Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
On optimality of adaptive finite element methods
with Dörfler marking
ESTIMATE
Christian Kreuzer, EFEF 2010
Joint work with
K. G. Siebert
Outline
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
1 The Adaptive Finite Element Method
2 Convergence Rate of the AFEM using Diverse Indicators
Outline
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
1 The Adaptive Finite Element Method
2 Convergence Rate of the AFEM using Diverse Indicators
Problem and Discretization
Optimality
of AFEM
Let Ω ⊂ Rd (d = 2, 3) be an open polygonal domain.
Christian
Kreuzer,
EFEF 2010
Problem:
Outline
AFEM
−∆u = f
in
Ω,
u=0
on
∂Ω.
ESTIMATE
Finite Elements
Let T0 be an initial, conforming triangulation of Ω.
We define T as the set of all conforming refinements of T0 , that can be
generated from T0 using iterative or recursive bisection.
For T ∈ T we define V(T ) as the space of continuous piecewise affine
finite elements over T .
Problem and Discretization
Optimality
of AFEM
Let Ω ⊂ Rd (d = 2, 3) be an open polygonal domain.
Christian
Kreuzer,
EFEF 2010
Problem:
Outline
AFEM
−∆u = f
in
Ω,
u=0
on
∂Ω.
ESTIMATE
Finite Elements
Let T0 be an initial, conforming triangulation of Ω.
We define T as the set of all conforming refinements of T0 , that can be
generated from T0 using iterative or recursive bisection.
For T ∈ T we define V(T ) as the space of continuous piecewise affine
finite elements over T .
Reliable and efficient estimators: Ritz Galerkin Solution U ∈ V(T )
|||u − U |||2Ω ≤ C1 ET2 (IT )
and
C2 ET2 (IT ) ≤ |||u − U |||2Ω + osc2T (IT , f ).
Adaptive Finite Element Method (AFEM)
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
Let T0 be an initial triangulation of Ω, k = 0.
SOLVE
Uk = SOLVE(Tk )
Computes the Ritz approximation in Vk = V(Tk ).
Adaptive Finite Element Method (AFEM)
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Let T0 be an initial triangulation of Ω, k = 0.
SOLVE
Uk = SOLVE(Tk )
Outline
AFEM
ESTIMATE
ESTIMATE
{Ek (I)}I∈Ik = ESTIMATE(Uk , Tk )
Computes error indicators on Ik .
Adaptive Finite Element Method (AFEM)
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Let T0 be an initial triangulation of Ω, k = 0.
SOLVE
Uk = SOLVE(Tk )
Outline
AFEM
ESTIMATE
ESTIMATE
{Ek (I)}I∈Ik = ESTIMATE(Uk , Tk )
MARK
Mk = MARK({Ek (I)}I∈Ik ) ⊂ Ik
Choose minimal Mk ⊂ Ik s.t. Ek (Mk ) ≥ θ Ek (Ik ).
Adaptive Finite Element Method (AFEM)
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Let T0 be an initial triangulation of Ω, k = 0.
SOLVE
Uk = SOLVE(Tk )
Outline
AFEM
ESTIMATE
ESTIMATE
{Ek (I)}I∈Ik = ESTIMATE(Uk , Tk )
MARK
Mk = MARK({Ek (I)}I∈Ik ) ⊂ Ik
REFINE
Tk+1 = REFINE(Tk , Mk )
Refine elements associated with marked indices Mk
Adaptive Finite Element Method (AFEM)
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Let T0 be an initial triangulation of Ω, k = 0.
SOLVE
Uk = SOLVE(Tk )
Outline
AFEM
ESTIMATE
ESTIMATE
{Ek (I)}I∈Ik = ESTIMATE(Uk , Tk )
MARK
Mk = MARK({Ek (I)}I∈Ik ) ⊂ Ik
REFINE
Tk+1 = REFINE(Tk , Mk )
Increment k and go to step SOLVE.
Approximation Rate
Optimality
of AFEM
Quantifies convergence speed in Degrees Of Freedom
Christian
Kreuzer,
EFEF 2010
Total Error
Outline
AFEM
ESTIMATE
|||u − U |||2Ω + ET2 (T ) ≈ |||u − U |||2Ω + osc2T (T , f ) =: Err(u − U, f, T )2
Approximation Rate
Optimality
of AFEM
Quantifies convergence speed in Degrees Of Freedom
Christian
Kreuzer,
EFEF 2010
Total Error
Outline
AFEM
|||u − U |||2Ω + ET2 (T ) ≈ |||u − U |||2Ω + osc2T (T , f ) =: Err(u − U, f, T )2
ESTIMATE
Assumption: Total Error of the problem can be approximated with rate s > 0.
