Convergence rates for tree approximation of gradients
Adaptive tree approximation
with finite elements
Andreas Veeser
Università degli Studi di Milano (Italy)
July 2015 / Cimpa School / Mumbai
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Outline
1
Basic notions in constructive approximation
2
Tree approximation
3
Mesh refinement with tree structure for PDEs
4
Approximation of gradients
5
Convergence rates for tree approximation of gradients
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Literature – 1
about Besov spaces:
G. G. Lorentz, R. A. DeVore, Constructive approximation,
Springer, 1993.
about interpolation of spaces: chapter 14 in
S. Brenner, R. Scott, The mathematical theory of finite
element methods, Springer, 2008.
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Literature – 2
What follows is the ’Besov’ version of Theorem 3 in
R. H. Nochetto, A. Veeser, Primer of Adaptive FEM, in:
Lecture Notes in Math 2040, Springer 2012
and a partial alternative for:
P. Binev, W. Dahmen, R. DeVore, P. Petrushev,
Approximation classes for adaptive methods, Serdica
Math. J. 28 (2002), 391–416.
F. D. Gaspoz, P. Morin, Approximation classes for adaptive
higher order finite element approximation, Math. Comp. 83
(2014), 2127–2160.
A. Veeser
FE tree approximation
Measuring regularity with Besov spaces
Convergence rates
Convergence rates for tree approximation of gradients
A simple example
We re-consider approximation of a continuous function on [0, 1]
with piecewise constants in the maximum norm.
The inequalities
1
E(v , P (I)) =
2
0
Z
max v − min v ≤ |v 0 | ≤ |I| sup |v 0 |
I
I
I
I
tell that the regularity ensuring the rate n−1 is, respectively, for
uniform
free
kv 0 kL∞ (0,1)
kv 0 kL1 (0,1)
breakpoints.
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Outline
1
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Shortcomings of weak derivatives and alternative?
Weak derivatives have the following shortcomings in measuring
regularity:
no fractional "number of derivatives"
defined only for p ≥ 1
Recall that, e.g., [Brezis ’11, Prop. 8.5] shows, for 1 < p ≤ ∞,
f ∈W 1,p (R) ⇐⇒
∃C ≥ 0 ∀h ∈ R kf (· + h) − v kLp (R) ≤ C|h|
where C = kf 0 kLp (R) .
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Difference operators
Let Ω ⊂ R be a domain. Given h ∈ Rd , set
Ωh := {x ∈ Ω | [x, x + h] ⊂ Ω}
and define
(
f (x + h) − f (x),
∆h f (x) := ∆h (f , x, Ω) :=
0,
and, for k ∈ N,
∆kh f = ∆h (∆kh −1 )f
in Ω.
Notice
P ∈ P` =⇒ ∆`+1
h P = 0.
A. Veeser
FE tree approximation
x ∈ Ωh ,
otherwise.
Measuring regularity with Besov spaces
Convergence rates
Convergence rates for tree approximation of gradients
Moduli of smoothness
Given 0 < p ≤ ∞ and t > 0, we define
ωk (f , t)p := ωk (f , t, Ω)p := sup ∆kh f |h|≤t
Lp (Ω)
.
We have
ωk (f , ·)p is increasing with ωk (f , 0)p = 0
ωk (·, t)p is a (quasi-)seminorm on Lp (Ω)
if
ωk (f , t)p =
(
o(t k )
o(t
if p ≥ 1,
k −1+ p1
) if 0 < p ≤ 1,
then f ∈ Pk −1 (Ω).
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Besov spaces
Let 0 < p ≤ ∞ and s > 0, 0 < q ≤ ∞.
R1
Choosing k = [s] + 1 and recalling 0 t ρ dt < ∞ ⇐⇒ ρ > −1,
define

