Explicit Upper Bounds for Dual Norms of Residuals
Explicit Upper Bounds for Dual Norms of Residuals
Outline
Explicit Upper Bounds for Dual Norms
of Residuals
R. Verfürth
Background, Goal and Tools
Notations and Assumptions
Residual A Posteriori Error Estimates
A. Veeser
Ruhr-Universität Bochum, Fakultät für Mathematik
Trace Inequalities
Università degli Studi di Milano, Dipartimento di Matematica
Poincaré Inequalities
Paris / October 13th, 2008
Singularly Perturbed Problems
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Explicit Upper Bounds for Dual Norms of Residuals
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Explicit Upper Bounds for Dual Norms of Residuals
Background, Goal and Tools
Background, Goal and Tools
Background
Goal
A posteriori error estimation consists of two steps:
Derive upper bounds for dual norms of residuals which are:
I
Prove the equivalence of the primal norm of the error and
of the dual norm of the residual.
I
fully explicit,
I
Derive easily computable error indicators that yield upper
and lower bounds for the dual norm of the residual.
guaranteed,
I
I
robust.
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Explicit Upper Bounds for Dual Norms of Residuals
Explicit Upper Bounds for Dual Norms of Residuals
Background, Goal and Tools
Notations and Assumptions
Tools
Domain and Norms
I
Ω ⊂ Rd open, bounded, connected
Decomposition of the residual into a regular and singular
part. (element and jump residuals)
I
ΓD Dirichlet boundary
I
µd , µd−1 Lebesgue measures on Rd and Rd−1
I
Partition of unity based on the nodal shape functions.
I
Lp (Ω) and k·kLp (Ω) standard Lebesgue space and norm
I
Trace inequalities with explicit constants.
I
W 1,p (Ω) standard Sobolev space equipped with k |∇·| kLp (Ω)
I
Poincaré inequalities with explicit constants.
I
W −1,p (Ω) dual space equipped with
k·kW −1,p0 (Ω) = sup h·, vi : k |∇v| kLp (Ω) = 1
I
Resumé: Reduce everything to Poincaré inequalities for convex
domains.
0
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Explicit Upper Bounds for Dual Norms of Residuals
Explicit Upper Bounds for Dual Norms of Residuals
Notations and Assumptions
Notations and Assumptions
Partition and Finite Element Spaces
I
T affine equivalent, admissible partition, may be
anisotropic
I
N , E vertices and faces corresponding to T
I
Σ skeleton (union of all faces) of T
I
Residual
0
R ∈ W −1,p (Ω) given residual satisfying:
I
1,0
S 1,0 (T ), SD
(T ) standard lowest order conforming finite
element spaces
I
λz , z ∈ N , standard lowest order nodal shape functions
I
ωz , σz union of all elements resp. faces with vertex z
p
I µd,z , µd−1,z , Lz ,
Ω
I
k·kLpz (Ω) measures, spaces and norms with
weight λz , i.e.
kvkpLp (Ω)
z
Z
=
R has an Lp -representation
Z
Z
hR, vi =
rv +
jv.
Σ
1,0
R vanishes on SD
(T ).
|v|p λz
Ω
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Explicit Upper Bounds for Dual Norms of Residuals
Explicit Upper Bounds for Dual Norms of Residuals
Residual A Posteriori Error Estimates
Residual A Posteriori Error Estimates
Localization
I
Basic Poincaré-type Inequalities
λz , z ∈ N , form a partition of unity:
X
X
hR, vi =
hR, λz vi, k∇vkpLp (Ω) =
k∇vkpLp (ω
z
z∈N
I
For every vertex z ∈ N there are constants cp (ωz ), cp (σz ) and
1,0
vz ∈ R with λz vz ∈ SD
(T ) and:
z)
z∈N
1,0
vz ∈ R arbitrary subject to the condition λz vz ∈ SD
(T ):
nX
hR, λz vi = hR, λz (v − vz )i
kv − vz kLpz (ωz ) ≤ cp (ωz )hz k∇vkLpz (ωz )
o1
p
p
≤ cp (σz )hz k∇vkLpz (ωz )
h⊥
kv
−
v
k
p
z L (E)
E
z
E⊂σz
I
Lp -representation of R:
Z
hR, λz (v − vz )i =
with hz the diameter of ωz and h⊥
E = µd,z (ωE )/µd−1,z (E).
Z
rλz (v − vz ) +
jλz (v − vz )
ωz
σz
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Explicit Upper Bounds for Dual Norms of Residuals
Explicit Upper Bounds for Dual Norms of Residuals
Residual A Posteriori Error Estimates
Residual A Posteriori Error Estimates
Vertex Oriented A Posteriori Error Estimate
Element Oriented A Posteriori Error Estimate
Set
n
hz o
cp (K) = max cp (ωz )
,
z∈NK
hK
n
o
hz
cp (E) = max cp (σz )
1
1
1−
z∈NE
p
hE p (h⊥
E)
Set
h
ηz = hz cp (ωz )krkLp0 (ω )
z
z
X ⊥ 1−p0
0
+ cp (σz )
(hE )
kjkp p0
p
Lz (E)
E⊂σz
then
then
kRkW −1,p0 (Ω) ≤
10 i
nX
0
ηzp
o 10
p
kRkW −1,p0 (Ω) ≤
.
nX
0
0
0
cp (K)p hpK krkpLp0 (K)
o 10
p
K∈T
z∈N
+
nX
cp (E)
p0
0
hE kjkpLp0 (E)
o 10
p
.