Approximation Rate
Optimality
of AFEM
Quantifies convergence speed in Degrees Of Freedom
Christian
Kreuzer,
EFEF 2010
Total Error
Outline
AFEM
|||u − U |||2Ω + ET2 (T ) ≈ |||u − U |||2Ω + osc2T (T , f ) =: Err(u − U, f, T )2
ESTIMATE
Assumption: Total Error of the problem can be approximated with rate s > 0.
Hence, a good AFEM should yield
´−s
`
Errk := Err(u − Uk , f, Tk ) 4 #Tk − #T0
Approximation Rate
Optimality
of AFEM
Quantifies convergence speed in Degrees Of Freedom
Christian
Kreuzer,
EFEF 2010
Total Error
Outline
AFEM
|||u − U |||2Ω + ET2 (T ) ≈ |||u − U |||2Ω + osc2T (T , f ) =: Err(u − U, f, T )2
ESTIMATE
Assumption: Total Error of the problem can be approximated with rate s > 0.
Hence, a good AFEM should yield
´−s
`
Errk := Err(u − Uk , f, Tk ) 4 #Tk − #T0
Remark
Generically oscillation is of higher order, hence for fine meshes T :
Err(u − U, f, T )2 ≈ |||u − U |||2Ω
Outline
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
1 The Adaptive Finite Element Method
2 Convergence Rate of the AFEM using Diverse Indicators
Residual based Estimator
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
Which indicators for optimal rates?
9
[Stevenson:07]
=
[CKNS:08]
⇒ Choose residual based estimators!
;
[Diening and Kreuzer:08]
Residual based Estimator
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
Which indicators for optimal rates?
9
[Stevenson:07]
=
[CKNS:08]
⇒ Choose residual based estimators!
;
[Diening and Kreuzer:08]
Organized element wise:
1/2
EbT2 (U, T ) := khT f k2L2 (T ) + khT J(U )k2L2 (∂T )
osc
c 2T
(f, T ) := khT (f −
fT )k2L2 (T )
∀T ∈ T .
∀T ∈ T ,
Main Auxiliary Results: Contraction and Local Discrete Upper Bound
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
Theorem ([CKNS:08], [Diening and Kreuzer:08])
b0 > 0, α ∈ (0, 1) such that
There exists C
dk ≤ C
b0 αk−` Err
d` .
Err
2
dk
Remark: |||u − Uk |||2Ω + γ Ebk2 (Tk ) ≈ Err
Main Auxiliary Results: Contraction and Local Discrete Upper Bound
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
Theorem ([CKNS:08], [Diening and Kreuzer:08])
b0 > 0, α ∈ (0, 1) such that
There exists C
dk ≤ C
b0 αk−` Err
d` .
Err
2
dk
Remark: |||u − Uk |||2Ω + γ Ebk2 (Tk ) ≈ Err
Theorem ([Stevenson:07], [CKNS:08])
Let T∗ ∈ T being a refinement of Tk with corresponding Ritz approximation U∗
b 1 EbT2 (Tk \ T∗ ).
|||Uk − U∗ |||2Ω ≤ D
k
Tk \ T∗ is the set of elements that have to be refined in order to obtain T∗ .
b1 = C
b1 .
Remark: Residual based estimator: D
[CKNS]: Optimal Rates for Residual based Estimator
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Let (u, f ) can be approximated with rate s > 0. Let T0 satisfy some initial
marking conditions [Binev, Dahmen, DeVore:04,Stevenson:08] and assume
Outline
AFEM
θ ∈ (0, θ∗ )
ESTIMATE
with
θ∗2 :=
b2
C
.
b1
1+D
Theorem ([Stevenson:07], [CKNS:08])
AFEM produces optimal rates, i.e.,
`
|||u − Uk |||2Ω + osc
c 2k
´1/2
“
´−s
θ2 ”−1/2 `
≤C 1− 2
#Tk − #T0
θ∗
The constant C depends on α and on C0 .
∀k ∈ N.
