Laplace a posteriori error estimates
in a unified framework
Martin Vohralík, INRIA Paris-Rocquencourt
Lecture I/IV
IIT Bombay, July 13–17, 2015
I Estimate Efficiency Applications Numerics R&B
Outline
1
Introduction
2
A guaranteed a posteriori error estimate
3
Polynomial-degree-robust local efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
2 / 39
I Estimate Efficiency Applications Numerics R&B
Outline
1
Introduction
2
A guaranteed a posteriori error estimate
3
Polynomial-degree-robust local efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
2 / 39
I Estimate Efficiency Applications Numerics R&B
What is an a posteriori error estimate
A posteriori error estimate
Let u be a weak solution of a PDE (−∆u = f in Ω ⊂ Rd ,
u = 0 on ∂Ω).
Let uh be its approximate numerical solution.
A priori error estimate: k∇(u − uh )k ≤ C(u)hk . Dependent
on u, not computable. Useful in theoretical assessment of
convergence.
A posteriori error estimate: k∇(u − uh )k ≤ Cη(uh ). Only
uses uh , computable. Great in practical calculation.
Usual form
Element indicators ηK (uh ), K ∈ Th .
Can be used to determine mesh elements with large error.
We can then refine these elements: mesh adaptivity.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
3 / 39
I Estimate Efficiency Applications Numerics R&B
What is an a posteriori error estimate
A posteriori error estimate
Let u be a weak solution of a PDE (−∆u = f in Ω ⊂ Rd ,
u = 0 on ∂Ω).
Let uh be its approximate numerical solution.
A priori error estimate: k∇(u − uh )k ≤ C(u)hk . Dependent
on u, not computable. Useful in theoretical assessment of
convergence.
A posteriori error estimate: k∇(u − uh )k ≤ Cη(uh ). Only
uses uh , computable. Great in practical calculation.
Usual form
Element indicators ηK (uh ), K ∈ Th .
Can be used to determine mesh elements with large error.
We can then refine these elements: mesh adaptivity.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
3 / 39
I Estimate Efficiency Applications Numerics R&B
What is an a posteriori error estimate
A posteriori error estimate
Let u be a weak solution of a PDE (−∆u = f in Ω ⊂ Rd ,
u = 0 on ∂Ω).
Let uh be its approximate numerical solution.
A priori error estimate: k∇(u − uh )k ≤ C(u)hk . Dependent
on u, not computable. Useful in theoretical assessment of
convergence.
A posteriori error estimate: k∇(u − uh )k ≤ Cη(uh ). Only
uses uh , computable. Great in practical calculation.
Usual form
Element indicators ηK (uh ), K ∈ Th .
Can be used to determine mesh elements with large error.
We can then refine these elements: mesh adaptivity.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
3 / 39
I Estimate Efficiency Applications Numerics R&B
What is an a posteriori error estimate
A posteriori error estimate
Let u be a weak solution of a PDE (−∆u = f in Ω ⊂ Rd ,
u = 0 on ∂Ω).
Let uh be its approximate numerical solution.
A priori error estimate: k∇(u − uh )k ≤ C(u)hk . Dependent
on u, not computable. Useful in theoretical assessment of
convergence.
A posteriori error estimate: k∇(u − uh )k ≤ Cη(uh ). Only
uses uh , computable. Great in practical calculation.
Usual form
Element indicators ηK (uh ), K ∈ Th .
Can be used to determine mesh elements with large error.
We can then refine these elements: mesh adaptivity.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
3 / 39
I Estimate Efficiency Applications Numerics R&B
What is an a posteriori error estimate
A posteriori error estimate
Let u be a weak solution of a PDE (−∆u = f in Ω ⊂ Rd ,
u = 0 on ∂Ω).
Let uh be its approximate numerical solution.
A priori error estimate: k∇(u − uh )k ≤ C(u)hk . Dependent
on u, not computable. Useful in theoretical assessment of
convergence.
A posteriori error estimate: k∇(u − uh )k ≤ Cη(uh ). Only
uses uh , computable. Great in practical calculation.
Usual form
Element indicators ηK (uh ), K ∈ Th .
Can be used to determine mesh elements with large error.
We can then refine these elements: mesh adaptivity.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
3 / 39
I Estimate Efficiency Applications Numerics R&B
What an a posteriori error estimate should fulfill
Optimal estimate for −∆u = f in Ω ⊂ Rd , u = 0 on ∂Ω
1/2
guaranteed upper bound: 

X
ηK (uh )2
k∇(u − uh )k ≤


K ∈Th
local efficiency:
ηK (uh ) ≤ Ceff k∇(u − uh )kTK
asymptotic exactness:
nP
2
K ∈Th ηK (uh )
∀K ∈ Th
o1/2
&1
k∇(u − uh )k
robustness: the three previous properties hold
independently of the parameters of the problem and of
their variation (size of Ω, shape of Ω, regularity of u, local
refinement of Th , sizes hK , polynomial degree of uh )
small evaluation cost of ηK (uh )
error components identification
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
4 / 39
I Estimate Efficiency Applications Numerics R&B
What an a posteriori error estimate should fulfill
Optimal estimate for −∆u = f in Ω ⊂ Rd , u = 0 on ∂Ω
1/2
guaranteed upper bound: 

X
ηK (uh )2
k∇(u − uh )k ≤


K ∈Th
local efficiency:
ηK (uh ) ≤ Ceff k∇(u − uh )kTK
asymptotic exactness:
nP
2
K ∈Th ηK (uh )
∀K ∈ Th
o1/2
&1
k∇(u − uh )k
robustness: the three previous properties hold
independently of the parameters of the problem and of
their variation (size of Ω, shape of Ω, regularity of u, local
refinement of Th , sizes hK , polynomial degree of uh )
small evaluation cost of ηK (uh )
error components identification
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
4 / 39
I Estimate Efficiency Applications Numerics R&B
What an a posteriori error estimate should fulfill
Optimal estimate for −∆u = f in Ω ⊂ Rd , u = 0 on ∂Ω
1/2
guaranteed upper bound: 

X
ηK (uh )2
k∇(u − uh )k ≤


K ∈Th
local efficiency:
ηK (uh ) ≤ Ceff k∇(u − uh )kTK
asymptotic exactness:
nP
2
K ∈Th ηK (uh )
∀K ∈ Th
o1/2
&1
k∇(u − uh )k
robustness: the three previous properties hold
independently of the parameters of the problem and of
their variation (size of Ω, shape of Ω, regularity of u, local
refinement of Th , sizes hK , polynomial degree of uh )
small evaluation cost of ηK (uh )
error components identification
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
4 / 39
I Estimate Efficiency Applications Numerics R&B
What an a posteriori error estimate should fulfill
Optimal estimate for −∆u = f in Ω ⊂ Rd , u = 0 on ∂Ω
1/2
guaranteed upper bound: 

X
ηK (uh )2
k∇(u − uh )k ≤


K ∈Th
local efficiency:
ηK (uh ) ≤ Ceff k∇(u − uh )kTK
asymptotic exactness:
nP
2
K ∈Th ηK (uh )
∀K ∈ Th
o1/2
&1
k∇(u − uh )k
robustness: the three previous properties hold
independently of the parameters of the problem and of
their variation (size of Ω, shape of Ω, regularity of u, local
refinement of Th , sizes hK , polynomial degree of uh )
small evaluation cost of ηK (uh )
error components identification
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
4 / 39
I Estimate Efficiency Applications Numerics R&B
What an a posteriori error estimate should fulfill
Optimal estimate for −∆u = f in Ω ⊂ Rd , u = 0 on ∂Ω
1/2
guaranteed upper bound: 

X
ηK (uh )2
k∇(u − uh )k ≤


K ∈Th
local efficiency:
ηK (uh ) ≤ Ceff k∇(u − uh )kTK
asymptotic exactness:
nP
2
K ∈Th ηK (uh )
∀K ∈ Th
o1/2
&1
k∇(u − uh )k
robustness: the three previous properties hold
independently of the parameters of the problem and of
their variation (size of Ω, shape of Ω, regularity of u, local
refinement of Th , sizes hK , polynomial degree of uh )
small evaluation cost of ηK (uh )
error components identification
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
4 / 39
I Estimate Efficiency Applications Numerics R&B
What an a posteriori error estimate should fulfill
Optimal estimate for −∆u = f in Ω ⊂ Rd , u = 0 on ∂Ω
1/2
guaranteed upper bound: 

X
ηK (uh )2
k∇(u − uh )k ≤


K ∈Th
local efficiency:
ηK (uh ) ≤ Ceff k∇(u − uh )kTK
asymptotic exactness:
nP
2
K ∈Th ηK (uh )
∀K ∈ Th
o1/2
&1
k∇(u − uh )k
robustness: the three previous properties hold
independently of the parameters of the problem and of
their variation (size of Ω, shape of Ω, regularity of u, local
refinement of Th , sizes hK , polynomial degree of uh )
small evaluation cost of ηK (uh )
error components identification
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
4 / 39
I Estimate Efficiency Applications Numerics R&B
Outline
1
Introduction
2
A guaranteed a posteriori error estimate
3
Polynomial-degree-robust local efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
4 / 39
I Estimate Efficiency Applications Numerics R&B
Model problem
Model problem
−∆u = f
u=0
in Ω,
on ∂Ω
Weak formulation
Find u ∈ H01 (Ω) such that
(∇u, ∇v ) = (f , v )
∀v ∈ H01 (Ω)
Properties of the weak solution
u ∈ H01 (Ω) (constraint)
σ := −∇u (constitutive relation)
∇·σ = f (equilibrium)
σ ∈ H(div, Ω) (constraint)
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
5 / 39
I Estimate Efficiency Applications Numerics R&B
Model problem
Model problem
−∆u = f
u=0
in Ω,
on ∂Ω
Weak formulation
Find u ∈ H01 (Ω) such that
(∇u, ∇v ) = (f , v )
∀v ∈ H01 (Ω)
Properties of the weak solution
u ∈ H01 (Ω) (constraint)
σ := −∇u (constitutive relation)
∇·σ = f (equilibrium)
σ ∈ H(div, Ω) (constraint)
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
5 / 39
I Estimate Efficiency Applications Numerics R&B
Model problem
Model problem
−∆u = f
u=0
in Ω,
on ∂Ω
Weak formulation
Find u ∈ H01 (Ω) such that
(∇u, ∇v ) = (f , v )
∀v ∈ H01 (Ω)
Properties of the weak solution
u ∈ H01 (Ω) (constraint)
σ := −∇u (constitutive relation)
∇·σ = f (equilibrium)
σ ∈ H(div, Ω) (constraint)
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
5 / 39
I Estimate Efficiency Applications Numerics R&B
Exact solution and flux
1.5
1.8
-exact flux
exact solution
1.6
1.0
1.4
1.2
0.5
1.0
0.0
0.8
0.6
-0.5
0.4
0.2
-1.0
0.0
-0.2
-1.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Solution u is continuous
M. Vohralík
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Flux σ := −K∇u is continuous
I/IV Laplace a posteriori error estimates in a unified framework
6 / 39
I Estimate Efficiency Applications Numerics R&B
Approximate solution and flux
1.8
1.5
exact solution
approximate solution
1.6
-exact flux
-approximate flux
1.0
1.4
1.2
0.5
1.0
0.8
0.0
0.6
-0.5
0.4
0.2
-1.0
0.0
-0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Approximate solution uh is not
necessarily continuous
M. Vohralík
-1.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Approximate flux −K∇uh is not
necessarily continuous
I/IV Laplace a posteriori error estimates in a unified framework
7 / 39
I Estimate Efficiency Applications Numerics R&B
Potential and flux reconstructions
1.8
1.5
exact solution
approximate solution
postprocessed solution
1.6
-exact flux
-approximate flux
-postprocessed flux
1.0
1.4
1.2
0.5
1.0
0.8
0.0
0.6
-0.5
0.4
0.2
-1.0
0.0
-0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Potential reconstruction
M. Vohralík
1.0
-1.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Flux reconstruction
I/IV Laplace a posteriori error estimates in a unified framework
8 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Theorem (A guaranteed a posteriori error estimate, Prager and Synge
(1947), Dari, Durán, Padra, and Vampa (1996), Ainsworth (2005), Kim (2007))
Let u ∈ H01 (Ω) be the weak solution;
uh ∈ H 1 (Th ) := {v ∈ L2 (Ω), v |K ∈ H 1 (K ) ∀K ∈ Th } be
arbitrary;
sh ∈ H01 (Ω) and σh ∈ H(div, Ω) be such that
(∇·σh , 1)K = (f , 1)K for all K ∈ Th .
