Estimates Balancing components Efficiency
Efficiency
Theorem (Efficiency)
Let Assumptions B and C hold. Then, for all 1 ≤ n ≤ N,
n
n
ηsp
+ ηtm
. ku − uhτ kY (In ) + J n (uhτ ) + Efn .
Notation
jump seminorm:
J n (uhτ )2 := τ n
X
e∈E n−1
he−1 k[[uhn−1 ]]k2e + τ n
(Efn ) space-time data oscillation term
X
e∈E n
he−1 k[[uhn ]]k2e
Comments
J n (uhτ ) . ku − uhτ kX (In ) if the jumps in uhτ have zero
mean values
J n can be added to error: (J n (uhτ) = J n (u − uhτ))
efficiency is local in time but only global in space
M. Vohralík
II/IV A posteriori error control for the heat equation
18 / 25
I Setting Est. & eff. Applications Numerics R&B
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
18 / 25
I Setting Est. & eff. Applications Numerics R&B
Applications
Space discretization
conforming finite elements
nonconforming finite elements
discontinuous Galerkin
various finite volumes
mixed finite elements
Time discretization
backward Euler
M. Vohralík
II/IV A posteriori error control for the heat equation
19 / 25
I Setting Est. & eff. Applications Numerics R&B
Applications
Space discretization
conforming finite elements
nonconforming finite elements
discontinuous Galerkin
various finite volumes
mixed finite elements
Time discretization
backward Euler
M. Vohralík
II/IV A posteriori error control for the heat equation
19 / 25
I Setting Est. & eff. Applications Numerics R&B
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
19 / 25
I Setting Est. & eff. Applications Numerics R&B
Numerical experiment
Numerical experiment
exact solution u = ex+y +t−3 on square domain Ω =]0, 3[2 ,
T = 1.5 or T = 3
square meshes: 10 × 10, 30 × 30, 90 × 90
time steps: 0.3, 0.1, 0.3333
vertex-centered finite volumes
M. Vohralík
II/IV A posteriori error control for the heat equation
20 / 25
I Setting Est. & eff. Applications Numerics R&B
Error, estimate, and effectivity index, T = 1.5
2
5.56
10
effectivity ind. un.
error un.
estimate un.
5.55
Dual error effectivity index
5.54
1
Dual error
10
0
10
5.53
5.52
5.51
5.5
5.49
5.48
5.47
−1
10
2
10
3
4
5
10
10
10
Total numer of space−time unknowns
Dual error and estimator
M. Vohralík
6
10
5.46 2
10
3
4
5
10
10
10
Total numer of space−time unknowns
6
10
Effectivity index
II/IV A posteriori error control for the heat equation
21 / 25
I Setting Est. & eff. Applications Numerics R&B
Error, estimate, and effectivity index, T = 3
3
5.56
10
effectivity ind. un.
error un.
estimate un.
Dual error effectivity index
5.55
2
Dual error
10
1
10
5.54
5.53
5.52
5.51
5.5
5.49
5.48
0
10 3
10
4
5
10
10
Total numer of space−time unknowns
Dual error and estimator
M. Vohralík
6
10
5.47 3
10
4
5
10
10
Total numer of space−time unknowns
6
10
Effectivity index
II/IV A posteriori error control for the heat equation
22 / 25
I Setting Est. & eff. Applications Numerics R&B
Outline
1
Introduction
2
Setting
Heat equation
Equivalence of error and residual
Discrete setting
3
A posteriori error estimates and their efficiency
Potential and flux reconstructions
A posteriori error estimates
Balancing the spatial and temporal error components
Efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
II/IV A posteriori error control for the heat equation
22 / 25
I Setting Est. & eff. Applications Numerics R&B
Previous results
Continuous finite elements
Bieterman and Babuška (1982), introduction
Eriksson and Johnson (1991), duality techniques
Picasso (1998), no derefinement allowed
Babuška, Feistauer, and Šolín (2001), continuous-in-time
discretization
Strouboulis, Babuška, and Datta (2003), guaranteed
estimates
Verfürth (2003), efficiency, robustness with respect to the
final time
Makridakis and Nochetto (2003), elliptic reconstruction
Bergam, Bernardi, and Mghazli (2005), efficiency
Lakkis and Makridakis (2006), elliptic reconstruction
M. Vohralík
II/IV A posteriori error control for the heat equation
23 / 25
I Setting Est. & eff. Applications Numerics R&B
Previous results
Finite volumes
Ohlberger (2001), non-energy-norm estimates
Amara, Nadau, and Trujillo (2004), energy-norm estimates
Discontinuous Galerkin finite elements
Sun and Wheeler (2005, 2006), non-energy-norm
estimates
Georgoulis and Lakkis (2009)
Nonconforming finite elements
Nicaise and Soualem (2005)
Mixed finite elements
Cascón, Ferragut, and Asensio (2006)
M. Vohralík
II/IV A posteriori error control for the heat equation
24 / 25
I Setting Est. & eff. Applications Numerics R&B
Previous results
Finite volumes
Ohlberger (2001), non-energy-norm estimates
Amara, Nadau, and Trujillo (2004), energy-norm estimates
Discontinuous Galerkin finite elements
Sun and Wheeler (2005, 2006), non-energy-norm
estimates
Georgoulis and Lakkis (2009)
Nonconforming finite elements
Nicaise and Soualem (2005)
Mixed finite elements
Cascón, Ferragut, and Asensio (2006)
M. Vohralík
II/IV A posteriori error control for the heat equation
24 / 25
I Setting Est. & eff. Applications Numerics R&B
Previous results
Finite volumes
Ohlberger (2001), non-energy-norm estimates
Amara, Nadau, and Trujillo (2004), energy-norm estimates
Discontinuous Galerkin finite elements
Sun and Wheeler (2005, 2006), non-energy-norm
estimates
Georgoulis and Lakkis (2009)
Nonconforming finite elements
Nicaise and Soualem (2005)
Mixed finite elements
Cascón, Ferragut, and Asensio (2006)
M. Vohralík
II/IV A posteriori error control for the heat equation
24 / 25
I Setting Est. & eff. Applications Numerics R&B
Previous results
Finite volumes
Ohlberger (2001), non-energy-norm estimates
Amara, Nadau, and Trujillo (2004), energy-norm estimates
Discontinuous Galerkin finite elements
Sun and Wheeler (2005, 2006), non-energy-norm
estimates
Georgoulis and Lakkis (2009)
Nonconforming finite elements
Nicaise and Soualem (2005)
Mixed finite elements
Cascón, Ferragut, and Asensio (2006)
M. Vohralík
II/IV A posteriori error control for the heat equation
24 / 25
I Setting Est. & eff. Applications Numerics R&B
Bibliography
Bibliography
E RN A., VOHRALÍK M., A posteriori error estimation based
on potential and flux reconstruction for the heat equation,
SIAM J. Numer. Anal. 48 (2010), 198–223.
Thank you for your attention!
M. Vohralík
II/IV A posteriori error control for the heat equation
25 / 25