[CKNS]: Optimal Rates for Residual based Estimator
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Let (u, f ) can be approximated with rate s > 0. Let T0 satisfy some initial
marking conditions [Binev, Dahmen, DeVore:04,Stevenson:08] and assume
Outline
AFEM
θ ∈ (0, θ∗ )
ESTIMATE
with
θ∗2 :=
b2
C
.
b1
1+D
Theorem ([Stevenson:07], [CKNS:08])
AFEM produces optimal rates, i.e.,
`
|||u − Uk |||2Ω + osc
c 2k
´1/2
“
´−s
θ2 ”−1/2 `
≤C 1− 2
#Tk − #T0
θ∗
The constant C depends on α and on C0 .
BUT: WHAT’S ABOUT OTHER ESTIMATORS?
∀k ∈ N.
Other Estimators: Hierarchical Estimator
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
With local side and element hat functions φσ and φT :
ESTIMATE
¸2
˙
ET2 (U, σ) := Res(U ), |||φφσσ|||
Ω
”
X “˙
¸2
Res(U ), |||φφTT|||
+
+ khT (f − fT )k2L2 (T )
Ω
T ⊂ωσ
osc2T (f, σ) :=
X
T ⊂ωσ
khT (f − fT )k2L2 (T ) ,
∀σ ∈ S.
Other Estimators: Estimator Based on Local Problems
[Morin, Nochetto, and Siebert:03]
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
Let φz , z ∈ N , be the Lagrange basis functions.
Wz :
Continuous quadratic finite elements in ωz with vanishing trace on ∂ωz .
R
If z ∈ N ∩ Ω, then additionally ωz ψφz = 0 for all ψ ∈ Wz .
ESTIMATE
For each vertex z ∈ N solve the linear problem
Z
˙
¸
ηz ∈ Wz :
∇ηz · ∇ψ φz dx = Res(U ), ψφz
∀ ψ ∈ Wz .
ωz
Organized by nodes
1
1
ET2 (U, z) := k∇ηz φz2 k2L2 (ωz ) + khz (f − fz )φz2 k2L2 (ωz ) ,
1
osc2T (f, z) := khz (f − fz )φz2 k2L2 (ωz ) .
Other Estimators: Recovery Based Estimator
[Bartels and Carstensen:02] and [Zienkiewicz and Zhu:87]
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
Define the averaging operator G : V → V(T )d by the nodal values
Z
1
(GV )(z) =
∇V dx, z ∈ N .
|ωz | ωz
Local error indicators
o
n
ET2 (U, z) := k∇U − GU k2L2 (ωz ) + h2z kf − fz k2L2 (ωz ) ,
osc2T (f, z) := khz (f − fz )k2L2 (ωz ) ,
z ∈ N.
z ∈ N,
Other Estimators: Equilibrated Residual Estimator [Braes et. al.]
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
Given z ∈ N find ξz ∈ RT−1 (T , z) with minimal L2 -norm such that
Z
1
f φz dx =: fz |T ,
in each T ⊂ ωz ,
div ξz |T = −
|T | T
Z
1
[[ξz ]]|σ =
J(U )φz do = J(U )|σ ,
on each σ ⊂ Σz ,
2
σ
ξz · ν = 0,
(1)
on ∂ωz ,
where RT−1 (T , z) denotes the local broken Raviart-Thomas space
o
n
RT−1 (T , z) := ~g ∈ L2 (ωz )2 | ~g|T (x) = ~a + bx, ~a ∈ R2 , b ∈ R ∀T ⊂ ωz .
Local error indicators
n
o
ET2 (U, z) := kξz k2L2 (ωz ) + h2z kf − fz k2L2 (ωz ) ,
osc2T (f, z) := khz (f − fz )k2L2 (ωz ) ,
z ∈ N.
z ∈ N,
Notation: Index-Set I = T , I = S, or I = N
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
AFEM
ESTIMATE
Relations between triangulation T and corresponding index set I:
T (I 0 ) := {T | I ⊂ T for some I ∈ I 0 }
I 0 ⊂ I,
I(T 0 ) := {I | I ⊂ T for some T ∈ T 0 }
T0 ⊂T.
I(T )
I(T )
T
T (σ)
T
σ
T (z)
z
Local Equivalence
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
All indicators are locally equivalent to the residual based indicators:
Partially in [Verfürth:1996].