Then
k∇(u − uh )k2 ≤
X
K ∈Th
+
2
hK
k∇uh + σh kK +
kf − ∇·σh kK
|
{z
} |π
{z
}
constitutive relation
X
K ∈Th
M. Vohralík
equilibrium
k∇(uh − sh )k2K .
{z
}
|
constraint
I/IV Laplace a posteriori error estimates in a unified framework
9 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Theorem (A guaranteed a posteriori error estimate, Prager and Synge
(1947), Dari, Durán, Padra, and Vampa (1996), Ainsworth (2005), Kim (2007))
Let u ∈ H01 (Ω) be the weak solution;
uh ∈ H 1 (Th ) := {v ∈ L2 (Ω), v |K ∈ H 1 (K ) ∀K ∈ Th } be
arbitrary;
sh ∈ H01 (Ω) and σh ∈ H(div, Ω) be such that
(∇·σh , 1)K = (f , 1)K for all K ∈ Th .
Then
k∇(u − uh )k2 ≤
X
K ∈Th
+
2
hK
k∇uh + σh kK +
kf − ∇·σh kK
|
{z
} |π
{z
}
constitutive relation
X
K ∈Th
M. Vohralík
equilibrium
k∇(uh − sh )k2K .
{z
}
|
constraint
I/IV Laplace a posteriori error estimates in a unified framework
9 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Proof.
define s ∈ H01 (Ω) by
∀v ∈ H01 (Ω)
(∇s, ∇v ) = (∇uh , ∇v )
develop (Pythagoras)
k∇(u − uh )k2 = k∇(u − s)k2 + k∇(s − uh )k2
projection definition of s:
k∇(s − uh )k2 =
min k∇(v − uh )k2
v ∈H01 (Ω)
|
{z
distance of uh to H01 (Ω)
}
dual norm characterization, definition of s:
k∇(u − s)k2 =
sup
|
M. Vohralík
(∇(u − ), ∇ϕ)2
ϕ∈H01 (Ω); k∇ϕk=1
{z
dual norm of the residual
}
I/IV Laplace a posteriori error estimates in a unified framework
10 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Proof.
define s ∈ H01 (Ω) by
∀v ∈ H01 (Ω)
(∇s, ∇v ) = (∇uh , ∇v )
develop (Pythagoras)
k∇(u − uh )k2 = k∇(u − s)k2 + k∇(s − uh )k2
projection definition of s:
k∇(s − uh )k2 =
min k∇(v − uh )k2
v ∈H01 (Ω)
|
{z
distance of uh to H01 (Ω)
}
dual norm characterization, definition of s:
k∇(u − s)k2 =
sup
|
M. Vohralík
(∇(u − ), ∇ϕ)2
ϕ∈H01 (Ω); k∇ϕk=1
{z
dual norm of the residual
}
I/IV Laplace a posteriori error estimates in a unified framework
10 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Proof.
define s ∈ H01 (Ω) by
∀v ∈ H01 (Ω)
(∇s, ∇v ) = (∇uh , ∇v )
develop (Pythagoras)
k∇(u − uh )k2 = k∇(u − s)k2 + k∇(s − uh )k2
projection definition of s:
k∇(s − uh )k2 =
min k∇(v − uh )k2
v ∈H01 (Ω)
|
{z
distance of uh to H01 (Ω)
}
dual norm characterization, definition of s:
k∇(u − s)k2 =
sup
|
M. Vohralík
(∇(u − ), ∇ϕ)2
ϕ∈H01 (Ω); k∇ϕk=1
{z
dual norm of the residual
}
I/IV Laplace a posteriori error estimates in a unified framework
10 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Proof.
define s ∈ H01 (Ω) by
∀v ∈ H01 (Ω)
(∇s, ∇v ) = (∇uh , ∇v )
develop (Pythagoras)
k∇(u − uh )k2 = k∇(u − s)k2 + k∇(s − uh )k2
projection definition of s:
k∇(s − uh )k2 =
min k∇(v − uh )k2
v ∈H01 (Ω)
|
{z
distance of uh to H01 (Ω)
}
dual norm characterization, definition of s:
k∇(u − s)k2 =
sup
|
M. Vohralík
(∇(u − s), ∇ϕ)2
ϕ∈H01 (Ω); k∇ϕk=1
{z
dual norm of the residual
}
I/IV Laplace a posteriori error estimates in a unified framework
10 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Proof.
define s ∈ H01 (Ω) by
∀v ∈ H01 (Ω)
(∇s, ∇v ) = (∇uh , ∇v )
develop (Pythagoras)
k∇(u − uh )k2 = k∇(u − s)k2 + k∇(s − uh )k2
projection definition of s:
k∇(s − uh )k2 =
min k∇(v − uh )k2
v ∈H01 (Ω)
|
{z
distance of uh to H01 (Ω)
}
dual norm characterization, definition of s:
k∇(u − s)k2 =
sup
|
M. Vohralík
(∇(u − uh ), ∇ϕ)2
ϕ∈H01 (Ω); k∇ϕk=1
{z
dual norm of the residual
}
I/IV Laplace a posteriori error estimates in a unified framework
10 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Proof (continuation).
nonconformity upper bound:
min k∇(v − uh )k ≤ k∇(uh − sh )k
v ∈H01 (Ω)
weak solution definition, equilibrated flux, Green theorem:
(∇(u − uh ), ∇ϕ) = (f , ϕ) − (∇uh , ∇ϕ)
= (f − ∇·σh , ϕ) − (∇uh + σh , ∇ϕ)
Cauchy–Schwarz and Poincaré inequalities, equilibration:
X
−(∇uh + σh , ∇ϕ) ≤
k∇uh + σh kK k∇ϕkK ,
K ∈Th
(f − ∇·σh , ϕ) =
≤
M. Vohralík
X
K ∈Th
(f − ∇·σh , ϕ − ϕK )K
X hK
kf − ∇·σh kK k∇ϕkK
π
K ∈Th
I/IV Laplace a posteriori error estimates in a unified framework
11 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Proof (continuation).
nonconformity upper bound:
min k∇(v − uh )k ≤ k∇(uh − sh )k
v ∈H01 (Ω)
weak solution definition, equilibrated flux, Green theorem:
(∇(u − uh ), ∇ϕ) = (f , ϕ) − (∇uh , ∇ϕ)
= (f − ∇·σh , ϕ) − (∇uh + σh , ∇ϕ)
Cauchy–Schwarz and Poincaré inequalities, equilibration:
X
−(∇uh + σh , ∇ϕ) ≤
k∇uh + σh kK k∇ϕkK ,
K ∈Th
(f − ∇·σh , ϕ) =
≤
M. Vohralík
X
K ∈Th
(f − ∇·σh , ϕ − ϕK )K
X hK
kf − ∇·σh kK k∇ϕkK
π
K ∈Th
I/IV Laplace a posteriori error estimates in a unified framework
11 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Proof (continuation).
nonconformity upper bound:
min k∇(v − uh )k ≤ k∇(uh − sh )k
v ∈H01 (Ω)
weak solution definition, equilibrated flux, Green theorem:
(∇(u − uh ), ∇ϕ) = (f , ϕ) − (∇uh , ∇ϕ)
= (f − ∇·σh , ϕ) − (∇uh + σh , ∇ϕ)
Cauchy–Schwarz and Poincaré inequalities, equilibration:
X
−(∇uh + σh , ∇ϕ) ≤
k∇uh + σh kK k∇ϕkK ,
K ∈Th
(f − ∇·σh , ϕ) =
≤
M. Vohralík
X
K ∈Th
(f − ∇·σh , ϕ − ϕK )K
X hK
kf − ∇·σh kK k∇ϕkK
π
K ∈Th
I/IV Laplace a posteriori error estimates in a unified framework
11 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Proof (continuation).
nonconformity upper bound:
min k∇(v − uh )k ≤ k∇(uh − sh )k
v ∈H01 (Ω)
weak solution definition, equilibrated flux, Green theorem:
(∇(u − uh ), ∇ϕ) = (f , ϕ) − (∇uh , ∇ϕ)
= (f − ∇·σh , ϕ) − (∇uh + σh , ∇ϕ)
Cauchy–Schwarz and Poincaré inequalities, equilibration:
X
−(∇uh + σh , ∇ϕ) ≤
k∇uh + σh kK k∇ϕkK ,
K ∈Th
(f − ∇·σh , ϕ) =
≤
M. Vohralík
X
K ∈Th
(f − ∇·σh , ϕ − ϕK )K
X hK
kf − ∇·σh kK k∇ϕkK
π
K ∈Th
I/IV Laplace a posteriori error estimates in a unified framework
11 / 39
I Estimate Efficiency Applications Numerics R&B
A posteriori error estimate
Proof (continuation).