E∈E
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Explicit Upper Bounds for Dual Norms of Residuals
Explicit Upper Bounds for Dual Norms of Residuals
Residual A Posteriori Error Estimates
Trace Inequalities
Typical Values of Constants
cp (ωz )
cp (σz )
cp (K)
cp (E)
ωz convex
z 6∈ ΓD z ∈ ΓD
0.3
1.0
0.7
1.7
0.7
2.2
√
√
2.4 λ 5.5 λ
Vector Field
•
ωz not convex
z 6∈ ΓD z ∈ ΓD
0.7
2.0
1.4
3.4
1.4λ
4.1λ
√
√
5.8 λ 13.5 λ
A
A
A
A
A
A
•
?
where λ measures the anisotropy
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Explicit Upper Bounds for Dual Norms of Residuals
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Explicit Upper Bounds for Dual Norms of Residuals
Trace Inequalities
Trace Inequalities
Vector Field
Trace Identity
The vector field
(
x − aK,E
γK,E = (x−aK,E )·nK,E
mK,E ·nK,E
The Gauß theorem applied to wγK,E and λz wγK,E yields:
Z
Z
1
1
w=
w
µd−1 (E) E
µd (K) K
Z
1
+
γK,E · ∇w,
νK µd (K) K
Z
Z
1
1
λz w =
λz w
µd−1,z (E) E
µd,z (K) K
Z
1
+
λz γK,E · ∇w.
(νK + 1)µd,z (K) K
if K is a simplex,
mK,E
if K is a parallelepiped,
has the properties:
div γK,E = νK
(
d if K is a simplex,
=
1 if K is a parallelepiped,
γK,E · nK = 0
νK µd (K)
γK,E · nK =
µd−1 (E)
kγK,E kL∞ (K) ≤ hK .
on ∂K \ E,
on E,
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Explicit Upper Bounds for Dual Norms of Residuals
Explicit Upper Bounds for Dual Norms of Residuals
Trace Inequalities
Trace Inequalities
Trace Inequalities
Contribution of the Skeleton
The trace identity applied to
1
µd−1,z (E)
kukqLq (E) ≤
z
|u|q
yields:
1
kukqLq (K)
z
µd,z (K)
qhK
+
kukq−1
k∇ukLqz (K) .
Lqz (K)
(νK + 1)µd,z (K)
The trace inequality for σz implies:
o p1
p
hK
max
cp (ωz ) K⊂ωz (νK + 1)hz
1
n o
n
o
p p
1− 1
≤ d 1+
max cp (ωz ), cp (ωz ) p .
2
n cp (σz ) ≤ cp (ωz ) d 1 +
The above trace inequality applied to the elements in ωz and
their faces gives:
X
q
q
h⊥
E kukLq (E) ≤ dkukLq (ω )
z
z
z
E⊂σz
+ d max
K⊂ωz
qhK
kukq−1
k∇ukLqz (ωz ) .
Lqz (ωz )
νK + 1
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Explicit Upper Bounds for Dual Norms of Residuals
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Explicit Upper Bounds for Dual Norms of Residuals
Poincaré Inequalities
Poincaré Inequalities
Tasks
Vertices on the Dirichlet Boundary
The trace inequality for elements and faces yields for all
Dirichlet vertices
We must bound
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cp (ωz ) for z ∈ ΓD (Friedrich’s inequality),
I
cp (ωz ) for z 6∈ ΓD and convex ωz ,
I
cp (ωz ) for z 6∈ ΓD and non-convex ωz .
I
Set
CP,p (ωz ) =
sup
inf
u∈W 1,p (ωz ) c∈R
ku − ckLpz (ωz )
hz k∇ukLpz (ωz )
h
h⊥ 1 i
p
z
cp (ωz ) ≤ 1 + Mz max
CP,p (ωz )
⊥
E⊂ΓD,z h
E
h⊥ 1 i
h
hωE
p
z
+ Mz max
⊥
E⊂ΓD,z hz (νωE + 1) h
E
.
where Mz ≤ maxK⊂ωz (d + 1 − νK ) is the maximal number of
Dirichlet faces per element, ΓD,z = ΓD ∩ ∂ωz and
h⊥
z = µd,z (ωz )/µd−1,z (ΓD,z ).
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Explicit Upper Bounds for Dual Norms of Residuals
Explicit Upper Bounds for Dual Norms of Residuals
Poincaré Inequalities
Poincaré Inequalities
Convex Domains
Non-Convex Domains
If ωz is not convex we have
1− p1
CP,p (ωz ) ≤ 4CP,p (nz − 1)
If ωz is convex a result of Chua and Wheeden implies
CP,p (ωz ) ≤ CP,p ≤
1

2
and
for p = 1,
for p = 2,
1
π


2
d
CP,p (ωz ) ≤ 12dCP,p κzp
1
pp
2
for general p.
with κz =
3d
2
Kp,d (x) ≤
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Explicit Upper Bounds for Dual Norms of Residuals
Singularly Perturbed Problems
Necessary Modifications
Replace the primal norm by
1
k|vk| = εk |∇·| k2L2 (Ω) + k·k2L2 (Ω) 2 .
I
Replace the dual norm by k|·k|∗ = sup{h·, vi : k|vk| = 1}.
1
Replace hz by min ε− 2 hz , 1 .
Replace c2 (ωz ) by max c2 (ωz ), 1 .
I
I
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−1
1
2
+
p p1
1
K
(κ
),
max κ−d
p,d z
z
2
dCP,p
maxy∈∂ωz |y − z|/ miny∈∂ωz |y − z| and
(
I
1
p i p1
hKi µd (ωz ) p
+
max
2 2CP,p 1≤i≤nz µ (K ) p1 h
z
i
d
h1
1 p−1 p−1 max{p,d}
x
d |d−p|
d−1
1
d ln x
if p 6= d,
if p = d.
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