Outline
AFEM
ESTIMATE
`
´
ĉ ET2 (I 0 ) ≤ EbT2 T (I 0 )
`
´
c̃ EbT2 (T 0 ) ≤ ET2 I(T 0 )
∀ I0 ⊂ I
∀T 0 ⊂ T ;
This yields a Dörfler property for the residual estimator with θ̂ = ĉ c̃ θ:
`
´
Ebk2 T (Mk ) ≥ ĉ Ek2 (Mk ) ≥ ĉ θ2 Ek2 (Ik ) ≥ ĉ c̃ θ2 Ebk2 (Tk ) = θ̂2 Ebk2 (Tk )
Local Equivalence
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
All indicators are locally equivalent to the residual based indicators:
Partially in [Verfürth:1996].
Outline
AFEM
ESTIMATE
`
´
ĉ ET2 (I 0 ) ≤ EbT2 T (I 0 )
`
´
c̃ EbT2 (T 0 ) ≤ ET2 I(T 0 )
∀ I0 ⊂ I
∀T 0 ⊂ T ;
This yields a Dörfler property for the residual estimator with θ̂ = ĉ c̃ θ:
`
´
Ebk2 T (Mk ) ≥ ĉ Ek2 (Mk ) ≥ ĉ θ2 Ek2 (Ik ) ≥ ĉ c̃ θ2 Ebk2 (Tk ) = θ̂2 Ebk2 (Tk )
`
´
⇒ Ebk2 T (Mk ) ≥ θ̂2 Ebk2 (Tk )
Local Equivalence – Error Reduction
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
All indicators are locally equivalent to the residual based indicators:
Partially in [Verfürth:1996].
Outline
AFEM
ESTIMATE
`
´
ĉ ET2 (I 0 ) ≤ EbT2 T (I 0 )
`
´
c̃ EbT2 (T 0 ) ≤ ET2 I(T 0 )
∀ I0 ⊂ I
∀T 0 ⊂ T ;
This yields a Dörfler property for the residual estimator with θ̂ = ĉ c̃ θ:
`
´
Ebk2 T (Mk ) ≥ ĉ Ek2 (Mk ) ≥ ĉ θ2 Ek2 (Ik ) ≥ ĉ c̃ θ2 Ebk2 (Tk ) = θ̂2 Ebk2 (Tk )
`
´
⇒ Ebk2 T (Mk ) ≥ θ̂2 Ebk2 (Tk )
Implies error reduction for the ’residual’ total error; [CKNS:08], [Diening
and Kreuzer:08]:
dk ≤ C0 αk−` Err
d` (T` )
Err
Local Equivalence – Error Reduction
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
All indicators are locally equivalent to the residual based indicators:
Partially in [Verfürth:1996].
Outline
AFEM
ESTIMATE
`
´
ĉ ET2 (I 0 ) ≤ EbT2 T (I 0 )
`
´
c̃ EbT2 (T 0 ) ≤ ET2 I(T 0 )
∀ I0 ⊂ I
∀T 0 ⊂ T ;
This yields a Dörfler property for the residual estimator with θ̂ = ĉ c̃ θ:
`
´
Ebk2 T (Mk ) ≥ ĉ Ek2 (Mk ) ≥ ĉ θ2 Ek2 (Ik ) ≥ ĉ c̃ θ2 Ebk2 (Tk ) = θ̂2 Ebk2 (Tk )
`
´
⇒ Ebk2 T (Mk ) ≥ θ̂2 Ebk2 (Tk )
Implies error reduction for the ’residual’ total error; [CKNS:08], [Diening
and Kreuzer:08]:
dk ≤ C0 αk−` Err
d` (T` )
Errk ≈ Err
Local Equivalence
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
All indicators are locally equivalent to the residual based indicators:
Partially in [Verfürth:1996].