nonconformity upper bound:
min k∇(v − uh )k ≤ k∇(uh − sh )k
v ∈H01 (Ω)
weak solution definition, equilibrated flux, Green theorem:
(∇(u − uh ), ∇ϕ) = (f , ϕ) − (∇uh , ∇ϕ)
= (f − ∇·σh , ϕ) − (∇uh + σh , ∇ϕ)
Cauchy–Schwarz and Poincaré inequalities, equilibration:
X
−(∇uh + σh , ∇ϕ) ≤
k∇uh + σh kK k∇ϕkK ,
K ∈Th
(f − ∇·σh , ϕ) =
≤
M. Vohralík
X
K ∈Th
(f − ∇·σh , ϕ − ϕK )K
X hK
kf − ∇·σh kK k∇ϕkK
π
K ∈Th
I/IV Laplace a posteriori error estimates in a unified framework
11 / 39
I Estimate Efficiency Applications Numerics R&B
Global potential and flux reconstructions
Ideally
σh := arg
min
vh ∈Vh , ∇·vh =ΠQh f
k∇uh + vh k
sh := arg min k∇(uh − vh )k
vh ∈Vh
Vh ⊂ H(div, Ω), Qh ⊂ L2 (Ω), Vh ⊂ H01 (Ω)
too expensive, global minimization problems (the
hypercircle method)
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
12 / 39
I Estimate Efficiency Applications Numerics R&B
Local potential and flux reconstructions
Partition of unity localization
σha := arg
min
kψa ∇uh + vh kωa
vh ∈Vah , ∇·vh =?
sha := arg mina k∇(ψa uh − vh )kωa
vh ∈Vh
cut-off by hat basis functions ψa ∈ P1 (Th ) ∩ H 1 (Ω)
X
X
a
σh :=
σha , sh :=
sha
4
a∈Vh
local minimizations
a∈Vh
a3
a5
a ∈ Vh
a2
a1
M. Vohralík
patch ωa
ψa (a) = 1, ψa (a∗ ) = 0
I/IV Laplace a posteriori error estimates in a unified framework
13 / 39
I Estimate Efficiency Applications Numerics R&B
Local potential and flux reconstructions
Partition of unity localization
σha := arg
min
kψa ∇uh + vh kωa
vh ∈Vah , ∇·vh =?
sha := arg mina k∇(ψa uh − vh )kωa
vh ∈Vh
cut-off by hat basis functions ψa ∈ P1 (Th ) ∩ H 1 (Ω)
X
X
a
σh :=
σha , sh :=
sha
4
a∈Vh
local minimizations
a∈Vh
a3
a5
a ∈ Vh
a2
a1
M. Vohralík
patch ωa
ψa (a) = 1, ψa (a∗ ) = 0
I/IV Laplace a posteriori error estimates in a unified framework
13 / 39
I Estimate Efficiency Applications Numerics R&B
Potential reconstruction
Potential uh
Potential reconstruction sh
n (à gauche) et solution conforme sn (à droite)
Figure
5 – Solution
post-traitée
h
igure
5 – Solution
post-traitée
unh (àugauche)
et solution conforme snh (à hdroite)
M. Vohralík
I/IV Laplace a posteriori errornestimates
n in a unified
1 framework
14 / 39
I Estimate Efficiency Applications Numerics R&B
Equilibrated
flux
reconstruction
Estimation a posteriori
Reconstructions pour les schémas VF Estimation a posteriori
tions des flux
Reconstructions pour les schémas
Reconstructions des flux
Flux −∇u
Flux reconstruction σh
ale :
Forme
générale :
h
onstantes par morceaux
θ h,τenettemps
ψ h,τ : constantes par morceaux en temps
ψ nh := ψ h,τ |I n ∈ H(div, Ω),
θ nh ∀1
:= ≤
θ h,τ
n |≤
ψ nh := ψ h,τ |I n ∈ H(div, Ω), ∀1 ≤ n ≤ N.
I n ,N.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
15 / 39
I Estimate Efficiency Applications Numerics R&B
Local equilibrated flux reconstruction
Assumption A (Galerkin orthogonality)
There holds uh ∈ H 1 (Th ) and
(∇uh , ∇ψa )ωa = (f , ψa )ωa
∀a ∈ Vhint .
Vah × Qha : MFE spaces (hom. Neumann BC for a ∈ Vhint ,
homogeneous Dirichlet BC on ∂ωa ∩ ∂Ω for a ∈ Vhext )
Definition (Constr. of σh , Destuynder and Métivet (1999) & Braess and Schöberl (2008))
Let Assumption A be satisfied. For each a ∈ Vh , prescribe
σha ∈ Vah and r̄ha ∈ Qha by solving the local mixed FE problem
σha := arg
min
h
(σha , vh )ωa
− (r̄ha , ∇·vh )ωa
(∇·σha , qh )ωa
M. Vohralík
kψa ∇uh + vh kωa
vh ∈Vah , ∇·vh =ΠQ a (ψa f −∇ψa ·∇uh )
m
= −(ψa ∇uh , vh )ωa
= (ψa f − ∇ψa ·∇uh , qh )ωa
∀vh ∈ Vah ,
∀qh ∈ Qha .
I/IV Laplace a posteriori error estimates in a unified framework
16 / 39
I Estimate Efficiency Applications Numerics R&B
Local equilibrated flux reconstruction
Assumption A (Galerkin orthogonality)
There holds uh ∈ H 1 (Th ) and
(∇uh , ∇ψa )ωa = (f , ψa )ωa
∀a ∈ Vhint .
Vah × Qha : MFE spaces (hom. Neumann BC for a ∈ Vhint ,
homogeneous Dirichlet BC on ∂ωa ∩ ∂Ω for a ∈ Vhext )
Definition (Constr. of σh , Destuynder and Métivet (1999) & Braess and Schöberl (2008))
Let Assumption A be satisfied. For each a ∈ Vh , prescribe
σha ∈ Vah and r̄ha ∈ Qha by solving the local mixed FE problem
σha := arg
min
h
(σha , vh )ωa
− (r̄ha , ∇·vh )ωa
(∇·σha , qh )ωa
M. Vohralík
kψa ∇uh + vh kωa
vh ∈Vah , ∇·vh =ΠQ a (ψa f −∇ψa ·∇uh )
m
= −(ψa ∇uh , vh )ωa
= (ψa f − ∇ψa ·∇uh , qh )ωa
∀vh ∈ Vah ,
∀qh ∈ Qha .
I/IV Laplace a posteriori error estimates in a unified framework
16 / 39
I Estimate Efficiency Applications Numerics R&B
Local equilibrated flux reconstruction
Assumption A (Galerkin orthogonality)
There holds uh ∈ H 1 (Th ) and
(∇uh , ∇ψa )ωa = (f , ψa )ωa
∀a ∈ Vhint .
Vah × Qha : MFE spaces (hom. Neumann BC for a ∈ Vhint ,
homogeneous Dirichlet BC on ∂ωa ∩ ∂Ω for a ∈ Vhext )
Definition (Constr. of σh , Destuynder and Métivet (1999) & Braess and Schöberl (2008))
Let Assumption A be satisfied. For each a ∈ Vh , prescribe
σha ∈ Vah and r̄ha ∈ Qha by solving the local mixed FE problem
σha := arg
min
h
(σha , vh )ωa
− (r̄ha , ∇·vh )ωa
(∇·σha , qh )ωa
M. Vohralík
kψa ∇uh + vh kωa
vh ∈Vah , ∇·vh =ΠQ a (ψa f −∇ψa ·∇uh )
m
= −(ψa ∇uh , vh )ωa
= (ψa f − ∇ψa ·∇uh , qh )ωa
∀vh ∈ Vah ,
∀qh ∈ Qha .
I/IV Laplace a posteriori error estimates in a unified framework
16 / 39
I Estimate Efficiency Applications Numerics R&B
Local equilibrated flux reconstruction
Assumption A (Galerkin orthogonality)
There holds uh ∈ H 1 (Th ) and
(∇uh , ∇ψa )ωa = (f , ψa )ωa
∀a ∈ Vhint .
Vah × Qha : MFE spaces (hom. Neumann BC for a ∈ Vhint ,
homogeneous Dirichlet BC on ∂ωa ∩ ∂Ω for a ∈ Vhext )
Definition (Constr. of σh , Destuynder and Métivet (1999) & Braess and Schöberl (2008))
Let Assumption A be satisfied. For each a ∈ Vh , prescribe
σha ∈ Vah and r̄ha ∈ Qha by solving the local mixed FE problem
σha := arg
min
h
(σha , vh )ωa
− (r̄ha , ∇·vh )ωa
(∇·σha , qh )ωa
M. Vohralík
kψa ∇uh + vh kωa
vh ∈Vah , ∇·vh =ΠQ a (ψa f −∇ψa ·∇uh )
m
= −(ψa ∇uh , vh )ωa
= (ψa f − ∇ψa ·∇uh , qh )ωa
∀vh ∈ Vah ,
∀qh ∈ Qha .
I/IV Laplace a posteriori error estimates in a unified framework
16 / 39
I Estimate Efficiency Applications Numerics R&B
Comments
H(div, Ω)-conformity
P
σha ∈ H(div, Ω) ⇒ σh = a∈Vh σha ∈ H(div, Ω)
Neumann compatibility condition
for a ∈ Vhint , one needs
(ψa f − ∇ψa ·∇uh , 1)ωa = 0
but Assumption A gives
0 = (f , ψa )ωa − (∇uh , ∇ψa )ωa = (ψa f − ∇ψa ·∇uh , 1)ωa
Divergence
Neumann compatibility condition gives
∇·σha |K = ΠQh (ψa f − ∇ψa ·∇uh )|K
∀K ∈ Th
P
the
σh |K = a∈VK σha |K and the partition of unity
P fact that
a
a∈VK ψ |K = 1|K yield
M. Vohralík
∇·σh |K = ΠQh f |K
∀K ∈ Th
I/IV Laplace a posteriori error estimates in a unified framework
17 / 39
I Estimate Efficiency Applications Numerics R&B
Comments
H(div, Ω)-conformity
P
σha ∈ H(div, Ω) ⇒ σh = a∈Vh σha ∈ H(div, Ω)
Neumann compatibility condition
for a ∈ Vhint , one needs
(ψa f − ∇ψa ·∇uh , 1)ωa = 0
but Assumption A gives
0 = (f , ψa )ωa − (∇uh , ∇ψa )ωa = (ψa f − ∇ψa ·∇uh , 1)ωa
Divergence
Neumann compatibility condition gives
∇·σha |K = ΠQh (ψa f − ∇ψa ·∇uh )|K
∀K ∈ Th
P
the
σh |K = a∈VK σha |K and the partition of unity
P fact that
a
a∈VK ψ |K = 1|K yield
M. Vohralík
∇·σh |K = ΠQh f |K
∀K ∈ Th
I/IV Laplace a posteriori error estimates in a unified framework
17 / 39
I Estimate Efficiency Applications Numerics R&B
Comments
H(div, Ω)-conformity
P
σha ∈ H(div, Ω) ⇒ σh = a∈Vh σha ∈ H(div, Ω)
Neumann compatibility condition
for a ∈ Vhint , one needs
(ψa f − ∇ψa ·∇uh , 1)ωa = 0
but Assumption A gives
0 = (f , ψa )ωa − (∇uh , ∇ψa )ωa = (ψa f − ∇ψa ·∇uh , 1)ωa
Divergence
Neumann compatibility condition gives
∇·σha |K = ΠQh (ψa f − ∇ψa ·∇uh )|K
∀K ∈ Th
P
the
σh |K = a∈VK σha |K and the partition of unity
P fact that
a
a∈VK ψ |K = 1|K yield
M. Vohralík
∇·σh |K = ΠQh f |K
∀K ∈ Th
I/IV Laplace a posteriori error estimates in a unified framework
17 / 39
I Estimate Efficiency Applications Numerics R&B
Local potential reconstruction
Vha : FE space (hom. Dirichlet BC on ∂ωa for all a ∈ Vh )
Definition (Construction of sh , ≈ Carstensen and Merdon (2013))
Let uh ∈ H 1 (Th ). For each a ∈ Vh , prescribe sha ∈ Vha by solving
the local conforming finite element problem
sha := arg mina k∇(ψa uh − vh )kωa
vh ∈Vh
m
(∇sha , ∇vh )ωa = (∇(ψa uh ), ∇vh )ωa
M. Vohralík
∀vh ∈ Vha .