−s

q = ∞,
supt>0 t ωk (f , t)p ,
1
Z
∞
|f |Bqs (Lp (Ω)) :=
q dt q

t −s ωk (f , t)p
0 < q < ∞,

t
0
and
Bqs Lp (Ω) := {f ∈ Lp (Ω) | |f |Bqs (Lp (Ω)) < ∞},
kf kBqs (Lp (Ω)) := kf kLp (Ω) + |f |Bqs (Lp (Ω)) .
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Embeddings
Fix 0 < p ≤ ∞ and let s1 , s2 , s > 0 and 0 < q1 , q2 ≤ ∞.
We have
s1 > s2 =⇒ Bqs11 Lp (Ω) ⊂ Bqs22 Lp (Ω)
and
q1 ≤ q2 =⇒ Bqs1 Lp (Ω) ⊂ Bqs2 Lp (Ω)
Moreover, if Ω is Lipschitz and 0 < τ ≤ ∞ then
s−
d
d
≥ − =⇒ Bτs Lτ (Ω) ⊂ Lp (Ω).
τ
p
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Bramble-Hilbert lemma
Let 0 < p < ∞ and 0 < s ≤ ` + 1, 0 < τ < ∞ such that
δ := s −
d
d
+ ≥ 0.
τ
p
Then, for any simplex K ,
inf kv − PkLp (K ) 4 diam(K )δ |v |Bτs (Lτ (K )) ;
P∈P`
the hidden constant depends on p, s, τ , d and the shape
coefficient of K .
See [Gaspoz/Morin ’14, Lemma 4.15]
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
More smoothness moduli
Define
"
wk (f , t, Ω)p,q
1
:=
(2t)d
Z
[−t,t]d
q
k
∆h (f , ·, Ω) p
L (Ω)
#1/q
dh
and notice that if Ω1 , Ω2 is a partition of Ω, then
wk (f , t, Ω1 )p,p + wk (f , t, Ω2 )p,p ≤ wk (f , t, Ω1 ∪ Ω2 )p,p
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Setadditivity of Besov seminorms
Given a simplicial mesh M of Ω, we have
wk (f , t, K )p,q ≈ ωk (f , t, K )p
for all K ∈ M and t 4 diam K ; the hidden constants depend on
k , p, q, and the shape coefficient σ(M).
Therefore
X
|v |pB s (Lp (K )) 4 |v |pB s (Lp (Ω))
p
p
K ∈M
See [Gaspoz/Morin ’14, Corollary 4.3 and Lemma 4.10].
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Outline
1
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Besov regularity and convergence rates
Let M0 be an admissible (in 2d: with coinciding edge labeling)
initial mesh of Ω and MTn ,conf the corresponding set of all
conforming refinements with at most n simplex bisections.
Then
v ∈ Bτs Lτ (Ω)
with s −
d
d
>1−
τ
2
implies the following decay rate for the best H01 -error:
`
E v, P
(MTn ,conf )
min(s,r +1)−1
d
= O n−
A. Veeser
FE tree approximation
.
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Generating a mesh with maximum strategy
Let t > 0 to be chosen later. Taking
2
(K ) = E P` (K ) = inf k∇(v − P)k2K ,
P∈P`
run
M := M0
while {K ∈ M | (K ) > t} =
6 ∅
pick K ∈ M with (K ) > t
M := rec-bisect(M, K )
Mt := M
Collect the picked elements in M?t ; we have to bound #M?t .
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
A bound for the local best error
Using the Sobolev embedding, we may assume s ≤ ` + 1 with
changing τ but without changing
d
d
δ := s − − 1 −
> 0.
τ
2
Using the Bramble-Hilbert lemma, we derive
inf k∇(v − P)kK 4
P∈P`
inf
Q∈[P`−1 ]d
k∇v − QkK
4 diam(K )δ |∇v |Bτs (Lτ (K ))
A. Veeser
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Counting requested bisections by generations – 1
Denote by g(K ) the generation of K ∈ T . Given j ∈ N0 ,
consider
Lj := {K ∈ M?t | g(K ) = j},
whose elements are disjoint.
Therefore, with δ̃ = δ/d > 0,
X
X τ
t τ /2 #Lj ≤
(K )τ /2 4 2−δ̃τ j
|v |Bτs (Lτ (K )) 4 2−δ̃τ j |v |τBτs (Lτ (Ω)) ,
K ∈Lj
K ∈Lj
ie
#Lj 4 |v |τBτs (Lτ (Ω)) t −τ /2 2−δ̃τ j .
A. Veeser
FE tree approximation
Measuring regularity with Besov spaces
Convergence rates
Convergence rates for tree approximation of gradients
Counting requested bisections by generations – 2
Independently of t, we have
2−j #Lj 4 |Ω|,
ie #Lj 4 |Ω| 2j .
In summary,
#M?t =
∞
X
#Lj 4
j=0
∞
X
n
o
min 2j , |v |τBτs (Lτ (Ω)) t −τ /2 2−δ̃τ j
j=0
τ
4 |v |τBτs (Lτ (Ω)) t −τ /2 2−δ̃τ k 4 |v |Bτs (Lτ (Ω)) t −1/2 1+δ̃τ
with k satisfying
2k ≈ |v |τBτs (Lτ (Ω)) t −τ /2 2−δ̃τ k
A. Veeser
ie
1
2k ≈ |v |τBτs (Lτ (Ω)) t −τ /2 1+δ̃τ
FE tree approximation
Convergence rates for tree approximation of gradients
Measuring regularity with Besov spaces
Convergence rates
Choosing t
Consequently, the above algorithm terminates and, in view of
the cost of maintaining conformity, we have
τ
#Mt − #M0 4 #M?t 4 |v |Bτs (Lτ (Ω)) t −1/2 1+δ̃τ
Choosing

1+δ̃τ 2
τ
2
 ,
t = C |v |Bτs (Lτ (Ω))
n

we have #Mt − #M0 ≤ n/2 and therefore
#Mt ≤ (#Mt − #M0 ) + #M0 ≤ n
for all n ≥ 2#M0 .
A. Veeser
FE tree approximation
Measuring regularity with Besov spaces
Convergence rates
Convergence rates for tree approximation of gradients
Error bound
The choice of t yields
1/2
E S0 (Mt ) 4 E P` (Mt ) 4 #Mt t 1/2
4 |v |Bτs (Lτ (Ω)) n
because
1
2
−
1
τ
− δ̃
4 |v |Bτs (Lτ (Ω)) n−
1 1
s−1
− − δ̃ = −
2 τ
d
thanks to
δ̃ =
δ
s
1 1 1
= − − + .
d
d
τ
d
2
A. Veeser
FE tree approximation
s−1
d