Outline
AFEM
ESTIMATE
`
´
ĉ ET2 (I 0 ) ≤ EbT2 T (I 0 )
`
´
c̃ EbT2 (T 0 ) ≤ ET2 I(T 0 )
∀ I0 ⊂ I
∀T 0 ⊂ T ;
This yields a Dörfler property for the residual estimator with θ̂ = ĉ c̃ θ:
`
´
Ebk2 T (Mk ) ≥ ĉ Ek2 (Mk ) ≥ ĉ θ2 Ek2 (Ik ) ≥ ĉ c̃ θ2 Ebk2 (Tk ) = θ̂2 Ebk2 (Tk )
`
´
⇒ Ebk2 T (Mk ) ≥ θ̂2 Ebk2 (Tk )
Implies error reduction for the ’residual’ total error; [CKNS:08], [Diening
and Kreuzer:08]:
dk ≤ C0 αk−` Err
d` (T` ) ≈ αk−` Err`
Errk ≈ Err
Local Equivalence – Local Upper Bound
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
All indicators are locally equivalent to the residual based indicators:
Partially in [Verfürth:1996].
Outline
AFEM
ESTIMATE
`
´
ĉ ET2 (I 0 ) ≤ EbT2 T (I 0 )
`
´
c̃ EbT2 (T 0 ) ≤ ET2 I(T 0 )
∀ I0 ⊂ I
∀T 0 ⊂ T ;
Discrete local upper bound can be achieved via equivalence of estimators
as well:
Local Equivalence – Local Upper Bound
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
All indicators are locally equivalent to the residual based indicators:
Partially in [Verfürth:1996].
Outline
AFEM
ESTIMATE
`
´
ĉ ET2 (I 0 ) ≤ EbT2 T (I 0 )
`
´
c̃ EbT2 (T 0 ) ≤ ET2 I(T 0 )
∀ I0 ⊂ I
∀T 0 ⊂ T ;
Discrete local upper bound can be achieved via equivalence of estimators
as well:
Let T ∈ T and T∗ be a refinement of T , then
`
´
`
´
b1 EbT2 (T \ T∗ ) ≤ c̃−1 D
b 1 ET2 I(T \ T∗ ) = D1 ET2 I(T \ T∗ )
|||U − U∗ |||2Ω ≤ C
Local Equivalence – Local Upper Bound
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
All indicators are locally equivalent to the residual based indicators:
Partially in [Verfürth:1996].
Outline
AFEM
ESTIMATE
`
´
ĉ ET2 (I 0 ) ≤ EbT2 T (I 0 )
`
´
c̃ EbT2 (T 0 ) ≤ ET2 I(T 0 )
∀ I0 ⊂ I
∀T 0 ⊂ T ;
Discrete local upper bound can be achieved via equivalence of estimators
as well:
Let T ∈ T and T∗ be a refinement of T , then
`
´
`
´
b1 EbT2 (T \ T∗ ) ≤ c̃−1 D
b 1 ET2 I(T \ T∗ ) = D1 ET2 I(T \ T∗ )
|||U − U∗ |||2Ω ≤ C
Drawback: bad constant D1 in definition of θ∗ =
C2
.
1 + D1
Local Equivalence – Local Upper Bound
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
All indicators are locally equivalent to the residual based indicators:
Partially in [Verfürth:1996].
Outline
AFEM
ESTIMATE
`
´
ĉ ET2 (I 0 ) ≤ EbT2 T (I 0 )
`
´
c̃ EbT2 (T 0 ) ≤ ET2 I(T 0 )
∀ I0 ⊂ I
∀T 0 ⊂ T ;
Discrete local upper bound can be achieved via equivalence of estimators
as well:
Let T ∈ T and T∗ be a refinement of T , then
`
´
`
´
b1 EbT2 (T \ T∗ ) ≤ c̃−1 D
b 1 ET2 I(T \ T∗ ) = D1 ET2 I(T \ T∗ )
|||U − U∗ |||2Ω ≤ C
Drawback: bad constant D1 in definition of θ∗ =
C2
.
1 + D1
Better: Try to directly Calculate discrete local upper bound
Other Estimators: Locally Conditioned Robust Estimators
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
Estimator for Equations with discontinuous diffusion coefficients
[Petzold:02]
− div k(x)∇u = f
Coefficients k positive, pw constant with quasi-monotone jumps on T0 .
AFEM
ESTIMATE
in Ω
ET2 (U, T ) :=
h2
T
k|T
kf k2L2 (T ) +
hT
k|T
kJ(U )k2L2 (∂T )
∀T ∈ T ,
Other Estimators: Locally Conditioned Robust Estimators
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
Estimator for Equations with discontinuous diffusion coefficients
[Petzold:02]
− div k(x)∇u = f
Coefficients k positive, pw constant with quasi-monotone jumps on T0 .