I/IV Laplace a posteriori error estimates in a unified framework
18 / 39
I Estimate Efficiency Applications Numerics R&B
Local potential reconstruction
Vha : FE space (hom. Dirichlet BC on ∂ωa for all a ∈ Vh )
Definition (Construction of sh , ≈ Carstensen and Merdon (2013))
Let uh ∈ H 1 (Th ). For each a ∈ Vh , prescribe sha ∈ Vha by solving
the local conforming finite element problem
sha := arg mina k∇(ψa uh − vh )kωa
vh ∈Vh
m
(∇sha , ∇vh )ωa = (∇(ψa uh ), ∇vh )ωa
M. Vohralík
∀vh ∈ Vha .
I/IV Laplace a posteriori error estimates in a unified framework
18 / 39
I Estimate Efficiency Applications Numerics R&B
Local potential reconstruction
Vha : FE space (hom. Dirichlet BC on ∂ωa for all a ∈ Vh )
Definition (Construction of sh , ≈ Carstensen and Merdon (2013))
Let uh ∈ H 1 (Th ). For each a ∈ Vh , prescribe sha ∈ Vha by solving
the local conforming finite element problem
sha := arg mina k∇(ψa uh − vh )kωa
vh ∈Vh
m
(∇sha , ∇vh )ωa = (∇(ψa uh ), ∇vh )ωa
M. Vohralík
∀vh ∈ Vha .
I/IV Laplace a posteriori error estimates in a unified framework
18 / 39
I Estimate Efficiency Applications Numerics R&B
Equivalent local potential reconstruction (d = 2)
Vah × Qha : MFE spaces (hom. Neumann BC for all a ∈ Vh )
Definition (Equivalent construction of sh )
For each a ∈ Vh , prescribe σha ∈ Vah and r̄ha ∈ Qha by solving the
local mixed finite element problem
(σha , vh )ωa − (r̄ha , ∇·vh )ωa = −(R π2 ∇(ψa uh ), vh )ωa
(∇·σha , qh )ωa = (0, qh )ωa
Set
∀vh ∈ Vah ,
∀qh ∈ Qha .
−R π2 ∇sha = σha ,
sha = 0 on ∂ωa ,
X
sh :=
sha .
a∈Vh
Remark
Same problem as for flux, only RHS/BC different.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
19 / 39
I Estimate Efficiency Applications Numerics R&B
Equivalent local potential reconstruction (d = 2)
Vah × Qha : MFE spaces (hom. Neumann BC for all a ∈ Vh )
Definition (Equivalent construction of sh )
For each a ∈ Vh , prescribe σha ∈ Vah and r̄ha ∈ Qha by solving the
local mixed finite element problem
(σha , vh )ωa − (r̄ha , ∇·vh )ωa = −(R π2 ∇(ψa uh ), vh )ωa
(∇·σha , qh )ωa = (0, qh )ωa
Set
∀vh ∈ Vah ,
∀qh ∈ Qha .
−R π2 ∇sha = σha ,
sha = 0 on ∂ωa ,
X
sh :=
sha .
a∈Vh
Remark
Same problem as for flux, only RHS/BC different.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
19 / 39
I Estimate Efficiency Applications Numerics R&B
Equivalent local potential reconstruction (d = 2)
Vah × Qha : MFE spaces (hom. Neumann BC for all a ∈ Vh )
Definition (Equivalent construction of sh )
For each a ∈ Vh , prescribe σha ∈ Vah and r̄ha ∈ Qha by solving the
local mixed finite element problem
(σha , vh )ωa − (r̄ha , ∇·vh )ωa = −(R π2 ∇(ψa uh ), vh )ωa
(∇·σha , qh )ωa = (0, qh )ωa
Set
∀vh ∈ Vah ,
∀qh ∈ Qha .
−R π2 ∇sha = σha ,
sha = 0 on ∂ωa ,
X
sh :=
sha .
a∈Vh
Remark
Same problem as for flux, only RHS/BC different.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
19 / 39
I Estimate Efficiency Applications Numerics R&B
Equivalent local potential reconstruction (d = 2)
Vah × Qha : MFE spaces (hom. Neumann BC for all a ∈ Vh )
Definition (Equivalent construction of sh )
For each a ∈ Vh , prescribe σha ∈ Vah and r̄ha ∈ Qha by solving the
local mixed finite element problem
(σha , vh )ωa − (r̄ha , ∇·vh )ωa = −(R π2 ∇(ψa uh ), vh )ωa
(∇·σha , qh )ωa = (0, qh )ωa
Set
∀vh ∈ Vah ,
∀qh ∈ Qha .
−R π2 ∇sha = σha ,
sha = 0 on ∂ωa ,
X
sh :=
sha .
a∈Vh
Remark
Same problem as for flux, only RHS/BC different.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
19 / 39
I Estimate Efficiency Applications Numerics R&B
Outline
1
Introduction
2
A guaranteed a posteriori error estimate
3
Polynomial-degree-robust local efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
19 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, flux reconstruction
Theorem (Cont. efficiency Carstensen & Funken (1999), Braess, Pillwein, & Schöberl (2009))
Let u be the weak solution and let uh ∈ H 1 (Th ) be arbitrary. Let
a ∈ Vh and let ra ∈ H∗1 (ωa ) be such that
(∇ra , ∇v )ωa = −(ψa ∇uh , ∇v )ωa + (ψa f − ∇ψa ·∇uh , v )ωa
for all v ∈ H∗1 (ωa ), where
H∗1 (ωa ) := {v ∈ H 1 (ωa ); (v , 1)ωa = 0},
a ∈ Vhint ,
H∗1 (ωa ) := {v ∈ H 1 (ωa ); v = 0 on ∂ωa ∩ ∂Ω}, a ∈ Vhext .
Then there exists a constant Ccont,PF > 0 only depending on the
mesh shape-regularity parameter κT such that
k∇ra kωa ≤ Ccont,PF k∇(u − uh )kωa .
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
20 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, flux reconstruction
Theorem (Cont. efficiency Carstensen & Funken (1999), Braess, Pillwein, & Schöberl (2009))
Let u be the weak solution and let uh ∈ H 1 (Th ) be arbitrary. Let
a ∈ Vh and let ra ∈ H∗1 (ωa ) be such that
(∇ra , ∇v )ωa = −(ψa ∇uh , ∇v )ωa + (ψa f − ∇ψa ·∇uh , v )ωa
for all v ∈ H∗1 (ωa ), where
H∗1 (ωa ) := {v ∈ H 1 (ωa ); (v , 1)ωa = 0},
a ∈ Vhint ,
H∗1 (ωa ) := {v ∈ H 1 (ωa ); v = 0 on ∂ωa ∩ ∂Ω}, a ∈ Vhext .
Then there exists a constant Ccont,PF > 0 only depending on the
mesh shape-regularity parameter κT such that
k∇ra kωa ≤ Ccont,PF k∇(u − uh )kωa .
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
20 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, flux reconstruction
Theorem (Cont. efficiency Carstensen & Funken (1999), Braess, Pillwein, & Schöberl (2009))
Let u be the weak solution and let uh ∈ H 1 (Th ) be arbitrary. Let
a ∈ Vh and let ra ∈ H∗1 (ωa ) be such that
(∇ra , ∇v )ωa = −(ψa ∇uh , ∇v )ωa + (ψa f − ∇ψa ·∇uh , v )ωa
for all v ∈ H∗1 (ωa ), where
H∗1 (ωa ) := {v ∈ H 1 (ωa ); (v , 1)ωa = 0},
a ∈ Vhint ,
H∗1 (ωa ) := {v ∈ H 1 (ωa ); v = 0 on ∂ωa ∩ ∂Ω}, a ∈ Vhext .
Then there exists a constant Ccont,PF > 0 only depending on the
mesh shape-regularity parameter κT such that
k∇ra kωa ≤ Ccont,PF k∇(u − uh )kωa .
