Main result
Linear convergence wrt k
Convergence rates
Convergence rates of adaptive FEM
a ‘talk’ as last lecture of “Adaptive FEM”
July 21, 2013
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Outline
1
Main result
2
Linear convergence wrt k
3
Convergence rates
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Model problem
Let u ∈ H01 (Ω) be sth
∀ϕ ∈
H01 (Ω)
Z
A∇u · ∇ϕ = hf , ϕi
Ω
and assume that
Ω is a domain in R2
A ∈ W 1,∞ (Ω)2×2 sth
A(x) is symmetric and λ|ξ|2 ≤ A(x)ξ · ξ ≤ Λ|ξ|2
with 0 < λ < Λ
f ∈ L2 (Ω) is a load term.
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Main steps of adaptive FEM
Denote by k the iteration index of the global solves for the
Galerkin solution in
S(Tk ) := V : C 0 (Ω) → R | ∀T ∈ Tk V|T ∈ P1 (T ),
V|∂Ω = 0 .
Given A, f and starting with initial triangulation T0 , iterate
Uk := SOLVE(A, f , Tk )
{ηk (T )}T ∈Tk := ESTIMATE(Uk , A, f )
Mk := MARK(Tk , {ηk (T )}T ∈Tk )
Tk+1 := REFINE(Tk , Mk )
increment k
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Assumptions on AFEM
Assume that
T0 verifies matching assumption and REFINE is recursive
bisection (sth ...),
{ηk (T )}T is the standard residual estimator for
Z
k · kΩ :=
|∇ · |
2
1/2
Ω
(Dörfler marking or bulk chasing)
fix some δ ∈ (0, 1] and choose each Mk with minimal
cardinality sth.
X
X
ηk (T )2 ≥ δ 2
ηk (T )2
T ∈Mk
a ‘talk’ as last lecture of “Adaptive FEM”
T ∈Tk
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Rate optimality (Stevenson ’07)
(in the formulation of Cascon/Morin/Nochetto/Siebert ’08)
Moreover assume that, for some r > 0, there holds
min E (T ) + osc(A, f , UT ) ≤ Cn−r
#T ≤n
where E (T ) is the best error in S(T ) and osc is the oscillation
associated with the standard residual error estimator.
Then, for sufficiently small δ > 0, there holds
ku − Uk k ≤ C #Tk−r
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Remarks
For solutions with localized singularities (in particular
power-like ones), adaptive FEM outperform classical FEM.
The threshold of the marking parameter δ is proportional to
p
λ/Λ or the quotient of constants in lower and upper bound.
Thus, for anisotropic operators or bad estimators, rate
optimality is ensured only for small δ and so for many global
solves.
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Outline
1
Main result
2
Linear convergence wrt k
3
Convergence rates
For the rest of the presentation, assume that oscillation is
vanishing.
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Pythagoras: correction = reduction in energy norm
Galerkin orthogonality implies:
If T∗ is a refinement of T , then the corresponding Galerkin
solutions U∗ ∈ S(T∗ ) and U ∈ S(T ) satisfy
ku − U∗ k2A = ku − Uk2A − kU∗ − Uk2A
with
1/2
Z
kv kA :=
A∇v · ∇v
Ω
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
.
Main result
Linear convergence wrt k
Convergence rates
Lower bounds for local correction (Dörfler ’96)
For any T ∈ T sth each triangle in ωT has vertices of T∗ in its
interior, there holds the following lower bound:
√
ηT ≤ C ΛkU∗ − UkA;ωT
Pf.: Relate to
sup{hR(U), ϕi | ϕ ∈ S(T∗ ), supp ϕ ⊂ ω, kϕkA ≤ 1}
and construct cut-off functions in S(T∗ ).
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Linking Uk and Uk+1
Since on one hand,
1/2

C
ku − Uk kA ≤ √ 
ηk (T )2 
λ T ∈T
X
k
and, on the other hand,
1/2

c
ηk (T )2 
kUk − Uk+1 kA ≥ √ 
Λ T ∈M
X
k
we may require
X
ηk (T )2 ≥ δ 2
T ∈Mk
a ‘talk’ as last lecture of “Adaptive FEM”
X
ηk (T )2 .
T ∈Tk
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Linear convergence
If δ ∈ (0, 1] is fixed (and Tk+1 has interior vertices for Mk ), there
exists α ∈ (0, 1) sth
ku − Uk+1 k ≤ αku − Uk k
and so
ku − Uk k ≤ C αk ,
which (for another α) also holds with oscillation (cf.
Morin/Nochetto/Siebert ’02).
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Outline
1
Main result
2
Linear convergence wrt k
3
Convergence rates
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Upper bound for global correction (Stevenson ’07)
Let T∗ be a refinement of T and TR := T \ T∗ the triangles of T
to be refined. If U∗ ∈ S(T∗ ) and U ∈ S(T ) are the corresponding
Galerkin solutions, there holds

1/2
X
C
kU∗ − UkA ≤ √ 
ηk (T )2 
λ T ∈T
R
Pf.: Observe
kU∗ − UkA ≤ sup{hR, ϕi | Φ ∈ S(T∗ ), kϕkA ≤ 1}
and, with the help of the Scott-Zhang interpolation operator,
choose cz sth
X
ϕ−
cz ϕz = 0 for any T ∈ T ∩ T ∗
z∈T ∩V
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Error reduction and indicators
If
ku − U∗ kA ≤ µku − UkA
for any µ ∈ [0, 1), then
X
c2
(1 − µ2 )
ηT2 ≤ (1 − µ2 )ku − Uk2A ≤ ku − Uk2A − ku − U∗ k2A
Λ
T ∈T
= kU∗ − Uk2A ≤
C2 X 2
ηT
λ
T ∈TR
ie
δ2
X
ηk (T )2 ≤
T ∈Tk
X
ηk (T )2
T ∈TR
q p
with δ = Cc Λλ 1 − µ2 > 0.
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Linking to approximability of u
Assume that minT 0 ≤n E (T 0 ) ≤ Cn−r with r > 0.
Then there exists a partition Tµ such that
E (Tµ ) ≤ µku − Uk and
#Tµ ≤ C (µ, r )ku − Uk−1/r
Let T∗ be the minimal common refinement of Tµ and T . Then
#TR ≤ #T∗ − #T ≤ #Tµ ≤ C (µ, r )ku − Uk−1/r
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Dörfler marking
Choose Mk with minimal cardinality sth
X
X
ηk (T )2 ≥ δ 2
ηk (T )2
T ∈Tk
T ∈Mk
Then
#Mk ≤ #TR ≤ C ku − Uk k−1/r
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates
Main result
Linear convergence wrt k
Convergence rates
Iteration history
Using the matching assumption and linear convergence, we
conclude
#TK ≤ C
K
−1
X
K
−1
X
#Mk ≤ C
k=0
ku − Uk k−1/r
k=0
−1/r
≤ C (α, r )ku − UK k
ie
ku − UK k ≤ C #TK−r
a ‘talk’ as last lecture of “Adaptive FEM”
Convergence rates