AFEM
ESTIMATE
in Ω
ET2 (U, T ) :=
h2
T
k|T
kf k2L2 (T ) +
hT
k|T
kJ(U )k2L2 (∂T )
∀T ∈ T ,
Estimator for singularly perturbed reaction-diffusion equations
[Verfürth:1998]
−∆u + κu = f
1/2
in Ω
κ 1.
ET2 (U, T ) := kαT f k2L2 (T ) + kαT J(U )k2L2 (∂T ) ,
αT := min{hT , κ−1 }.
Other Estimators: Locally Conditioned Robust Estimators
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
Estimator for Equations with discontinuous diffusion coefficients
[Petzold:02]
− div k(x)∇u = f
Coefficients k positive, pw constant with quasi-monotone jumps on T0 .
AFEM
ESTIMATE
in Ω
ET2 (U, T ) :=
h2
T
k|T
kf k2L2 (T ) +
hT
k|T
kJ(U )k2L2 (∂T )
∀T ∈ T ,
Estimator for singularly perturbed reaction-diffusion equations
[Verfürth:1998]
−∆u + κu = f
1/2
in Ω
κ 1.
ET2 (U, T ) := kαT f k2L2 (T ) + kαT J(U )k2L2 (∂T ) ,
Everything works as before.
αT := min{hT , κ−1 }.
Other Estimators: Locally Conditioned Robust Estimators
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
Estimator for Equations with discontinuous diffusion coefficients
[Petzold:02]
− div k(x)∇u = f
Coefficients k positive, pw constant with quasi-monotone jumps on T0 .
AFEM
ESTIMATE
in Ω
ET2 (U, T ) :=
h2
T
k|T
kf k2L2 (T ) +
hT
k|T
kJ(U )k2L2 (∂T )
∀T ∈ T ,
Estimator for singularly perturbed reaction-diffusion equations
[Verfürth:1998]
−∆u + κu = f
in Ω
1/2
κ 1.
ET2 (U, T ) := kαT f k2L2 (T ) + kαT J(U )k2L2 (∂T ) ,
Everything works as before.
Advantage of robust estimators:
θ̂∗ =
b2
C
C2
b
1
+
D1
1 + D1
αT := min{hT , κ−1 }.
Other Estimators: Locally Conditioned Robust Estimators
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Outline
Estimator for Equations with discontinuous diffusion coefficients
[Petzold:02]
− div k(x)∇u = f
Coefficients k positive, pw constant with quasi-monotone jumps on T0 .
AFEM
ESTIMATE
in Ω
ET2 (U, T ) :=
h2
T
k|T
kf k2L2 (T ) +
hT
k|T
kJ(U )k2L2 (∂T )
∀T ∈ T ,
Estimator for singularly perturbed reaction-diffusion equations
[Verfürth:1998]
−∆u + κu = f
in Ω
κ 1.
1/2
ET2 (U, T ) := kαT f k2L2 (T ) + kαT J(U )k2L2 (∂T ) ,
Everything works as before.
Advantage of robust estimators:
θ̂∗ =
b2
C
C2
= θ∗ .
b
1
+
D1
1 + D1
If calculated without equivalence of indicators!
αT := min{hT , κ−1 }.
Optimal Rate
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Let (u, f ) can be approximated with rate s > 0. Let T0 satisfy some initial
marking conditions [Binev, Dahmen, DeVore:04, Stevenson:08] and assume
Outline
AFEM
ESTIMATE
θ ∈ (0, θ∗ )
with
θ∗2 :=
C2
.
1 + D1
Optimal Rate
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
Let (u, f ) can be approximated with rate s > 0. Let T0 satisfy some initial
marking conditions [Binev, Dahmen, DeVore:04, Stevenson:08] and assume
Outline
AFEM
ESTIMATE
θ ∈ (0, θ∗ )
with
θ∗2 :=
C2
.
1 + D1
Theorem ([Stevenson:07], [CKNS:08])
AFEM produces optimal rates, i.e.,
`
|||u − Uk |||2Ω + osc2k
´1/2
“
´−s
θ2 ”−1/2 `
≤C 1− 2
#Tk − #T0
θ∗
Constant C depends on s, θ and the equivalence of estimators.
∀k ∈ N.
Thank You For Your Attention!
Optimality
of AFEM
Christian
Kreuzer,
EFEF 2010
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