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
20 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, flux reconstruction
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
weak solution definition, ψa v ∈ H01 (ωa ):
(∇ra , ∇v )ωa = (f , ψa v )ωa − (∇uh , ∇(ψa v ))ωa
= (∇(u − uh ), ∇(ψa v ))ωa
Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa ≤ k∇(u − uh )kωa k∇(ψa v )kωa
triangle & Poincaré–Friedrichs inequality: 1
1
z }| { z }| {
k∇(ψa v )kωa ≤ k∇ψa k∞,ωa kv kωa + kψa k∞,ωa k∇v kωa
≤ k∇ψa k∞,ωa CPF,ωa hωa k∇v kωa +1
| {z }
{z
}
|
−1
≈hω
a
M. Vohralík
1
I/IV Laplace a posteriori error estimates in a unified framework
21 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, flux reconstruction
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
weak solution definition, ψa v ∈ H01 (ωa ):
(∇ra , ∇v )ωa = (f , ψa v )ωa − (∇uh , ∇(ψa v ))ωa
= (∇(u − uh ), ∇(ψa v ))ωa
Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa ≤ k∇(u − uh )kωa k∇(ψa v )kωa
triangle & Poincaré–Friedrichs inequality: 1
1
z }| { z }| {
k∇(ψa v )kωa ≤ k∇ψa k∞,ωa kv kωa + kψa k∞,ωa k∇v kωa
≤ k∇ψa k∞,ωa CPF,ωa hωa k∇v kωa +1
| {z }
{z
}
|
−1
≈hω
a
M. Vohralík
1
I/IV Laplace a posteriori error estimates in a unified framework
21 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, flux reconstruction
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
weak solution definition, ψa v ∈ H01 (ωa ):
(∇ra , ∇v )ωa = (f , ψa v )ωa − (∇uh , ∇(ψa v ))ωa
= (∇(u − uh ), ∇(ψa v ))ωa
Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa ≤ k∇(u − uh )kωa k∇(ψa v )kωa
triangle & Poincaré–Friedrichs inequality: 1
1
z }| { z }| {
k∇(ψa v )kωa ≤ k∇ψa k∞,ωa kv kωa + kψa k∞,ωa k∇v kωa
≤ k∇ψa k∞,ωa CPF,ωa hωa k∇v kωa +1
| {z }
|
{z
}
−1
≈hω
a
M. Vohralík
1
I/IV Laplace a posteriori error estimates in a unified framework
21 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, flux reconstruction
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
weak solution definition, ψa v ∈ H01 (ωa ):
(∇ra , ∇v )ωa = (f , ψa v )ωa − (∇uh , ∇(ψa v ))ωa
= (∇(u − uh ), ∇(ψa v ))ωa
Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa ≤ k∇(u − uh )kωa k∇(ψa v )kωa
triangle & Poincaré–Friedrichs inequality: 1
1
z }| { z }| {
k∇(ψa v )kωa ≤ k∇ψa k∞,ωa kv kωa + kψa k∞,ωa k∇v kωa
≤ k∇ψa k∞,ωa CPF,ωa hωa k∇v kωa +1
| {z }
|
{z
}
−1
≈hω
a
M. Vohralík
1
I/IV Laplace a posteriori error estimates in a unified framework
21 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, flux reconstruction
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
weak solution definition, ψa v ∈ H01 (ωa ):
(∇ra , ∇v )ωa = (f , ψa v )ωa − (∇uh , ∇(ψa v ))ωa
= (∇(u − uh ), ∇(ψa v ))ωa
Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa ≤ k∇(u − uh )kωa k∇(ψa v )kωa
triangle & Poincaré–Friedrichs inequality: 1
1
z }| { z }| {
k∇(ψa v )kωa ≤ k∇ψa k∞,ωa kv kωa + kψa k∞,ωa k∇v kωa
≤ k∇ψa k∞,ωa CPF,ωa hωa k∇v kωa +1
| {z }
|
{z
}
−1
≈hω
a
M. Vohralík
1
I/IV Laplace a posteriori error estimates in a unified framework
21 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Assumption B (Weak continuity)
There holds
h[[uh ]], 1ie = 0
∀e ∈ Eh .
Theorem (Continuous efficiency)
Let u be the weak solution and let uh ∈ H 1 (Th ) satisfying
Assumption B be arbitrary. Let a ∈ Vh and let ra ∈ H∗1 (ωa ) be
given by
∀v ∈ H∗1 (ωa ),
(∇ra , ∇v )ωa = −(R π2 ∇(ψa uh ), ∇v )ωa
where
H∗1 (ωa ) := {v ∈ H 1 (ωa ); (v , 1)ωa = 0}.
Then there exists Ccont,bPF > 0 only depending on κT such that
k∇ra kωa ≤ Ccont,bPF k∇(u − uh )kωa .
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
22 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Assumption B (Weak continuity)
There holds
h[[uh ]], 1ie = 0
∀e ∈ Eh .
Theorem (Continuous efficiency)
Let u be the weak solution and let uh ∈ H 1 (Th ) satisfying
Assumption B be arbitrary. Let a ∈ Vh and let ra ∈ H∗1 (ωa ) be
given by
∀v ∈ H∗1 (ωa ),
(∇ra , ∇v )ωa = −(R π2 ∇(ψa uh ), ∇v )ωa
where
H∗1 (ωa ) := {v ∈ H 1 (ωa ); (v , 1)ωa = 0}.
Then there exists Ccont,bPF > 0 only depending on κT such that
k∇ra kωa ≤ Ccont,bPF k∇(u − uh )kωa .
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
22 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Assumption B (Weak continuity)
There holds
h[[uh ]], 1ie = 0
∀e ∈ Eh .
Theorem (Continuous efficiency)
Let u be the weak solution and let uh ∈ H 1 (Th ) satisfying
Assumption B be arbitrary. Let a ∈ Vh and let ra ∈ H∗1 (ωa ) be
given by
∀v ∈ H∗1 (ωa ),
(∇ra , ∇v )ωa = −(R π2 ∇(ψa uh ), ∇v )ωa
where
H∗1 (ωa ) := {v ∈ H 1 (ωa ); (v , 1)ωa = 0}.
Then there exists Ccont,bPF > 0 only depending on κT such that
k∇ra kωa ≤ Ccont,bPF k∇(u − uh )kωa .
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
22 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Assumption B (Weak continuity)
There holds
h[[uh ]], 1ie = 0
∀e ∈ Eh .
Theorem (Continuous efficiency)
Let u be the weak solution and let uh ∈ H 1 (Th ) satisfying
Assumption B be arbitrary. Let a ∈ Vh and let ra ∈ H∗1 (ωa ) be
given by
∀v ∈ H∗1 (ωa ),
(∇ra , ∇v )ωa = −(R π2 ∇(ψa uh ), ∇v )ωa
where
H∗1 (ωa ) := {v ∈ H 1 (ωa ); (v , 1)ωa = 0}.
Then there exists Ccont,bPF > 0 only depending on κT such that
k∇ra kωa ≤ Ccont,bPF k∇(u − uh )kωa .
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
22 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
arbitrary ũ ∈ H 1 (ωa ) with (ũ, 1)ωa = (uh , 1)ωa if a ∈ Vhint and
ũ = 0 on ∂ωa ∩ ∂Ω if a ∈ Vhext :
(R π2 ∇(ψa ũ), ∇v )ωa = 0
definition of ra , insertion of ũ, Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa = (R π2 ∇(ψa (ũ−uh )), ∇v )ωa ≤ k∇(ψa (ũ−uh ))kωa
broken Poincaré–Friedrichs inequality:
k∇(ψa (ũ−uh ))kωa ≤ (1+CbPF,ωa hωa k∇ψa k∞,ωa )k∇(ũ−uh )kωa
define ũ := u − cnst for a ∈ Vhint and ũ := u for a ∈ Vhext
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
23 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
arbitrary ũ ∈ H 1 (ωa ) with (ũ, 1)ωa = (uh , 1)ωa if a ∈ Vhint and
ũ = 0 on ∂ωa ∩ ∂Ω if a ∈ Vhext :
(R π2 ∇(ψa ũ), ∇v )ωa = 0
definition of ra , insertion of ũ, Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa = (R π2 ∇(ψa (ũ−uh )), ∇v )ωa ≤ k∇(ψa (ũ−uh ))kωa
broken Poincaré–Friedrichs inequality:
k∇(ψa (ũ−uh ))kωa ≤ (1+CbPF,ωa hωa k∇ψa k∞,ωa )k∇(ũ−uh )kωa
define ũ := u − cnst for a ∈ Vhint and ũ := u for a ∈ Vhext
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
23 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
arbitrary ũ ∈ H 1 (ωa ) with (ũ, 1)ωa = (uh , 1)ωa if a ∈ Vhint and
ũ = 0 on ∂ωa ∩ ∂Ω if a ∈ Vhext :
(R π2 ∇(ψa ũ), ∇v )ωa = 0
definition of ra , insertion of ũ, Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa = (R π2 ∇(ψa (ũ−uh )), ∇v )ωa ≤ k∇(ψa (ũ−uh ))kωa
broken Poincaré–Friedrichs inequality:
k∇(ψa (ũ−uh ))kωa ≤ (1+CbPF,ωa hωa k∇ψa k∞,ωa )k∇(ũ−uh )kωa
define ũ := u − cnst for a ∈ Vhint and ũ := u for a ∈ Vhext
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
23 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
arbitrary ũ ∈ H 1 (ωa ) with (ũ, 1)ωa = (uh , 1)ωa if a ∈ Vhint and
ũ = 0 on ∂ωa ∩ ∂Ω if a ∈ Vhext :
(R π2 ∇(ψa ũ), ∇v )ωa = 0
definition of ra , insertion of ũ, Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa = (R π2 ∇(ψa (ũ−uh )), ∇v )ωa ≤ k∇(ψa (ũ−uh ))kωa
broken Poincaré–Friedrichs inequality:
k∇(ψa (ũ−uh ))kωa ≤ (1+CbPF,ωa hωa k∇ψa k∞,ωa )k∇(ũ−uh )kωa
define ũ := u − cnst for a ∈ Vhint and ũ := u for a ∈ Vhext
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
23 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
arbitrary ũ ∈ H 1 (ωa ) with (ũ, 1)ωa = (uh , 1)ωa if a ∈ Vhint and
ũ = 0 on ∂ωa ∩ ∂Ω if a ∈ Vhext :
(R π2 ∇(ψa ũ), ∇v )ωa = 0
definition of ra , insertion of ũ, Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa = (R π2 ∇(ψa (ũ−uh )), ∇v )ωa ≤ k∇(ψa (ũ−uh ))kωa
broken Poincaré–Friedrichs inequality:
k∇(ψa (ũ−uh ))kωa ≤ (1+CbPF,ωa hωa k∇ψa k∞,ωa )k∇(ũ−uh )kωa
define ũ := u − cnst for a ∈ Vhint and ũ := u for a ∈ Vhext
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
23 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
arbitrary ũ ∈ H 1 (ωa ) with (ũ, 1)ωa = (uh , 1)ωa if a ∈ Vhint and
ũ = 0 on ∂ωa ∩ ∂Ω if a ∈ Vhext :
(R π2 ∇(ψa ũ), ∇v )ωa = 0
definition of ra , insertion of ũ, Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa = (R π2 ∇(ψa (ũ−uh )), ∇v )ωa ≤ k∇(ψa (ũ−uh ))kωa
broken Poincaré–Friedrichs inequality:
k∇(ψa (ũ−uh ))kωa ≤ (1+CbPF,ωa hωa k∇ψa k∞,ωa )k∇(ũ−uh )kωa
define ũ := u − cnst for a ∈ Vhint and ũ := u for a ∈ Vhext
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
23 / 39
I Estimate Efficiency Applications Numerics R&B
Continuous efficiency, potential reconstruction (d = 2)
Proof.
dual norm characterization
k∇ra kωa =
sup
(∇ra , ∇v )ωa
v ∈H∗1 (ωa ); k∇v kωa =1
arbitrary ũ ∈ H 1 (ωa ) with (ũ, 1)ωa = (uh , 1)ωa if a ∈ Vhint and
ũ = 0 on ∂ωa ∩ ∂Ω if a ∈ Vhext :
(R π2 ∇(ψa ũ), ∇v )ωa = 0
definition of ra , insertion of ũ, Cauchy–Schwarz inequality:
(∇ra , ∇v )ωa = (R π2 ∇(ψa (ũ−uh )), ∇v )ωa ≤ k∇(ψa (ũ−uh ))kωa
broken Poincaré–Friedrichs inequality:
k∇(ψa (ũ−uh ))kωa ≤ (1+CbPF,ωa hωa k∇ψa k∞,ωa )k∇(ũ−uh )kωa
define ũ := u − cnst for a ∈ Vhint and ũ := u for a ∈ Vhext
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
23 / 39
I Estimate Efficiency Applications Numerics R&B
Mixed finite elements stability (d = 2)
Assumption C (Piecewise polynomials, data, and meshes)
The approximation uh and the datum f are piecewise
polynomial. The degrees of the MFE reconstructions σh and sh
are chosen correspondingly. The meshes Th are shape-regular.
Theorem (MFE stability / continuous right inverse of the
divergence operator Braess, Pillwein, and Schöberl (2009); Costabel and McIntosh (2010);
Demkowicz, Gopalakrishnan, and Schöberl (2012))
Let u be the weak solution and let uh , f , and the reconstructions
satisfy Assumption C. Then there exists a constant Cst > 0 only
depending on the shape-regularity parameter κT such that
kσha + τha kωa ≤ Cst kσ a + τha kωa = k∇ra kωa
with τha = ψa ∇uh for the flux reconstruction and
τha = R π2 ∇(ψa uh ) for the potential reconstruction.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
24 / 39
I Estimate Efficiency Applications Numerics R&B
Mixed finite elements stability (d = 2)
Assumption C (Piecewise polynomials, data, and meshes)
The approximation uh and the datum f are piecewise
polynomial. The degrees of the MFE reconstructions σh and sh
are chosen correspondingly. The meshes Th are shape-regular.
Theorem (MFE stability / continuous right inverse of the
divergence operator Braess, Pillwein, and Schöberl (2009); Costabel and McIntosh (2010);
Demkowicz, Gopalakrishnan, and Schöberl (2012))
Let u be the weak solution and let uh , f , and the reconstructions
satisfy Assumption C. Then there exists a constant Cst > 0 only
depending on the shape-regularity parameter κT such that
kσha + τha kωa ≤ Cst kσ a + τha kωa = k∇ra kωa
with τha = ψa ∇uh for the flux reconstruction and
τha = R π2 ∇(ψa uh ) for the potential reconstruction.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
24 / 39
I Estimate Efficiency Applications Numerics R&B
Mixed finite elements stability (d = 2)
Assumption C (Piecewise polynomials, data, and meshes)
The approximation uh and the datum f are piecewise
polynomial. The degrees of the MFE reconstructions σh and sh
are chosen correspondingly. The meshes Th are shape-regular.
Theorem (MFE stability / continuous right inverse of the
divergence operator Braess, Pillwein, and Schöberl (2009); Costabel and McIntosh (2010);
Demkowicz, Gopalakrishnan, and Schöberl (2012))
Let u be the weak solution and let uh , f , and the reconstructions
satisfy Assumption C. Then there exists a constant Cst > 0 only
depending on the shape-regularity parameter κT such that
kσha + τha kωa ≤ Cst kσ a + τha kωa = k∇ra kωa
with τha = ψa ∇uh for the flux reconstruction and
τha = R π2 ∇(ψa uh ) for the potential reconstruction.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
24 / 39
I Estimate Efficiency Applications Numerics R&B
Mixed finite elements stability (d = 2)
Assumption C (Piecewise polynomials, data, and meshes)
The approximation uh and the datum f are piecewise
polynomial. The degrees of the MFE reconstructions σh and sh
are chosen correspondingly. The meshes Th are shape-regular.
Theorem (MFE stability / continuous right inverse of the
divergence operator Braess, Pillwein, and Schöberl (2009); Costabel and McIntosh (2010);
Demkowicz, Gopalakrishnan, and Schöberl (2012))
Let u be the weak solution and let uh , f , and the reconstructions
satisfy Assumption C. Then there exists a constant Cst > 0 only
depending on the shape-regularity parameter κT such that
kσha + τha kωa ≤ Cst kσ a + τha kωa = k∇ra kωa
with τha = ψa ∇uh for the flux reconstruction and
τha = R π2 ∇(ψa uh ) for the potential reconstruction.
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
24 / 39
I Estimate Efficiency Applications Numerics R&B
Polynomial-degree-robust efficiency
Theorem (Polynomial-degree-robust efficiency)
Let u be the weak solution and let Assumptions A, B, and C
hold. Then
X
k∇uh + σh kK ≤ Cst Ccont,PF
k∇(u − uh )kωa ,
a∈VK
k∇(uh − sh )kK ≤ Cst Ccont,bPF
X
a∈VK
k∇(u − uh )kωa .
Remarks
Cst can be bounded by solving the local Neumann
problems by conforming FEs: find rha ∈ Vha ⊂ H∗1 (ωa ) s.t.
(∇rha , ∇vh )ωa = −(τha , ∇vh )ωa + (g a , vh )ωa
then Cst ≤ kτha + σha kωa /k∇rha kωa
⇒ maximal overestimation factor guaranteed
M. Vohralík
∀vh ∈ Vha ;
I/IV Laplace a posteriori error estimates in a unified framework
25 / 39
I Estimate Efficiency Applications Numerics R&B
Polynomial-degree-robust efficiency
Theorem (Polynomial-degree-robust efficiency)
Let u be the weak solution and let Assumptions A, B, and C
hold. Then
X
k∇uh + σh kK ≤ Cst Ccont,PF
k∇(u − uh )kωa ,
a∈VK
k∇(uh − sh )kK ≤ Cst Ccont,bPF
X
a∈VK
k∇(u − uh )kωa .
Remarks
Cst can be bounded by solving the local Neumann
problems by conforming FEs: find rha ∈ Vha ⊂ H∗1 (ωa ) s.t.
(∇rha , ∇vh )ωa = −(τha , ∇vh )ωa + (g a , vh )ωa
then Cst ≤ kτha + σha kωa /k∇rha kωa
⇒ maximal overestimation factor guaranteed
M. Vohralík
∀vh ∈ Vha ;
I/IV Laplace a posteriori error estimates in a unified framework
25 / 39
I Estimate Efficiency Applications Numerics R&B
Outline
1
Introduction
2
A guaranteed a posteriori error estimate
3
Polynomial-degree-robust local efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
25 / 39
I Estimate Efficiency Applications Numerics R&B
Conforming finite elements
Conforming finite elements
Find uh ∈ Vh such that
(∇uh , ∇vh ) = (f , vh )
∀vh ∈ Vh .
Vh := Pp (Th ) ∩ H01 (Ω), p ≥ 1
Assumption A: take vh = ψa
Vh ⊂ H01 (Ω): sh := uh , no need for Assumption B
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
26 / 39
I Estimate Efficiency Applications Numerics R&B
Conforming finite elements
Conforming finite elements
Find uh ∈ Vh such that
(∇uh , ∇vh ) = (f , vh )
∀vh ∈ Vh .
Vh := Pp (Th ) ∩ H01 (Ω), p ≥ 1
Assumption A: take vh = ψa
Vh ⊂ H01 (Ω): sh := uh , no need for Assumption B
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
26 / 39
I Estimate Efficiency Applications Numerics R&B
Nonconforming finite elements
Nonconforming finite elements
Find uh ∈ Vh such that
(∇uh , ∇vh ) = (f , vh )
∀vh ∈ Vh .
Vh := Pp (Th ), p ≥ 1, vh ∈ Vh satisfy
h[[vh ]], qh ie = 0
∀qh ∈ Pp−1 (e), ∀e ∈ Eh
Assumption A: take vh = ψa
Assumption B: building requirement for the space Vh
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
27 / 39
I Estimate Efficiency Applications Numerics R&B
Nonconforming finite elements
Nonconforming finite elements
Find uh ∈ Vh such that
(∇uh , ∇vh ) = (f , vh )
∀vh ∈ Vh .
Vh := Pp (Th ), p ≥ 1, vh ∈ Vh satisfy
h[[vh ]], qh ie = 0
∀qh ∈ Pp−1 (e), ∀e ∈ Eh
Assumption A: take vh = ψa
Assumption B: building requirement for the space Vh
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
27 / 39
I Estimate Efficiency Applications Numerics R&B
Discontinuous Galerkin finite elements
Discontinuous Galerkin finite elements
Find uh ∈ Vh such that
X
X
(∇uh , ∇vh )K −
{h{{∇uh }}·ne , [[vh ]]ie +θh{{∇vh }}·ne , [[uh ]]ie }
K ∈Th
+
X
e∈Eh
e∈Eh
hαhe−1 [[uh ]], [[vh ]]ie
= (f , vh )
∀vh ∈ Vh .
Vh := Pp (Th ), p ≥ 1
Assumption A: take vh = ψa for θ = 0, otherwise:
estimates for the discrete gradient
X
G(uh ) := ∇uh − θ
le ([[uh ]])
e∈Eh
2
jumps lifting operator le : L (e) → [P0 (Te )]2
(le ([[uh ]]), vh ) = h{{vh }}·ne , [[uh ]]ie
⇒ modified Galerkin orthogonality
M. Vohralík
(G(uh ), ∇ψa )ωa = (f , ψa )ωa
∀vh ∈ [P0 (Te )]2
∀a ∈ Vhint
I/IV Laplace a posteriori error estimates in a unified framework
28 / 39
I Estimate Efficiency Applications Numerics R&B
Discontinuous Galerkin finite elements
Discontinuous Galerkin finite elements
Find uh ∈ Vh such that
X
X
(∇uh , ∇vh )K −
{h{{∇uh }}·ne , [[vh ]]ie +θh{{∇vh }}·ne , [[uh ]]ie }
K ∈Th
+
X
e∈Eh
e∈Eh
hαhe−1 [[uh ]], [[vh ]]ie
= (f , vh )
∀vh ∈ Vh .
Vh := Pp (Th ), p ≥ 1
Assumption A: take vh = ψa for θ = 0, otherwise:
estimates for the discrete gradient
X
G(uh ) := ∇uh − θ
le ([[uh ]])
e∈Eh
2
jumps lifting operator le : L (e) → [P0 (Te )]2
(le ([[uh ]]), vh ) = h{{vh }}·ne , [[uh ]]ie
⇒ modified Galerkin orthogonality
M. Vohralík
(G(uh ), ∇ψa )ωa = (f , ψa )ωa
∀vh ∈ [P0 (Te )]2
∀a ∈ Vhint
I/IV Laplace a posteriori error estimates in a unified framework
28 / 39
I Estimate Efficiency Applications Numerics R&B
Discontinuous Galerkin finite elements
Discontinuous Galerkin finite elements
Find uh ∈ Vh such that
X
X
(∇uh , ∇vh )K −
{h{{∇uh }}·ne , [[vh ]]ie +θh{{∇vh }}·ne , [[uh ]]ie }
K ∈Th
+
X
e∈Eh
e∈Eh
hαhe−1 [[uh ]], [[vh ]]ie
= (f , vh )
∀vh ∈ Vh .
Vh := Pp (Th ), p ≥ 1
Assumption A: take vh = ψa for θ = 0, otherwise:
estimates for the discrete gradient
X
G(uh ) := ∇uh − θ
le ([[uh ]])
e∈Eh
2
jumps lifting operator le : L (e) → [P0 (Te )]2
(le ([[uh ]]), vh ) = h{{vh }}·ne , [[uh ]]ie
⇒ modified Galerkin orthogonality
M. Vohralík
(G(uh ), ∇ψa )ωa = (f , ψa )ωa
∀vh ∈ [P0 (Te )]2
∀a ∈ Vhint
I/IV Laplace a posteriori error estimates in a unified framework
28 / 39
I Estimate Efficiency Applications Numerics R&B
Discontinuous Galerkin finite elements: Assumption B
Nonsymmetric and incomplete versions
broken Poincaré–Friedrichs inequality with jumps:
k∇(ψa (ũ − uh ))kωa ≤ (1 + CbPF,ωa hωa k∇ψa k∞,ωa )k∇(u − uh )kωa
)1/2
(
X
+CbPF,ωa hωa k∇ψa k∞,ωa
he−1 kΠ0e [[uh ]]k2e
e∈Eh , a∈e
include the jump terms in the error and estimators
Symmetric version
discrete gradient G satisfies
(G(uh ), R π2 ∇ψa )ωa = 0
∀a ∈ Vh
modified potential reconstruction: local MFE problems with
τha := ψa R π2 G(uh ) and g a := (R π2 ∇ψa )·G(uh )
local efficiency
X
kG(uh − sh )kK ≤ Cst Ccont,P
kG(u − uh )kωa
a∈VK
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
29 / 39
I Estimate Efficiency Applications Numerics R&B
Discontinuous Galerkin finite elements: Assumption B
Nonsymmetric and incomplete versions
broken Poincaré–Friedrichs inequality with jumps:
k∇(ψa (ũ − uh ))kωa ≤ (1 + CbPF,ωa hωa k∇ψa k∞,ωa )k∇(u − uh )kωa
)1/2
(
X
+CbPF,ωa hωa k∇ψa k∞,ωa
he−1 kΠ0e [[uh ]]k2e
e∈Eh , a∈e
include the jump terms in the error and estimators
Symmetric version
discrete gradient G satisfies
(G(uh ), R π2 ∇ψa )ωa = 0
∀a ∈ Vh
modified potential reconstruction: local MFE problems with
τha := ψa R π2 G(uh ) and g a := (R π2 ∇ψa )·G(uh )
local efficiency
X
kG(uh − sh )kK ≤ Cst Ccont,P
kG(u − uh )kωa
a∈VK
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
29 / 39
I Estimate Efficiency Applications Numerics R&B
Mixed finite elements
Mixed finite elements
Find a couple (σh , ūh ) ∈ Vh × Qh such that
(σh , vh ) − (ūh , ∇·vh ) = 0
(∇·σh , qh ) = (f , qh )
∀vh ∈ Vh ,
∀qh ∈ Qh .
postprocessed solution uh ∈ Vh , Vh := Pp (Th ), p ≥ 1;
vh ∈ Vh satisfy
h[[vh ]], qh ie = 0
∀qh ∈ Pp0 (e), ∀e ∈ Eh
Assumption A: no need for flux reconstruction, σh comes
from the discretization
Assumption B satisfied, building requirement for the
space Vh
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
30 / 39
I Estimate Efficiency Applications Numerics R&B
Mixed finite elements
Mixed finite elements
Find a couple (σh , ūh ) ∈ Vh × Qh such that
(σh , vh ) − (ūh , ∇·vh ) = 0
(∇·σh , qh ) = (f , qh )
∀vh ∈ Vh ,
∀qh ∈ Qh .
postprocessed solution uh ∈ Vh , Vh := Pp (Th ), p ≥ 1;
vh ∈ Vh satisfy
h[[vh ]], qh ie = 0
∀qh ∈ Pp0 (e), ∀e ∈ Eh
Assumption A: no need for flux reconstruction, σh comes
from the discretization
Assumption B satisfied, building requirement for the
space Vh
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
30 / 39
I Estimate Efficiency Applications Numerics R&B
Outline
1
Introduction
2
A guaranteed a posteriori error estimate
3
Polynomial-degree-robust local efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
30 / 39
I Estimate Efficiency Applications Numerics R&B
Numerics: smooth test case
Model problem
in Ω :=]0, 1[2
−∆u = f
u = uD
on ∂Ω
Exact solution
u(x) = (c1 + c2 (1 − x1 ) + e−αx1 )(c1 + c2 (1 − x2 ) + e−αx2 )
c1 = −e−α ,
c2 = −1 − c1 ,
α = 10
Discretization
incomplete interior penalty discontinuous Galerkin method
unstructured nested triangular grids
uniform refinement
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
31 / 39
I Estimate Efficiency Applications Numerics R&B
Numerics: smooth test case
Model problem
in Ω :=]0, 1[2
−∆u = f
u = uD
on ∂Ω
Exact solution
u(x) = (c1 + c2 (1 − x1 ) + e−αx1 )(c1 + c2 (1 − x2 ) + e−αx2 )
c1 = −e−α ,
c2 = −1 − c1 ,
α = 10
Discretization
incomplete interior penalty discontinuous Galerkin method
unstructured nested triangular grids
uniform refinement
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
31 / 39
I Estimate Efficiency Applications Numerics R&B
Numerics: smooth test case
Model problem
in Ω :=]0, 1[2
−∆u = f
u = uD
on ∂Ω
Exact solution
u(x) = (c1 + c2 (1 − x1 ) + e−αx1 )(c1 + c2 (1 − x2 ) + e−αx2 )
c1 = −e−α ,
c2 = −1 − c1 ,
α = 10
Discretization
incomplete interior penalty discontinuous Galerkin method
unstructured nested triangular grids
uniform refinement
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
31 / 39
I Estimate Efficiency Applications Numerics R&B
Estimates, errors, and effectivity indices
h
p k∇(u −uh )k ku −uh kDG k∇uh +σh k k∇(uh −sh )k
h0 /1 1
1.21E+00
1.22E+00
1.24E+00
1.07E-01
h0 /2
6.18E-01
6.22E-01
6.38E-01
5.09E-02
(0.97)
(0.97)
(0.96)
(1.07)
h0 /4
3.12E-01
3.13E-01
3.22E-01
2.43E-02
(0.99)
(0.99)
(0.99)
(1.07)
h0 /8
1.56E-01
1.57E-01
1.61E-01
1.18E-02
(1.00)
(1.00)
(1.00)
(1.05)
h0 /1 2
1.50E-01
1.53E-01
1.49E-01
2.76E-02
h0 /2
3.85E-02
3.92E-02
3.83E-02
7.99E-03
(1.96)
(1.96)
(1.96)
(1.79)
h0 /4
9.70E-03
9.88E-03
9.68E-03
2.12E-03
(1.99)
(1.99)
(1.98)
(1.92)
h0 /8
2.43E-03
2.48E-03
2.43E-03
5.42E-04
(1.99)
(1.99)
(1.99)
(1.96)
h0 /1 3
1.32E-02
1.34E-02
1.29E-02
2.52E-03
h0 /2
1.67E-03
1.69E-03
1.65E-03
3.13E-04
(2.98)
(2.98)
(2.97)
(3.01)
h0 /4
2.11E-04
2.13E-04
2.09E-04
3.83E-05
(2.99)
(2.99)
(2.99)
(3.03)
h0 /8
2.64E-05
2.67E-05
2.61E-05
4.69E-06
(3.00)
(3.00)
(3.00)
(3.03)
h0 /1 4
9.36E-04
9.54E-04
9.05E-04
2.41E-04
h0 /2
5.93E-05
6.05E-05
5.77E-05
1.68E-05
(3.98)
(3.98)
(3.97)
(3.84)
h0 /4
3.72E-06
3.80E-06
3.63E-06
1.10E-06
(3.99)
(3.99)
(3.99)
(3.94)
h0 /8
2.33E-07
2.38E-07
2.27E-07
7.02E-08
(4.00)
(4.00)
(4.00)
(3.97)
h0 /1 5
5.41E-05
5.50E-05
5.22E-05
1.38E-05
h0 /2
1.70E-06
1.72E-06
1.65E-06
4.39E-07
(4.99)
(5.00)
(4.98)
(4.98)
h0 /4
5.32E-08
5.39E-08
5.19E-08
1.40E-08
(5.00)
(5.00)
(4.99)
(4.97)
h0 /8
1.66E-09
1.69E-09
1.62E-09
4.41E-10
(5.00)
(5.00)
(5.00)
(4.99)
M. Vohralík
ηosc
5.56E-02
7.02E-03
(2.99)
8.80E-04
(3.00)
1.10E-04
(3.00)
5.10E-03
3.22E-04
(3.98)
2.02E-05
(4.00)
1.26E-06
(4.00)
3.58E-04
1.13E-05
(4.99)
3.53E-07
(5.00)
1.10E-08
(5.00)
2.12E-05
3.36E-07
(5.98)
5.31E-09
(5.98)
8.30E-11
(6.00)
1.06E-06
9.35E-09
(6.82)
7.67E-11
(6.93)
5.99E-13
(7.00)
η
1.30E+00
6.47E-01
(1.01)
3.24E-01
(1.00)
1.62E-01
(1.00)
1.56E-01
3.94E-02
(1.98)
9.93E-03
(1.99)
2.49E-03
(1.99)
1.35E-02
1.70E-03
(3.00)
2.12E-04
(3.00)
2.66E-05
(3.00)
9.57E-04
6.04E-05
(3.99)
3.80E-06
(3.99)
2.38E-07
(4.00)
5.50E-05
1.72E-06
(5.00)
5.38E-08
(5.00)
1.68E-09
(5.00)
ηDG
1.31E+00
6.50E-01
(1.01)
3.25E-01
(1.00)
1.63E-01
(1.00)
1.59E-01
4.01E-02
(1.98)
1.01E-02
(1.99)
2.54E-03
(1.99)
1.37E-02
1.71E-03
(3.00)
2.15E-04
(3.00)
2.69E-05
(3.00)
9.74E-04
6.16E-05
(3.98)
3.87E-06
(3.99)
2.43E-07
(3.99)
5.58E-05
1.74E-06
(5.00)
5.45E-08
(5.00)
1.70E-09
(5.00)
eff
I eff IDG
1.07 1.07
1.05 1.05
1.04 1.04
1.04 1.04
1.04 1.04
1.03 1.02
1.02 1.02
1.02 1.02
1.03 1.03
1.01 1.01
1.01 1.01
1.01 1.01
1.02 1.02
1.02 1.02
1.02 1.02
1.02 1.02
1.02 1.02
1.01 1.01
1.01 1.01
1.01 1.01
I/IV Laplace a posteriori error estimates in a unified framework
32 / 39
I Estimate Efficiency Applications Numerics R&B
Numerics: singular test case & hp-adaptivity
Model problem
−∆u = 0
u = uD
Exact solution
in Ω := Ω :=]−1, 1[2 \[0, 1]2 ,
on ∂Ω
u(r , φ) = r 2/3 sin(2φ/3)
Discretization
incomplete interior penalty discontinuous Galerkin method
unstructured non-nested triangular grids
hp-adaptive refinement
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
33 / 39
I Estimate Efficiency Applications Numerics R&B
Numerics: singular test case & hp-adaptivity
Model problem
−∆u = 0
u = uD
Exact solution
in Ω := Ω :=]−1, 1[2 \[0, 1]2 ,
on ∂Ω
u(r , φ) = r 2/3 sin(2φ/3)
Discretization
incomplete interior penalty discontinuous Galerkin method
unstructured non-nested triangular grids
hp-adaptive refinement
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
33 / 39
I Estimate Efficiency Applications Numerics R&B
Numerics: singular test case & hp-adaptivity
Model problem
−∆u = 0
u = uD
Exact solution
in Ω := Ω :=]−1, 1[2 \[0, 1]2 ,
on ∂Ω
u(r , φ) = r 2/3 sin(2φ/3)
Discretization
incomplete interior penalty discontinuous Galerkin method
unstructured non-nested triangular grids
hp-adaptive refinement
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
33 / 39
I Estimate Efficiency Applications Numerics R&B
hp-adaptive refinement algorithms
Algorithm 2, error
Algorithm 2, estimate
Algorithm 1, estimate
Algorithm 2, estimate
0.1
0.1
0.01
0.01
0.001
0.001
0.0001
0.0001
1000
1000
10000
Algorithm 1 (only refinement)
and Algorithm 2 (refinement &
derefinement) wrt DoF
M. Vohralík
10000
Exponential convergence of
Algorithm 2 wrt DoF
I/IV Laplace a posteriori error estimates in a unified framework
34 / 39
I Estimate Efficiency Applications Numerics R&B
hp-refinement grids
level 1
level 5
level 12
hp
total view
P7
P6
P5
P4
P3
P2
−1.0Ε+00
0.0Ε+00
1.0Ε+00
−1.0Ε+00
0.0Ε+00
1.0Ε+00
−1.0Ε+00
0.0Ε+00
1.0Ε+00
zoom 10x
hp
P7
P6
P5
P4
P3
P2
−1.0Ε−01
0.0Ε+00
M. Vohralík
1.0Ε−01
−1.0Ε−01
0.0Ε+00
1.0Ε−01
−1.0Ε−01
0.0Ε+00
1.0Ε−01
I/IV Laplace a posteriori error estimates in a unified framework
35 / 39
I Estimate Efficiency Applications Numerics R&B
Estimates, errors, and effectivity indices
lev |Th |
0 114
1 122
2 139
3 165
4 174
5 199
6 237
7 285
8 338
9 372
10 426
11 453
12 469
13 463
14 458
15 440
16 435
17 434
18 419
19 410
DoF k∇(u − uh )k k∇uh + σh k
684
6.22E-02
6.63E-02
1180
4.28E-02
4.27E-02
1919
3.28E-02
3.37E-02
2573
2.32E-02
2.30E-02
2858
1.02E-02
1.01E-02
3351
6.27E-03
6.21E-03
3926
4.21E-03
4.23E-03
4537
2.84E-03
2.91E-03
5257
2.04E-03
2.19E-03
5658
1.21E-03
1.23E-03
6500
7.70E-04
7.69E-04
7010
4.95E-04
5.04E-04
7308
3.41E-04
3.47E-04
7286
2.42E-04
2.42E-04
7215
1.69E-04
1.69E-04
6955
1.29E-04
1.31E-04
7035
9.71E-05
9.91E-05
7167
8.52E-05
8.97E-05
6960
7.51E-05
7.97E-05
6838
6.06E-05
6.35E-05
M. Vohralík
ηosc
k∇(uh − sh )k
1.89E-15
4.48E-02
1.18E-14
3.08E-02
8.21E-14
2.09E-02
3.88E-13
1.50E-02
4.48E-13
8.22E-03
1.12E-12
4.81E-03
1.98E-12
3.15E-03
7.47E-12
2.13E-03
4.63E-11
1.45E-03
1.11E-11
9.07E-04
5.69E-11
5.55E-04
9.77E-11
3.97E-04
1.13E-10
2.55E-04
1.39E-10
1.73E-04
1.23E-10
1.19E-04
1.45E-10
9.21E-05
1.39E-10
6.89E-05
1.41E-10
5.76E-05
1.44E-10
5.00E-05
1.47E-10
3.87E-05
ηBC
3.81E-02
2.92E-02
2.12E-02
1.03E-02
9.19E-03
6.18E-03
3.29E-03
2.42E-03
1.32E-03
9.99E-04
6.95E-04
4.74E-04
2.88E-04
1.94E-04
1.53E-04
9.10E-05
7.63E-05
5.47E-05
4.15E-05
3.65E-05
η
1.05E-01
7.29E-02
5.36E-02
3.41E-02
1.99E-02
1.25E-02
7.66E-03
5.33E-03
3.51E-03
2.26E-03
1.46E-03
9.91E-04
6.40E-04
4.37E-04
3.17E-04
2.24E-04
1.74E-04
1.42E-04
1.21E-04
9.69E-05
I eff
1.69
1.70
1.64
1.47
1.96
2.00
1.82
1.88
1.72
1.87
1.89
2.00
1.88
1.81
1.88
1.73
1.79
1.67
1.60
1.60
I/IV Laplace a posteriori error estimates in a unified framework
36 / 39
I Estimate Efficiency Applications Numerics R&B
Outline
1
Introduction
2
A guaranteed a posteriori error estimate
3
Polynomial-degree-robust local efficiency
4
Applications
5
Numerical results
6
References and bibliography
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
36 / 39
I Estimate Efficiency Applications Numerics R&B
Previous results
Global flux reconstructions
Prager and Synge (1947):
k∇u + σh k2 + k∇(u − uh )k2 = k∇uh + σh k2
for any uh ∈ H01 (Ω) and any σh ∈ H(div, Ω) s.t. ∇·σh = f
Hlaváček, Haslinger, Nečas, and Lovíšek (1979), Repin
(1997), . . . : global construction of σh : unprecise/costly
Local flux reconstructions
Ladevèze and Leguillon (1983), equilibrated face fluxes
Destuynder and Métivet (1999), discrete flux σh
Luce and Wohlmuth (2004), local efficiency proof
Vejchodský (2006), equilibration–hypercircle approach
Kim (2007) & Ern, Nicaise, and Vohralík (2007),
discontinuous Galerkin method elementwise prescription
Braess and Schöberl (2008), Vohralík (2008), Ern and
Vohralík (2009), local Neumann MFE problems
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
37 / 39
I Estimate Efficiency Applications Numerics R&B
Previous results
Global flux reconstructions
Prager and Synge (1947):
k∇u + σh k2 + k∇(u − uh )k2 = k∇uh + σh k2
for any uh ∈ H01 (Ω) and any σh ∈ H(div, Ω) s.t. ∇·σh = f
Hlaváček, Haslinger, Nečas, and Lovíšek (1979), Repin
(1997), . . . : global construction of σh : unprecise/costly
Local flux reconstructions
Ladevèze and Leguillon (1983), equilibrated face fluxes
Destuynder and Métivet (1999), discrete flux σh
Luce and Wohlmuth (2004), local efficiency proof
Vejchodský (2006), equilibration–hypercircle approach
Kim (2007) & Ern, Nicaise, and Vohralík (2007),
discontinuous Galerkin method elementwise prescription
Braess and Schöberl (2008), Vohralík (2008), Ern and
Vohralík (2009), local Neumann MFE problems
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
37 / 39
I Estimate Efficiency Applications Numerics R&B
Previous results
Local potential reconstructions (uh 6∈ H01 (Ω))
Achdou, Bernardi, and Coquel (2003) & Karakashian and
Pascal (2003), by prescription
Carstensen and Merdon (2013), local Neumann MFE
problems
Unified frameworks
Ainsworth and Oden (1993)
Carstensen (2005)
Ainsworth (2010)
Ern and Vohralík (heat equation 2010, Stokes equation
2012, nonlinear Laplace equation 2013)
Polynomial-degree-robust estimates
Braess, Pillwein, and Schöberl (2009), conforming finite
elements
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
38 / 39
I Estimate Efficiency Applications Numerics R&B
Previous results
Local potential reconstructions (uh 6∈ H01 (Ω))
Achdou, Bernardi, and Coquel (2003) & Karakashian and
Pascal (2003), by prescription
Carstensen and Merdon (2013), local Neumann MFE
problems
Unified frameworks
Ainsworth and Oden (1993)
Carstensen (2005)
Ainsworth (2010)
Ern and Vohralík (heat equation 2010, Stokes equation
2012, nonlinear Laplace equation 2013)
Polynomial-degree-robust estimates
Braess, Pillwein, and Schöberl (2009), conforming finite
elements
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
38 / 39
I Estimate Efficiency Applications Numerics R&B
Previous results
Local potential reconstructions (uh 6∈ H01 (Ω))
Achdou, Bernardi, and Coquel (2003) & Karakashian and
Pascal (2003), by prescription
Carstensen and Merdon (2013), local Neumann MFE
problems
Unified frameworks
Ainsworth and Oden (1993)
Carstensen (2005)
Ainsworth (2010)
Ern and Vohralík (heat equation 2010, Stokes equation
2012, nonlinear Laplace equation 2013)
Polynomial-degree-robust estimates
Braess, Pillwein, and Schöberl (2009), conforming finite
elements
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
38 / 39
I Estimate Efficiency Applications Numerics R&B
Bibliography
Bibliography
E RN A., VOHRALÍK M., Polynomial-degree-robust a
posteriori estimates in a unified setting for conforming,
nonconforming, discontinuous Galerkin, and mixed
discretizations, SIAM J. Numer. Anal., 53 (2015),
1058–1081.
D OLEJŠÍ V., E RN A., VOHRALÍK M., hp-adaptation driven
by polynomial-degree-robust a posteriori error estimates
for elliptic problems, HAL Preprint 01165187, submitted for
publication.
Thank you for your attention!
M. Vohralík
I/IV Laplace a posteriori error estimates in a unified framework
39 / 39