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
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
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Literature
A. Veeser, Approximating gradients with piecewise
polynomial functions, Found. Comp. Math. (2015).
For a "generalization", covering also the L2 -norm:
F. Tantardini, A. Veeser, R. Verfürth, Localization of the
best error with finite elements in the reaction-diffusion
norm, Constr. Approx. (2015).
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Outline
1
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
H01 -conformity and piecewise polynomials
Let M be any conforming triangulation of Ω ⊂ Rd and ` the
maximal polynomial degree. Recall
P` (M) := v | ∀ K ∈ M v|K ∈ P` (K )
and take
S0 (M) = P` (M) ∩ H01 (Ω)
For any v ∈ P` (M) holds
v ∈ H01 (Ω) ⇐⇒ v ∈ C 0 (Ω̄) and v|∂Ω = 0
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Global and local best errors
Write k·kD for k·kL2 (D) . For fixed v ∈ H01 (Ω), consider
E S0 (M) := inf {k∇(v − s)kΩ | s ∈ S0 (M)}
and, for each element K ∈ M,
n
o
E P` (K ) := inf k∇(v − P)kK | P ∈ P` (K )
!1/2
X
2
E P` (M) :=
E P` (K )
K ∈M
The latter do not take into account the ‘conformity’ constraints.
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Shape coefficient
For any K ∈ M, set
ρ(K ) := sup r > 0 | ∃x Br (x) ⊂ K
and write
diam(K )
,
ρ(K )
K ∈M
σ(M) := max
for the shape coefficient of M.
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Face-connecteness
Assume that z ∈ K ∩ F where
z is any node of S(M) := P` (M) ∩ H 1 (Ω),
K is any element of M,
F is any (d − 1)-face of M
there is a path K0 , . . . , Km such that
K0 = K ,
Km ⊂ F ,
Ki ∩ Ki−1 is a (d − 1)-face of M.
yes
no
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Localization of best H01 -error
There is a constant Cde depending on σ(M) and ` such that
E P` (M) ≤ E S0 (M) ≤ Cde E P` (M)
does not hold for L2 -norm
no apriori error estimate; no regularity involved
‘conformity’ constraints ok; beyond asymptotics
approximately knowing the error of the Ritz projection is
almost fully parallel
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Outline
1
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Explicit trace and Poincaré inequalities
Given an element K and a side F of K , there holds
|F |
2
2
2
kwkF ≤
kwkK + kwkK diam K k∇wkK
|K |
d
(cf. Ainsworth ’07, for generalizatons cf. Veeser/Verfürth ’09)
If
R
K
w = 0, then
kwkK ≤
1
diam K k∇wkK
π
(cf. Payne/Weinberger ’60, . . . )
A. Veeser
FE tree approximation
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Approximation of gradients
An interpolation operator
Let {φz }z∈N be the nodal basis functions of S(M) and define a
projection onto S(M) by
X
vz φz
Πu :=
z∈N
with
(
PK (z)
vz = R
∗
Fz φz v
if z interior K
otherwise
R
R
and PK ∈ P` (K ) st k∇(v − PK )k = E(P` (K )) and K PK = K v ,
Fz a face containing z and sharing its type,
{φ∗z }z as in Scott/Zhang ’90
A. Veeser
FE tree approximation
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Approximation of gradients
Sketch of proof – 1
Write
E S0 (M) ≤ k∇(v − Πv )kΩ
and
k∇(v − Πv )kK ≤ k∇(v − PK )kK + k∇(PK − Πv )kK
and observe
X Z
k∇(Πv − PK )kK ≤
z∈∂K
A. Veeser
Fz
φ∗z (v
− PK ) k∇φz kK
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Sketch of proof – 2
If Fz ⊂ K , there holds
Z
C
∗
φz (v − PK ) ≤ kφ∗z kFz kv − PK kFz ≤
kv − PK kFz
|Fz |1/2
Fz
and
kv − PK kFz ≤
1
1
+
π 2 dπ
1/2
|Fz |1/2
diam K k∇(v − PK )kK
|K |1/2
Otherwise use face-connectedness . . .
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Outline
1
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
complete
Let M0 be an admissible (in 2d: with coinciding edge labels)
initial mesh of Ω and let T be the corresponding forest of
master trees.
Denote by MT the set of all meshes that can be generated from
M0 by simplex bisection and by MT ,conf its subfamily of
conforming (face-to-face) meshes.
Given any (possibly non-conforming) mesh M0 ∈ MT , denote
by complete(M0 ) the smallest conforming refinement of M0 .
Thanks to third part and Remark 7.1 of Binev/DeVore ’04, we
have
#complete(M0 ) − #M0 4 #M0 − #M0 .
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Mesh conformity in H01 -approximation
Denote by MTn and MTn ,conf the respective subfamilies with at
most n simplex bisection.
Combining the localization of the best H01 -error with the
inequality for complete gives:
There are c2 > 0 depending on M0 and ` such that
E S0 (MTn ,conf ) ≤ Cde E P` (MTc2 n )
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Setting for tree approximation
Let M0 be an initial mesh of Ω with coinciding edge labels and
T the corresponding forest of master trees.
For any K ∈ T , we set
2
(K ) = E v , P` (K ) =
inf
P∈P` (K )
k∇(v − P)k2K
and, for any M ∈ MT ,conf that is conforming, we have
X
2
E(M) =
(K ) ≈ E v , S0 (M) .
K ∈M
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Thresholding algorithm
M0t := ∅
for all K ∈ M0
if ẽ(K ) > t then grow(K )
Mt :=complete(M0t )
where t > 0 is given and grow(K ) is
(K1 , K2 ) := bisect(K )
for i = 1, 2
if ẽ(Ki ) > t then
grow(Ki )
else
M0t := M0t ∪ {Ki }
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Instance optimality
If #Mt ≥ 3#M0 , then
E S0 (Mt ) ≤ 2Cde E P` (MTc2 #Mt )
for some c2 > 0 depending on M0 and `.
Comparing with Céa Lemma, the approximant here is
determined without PDE,
near best in a nonlinear approximation space that, for
given n, is much bigger
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Applications
The above thresholding algorithm or variants may be used
to create benchmark for the adaptive solution of PDEs
to coarsen in an adaptive algorithm of the form
error reduction → sparsity adjustment
to approximate data in the PDE, e.g., in connection with
−∆ with
(K ) = diam(K )2 kf k2K
(but . . . )
to coarsen in the adaptive solution of evolutionary PDEs
(rather reaction-diffusion or L2 -norm)
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Outline
1
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Gradient conformity and best errors
Note
{∇P | P ∈ P` } = Q ∈ (P`−1 )d | ∂i Qj = ∂j Qi , i, j = 1, . . . , d
Recalling
E P` (K ) = inf k∇(v − P)kK | P ∈ P` (K ) ,
consider
E ∇v , P`−1 (K )d
L2
n
o
:= inf k∇v − QkK | Q ∈ P`−1 (K )d .
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
Equivalence . . .
There holds
E ∇v , P`−1 (K )d
L2
≤ E P` (K ) ≤ Cde E ∇v , P`−1 (K )d L2
where Cde depends only on ` and d
coupling of partial derivatives ok
approximately knowing the error of an
H 1 -seminorm-projection is almost fully parallel in terms of
the error of the L2 -projections
A. Veeser
FE tree approximation
Approximation of gradients
Localization of the best H01 -error
About the proof of localization
Tree approximation for gradients
Approximation and gradient conformity
. . . and about its proof
Denoting by I` v the averaged Taylor polynomial of Scott/Dupont
’80,
`
E P (K )
2
≤
=
d
X
i=1
d
X
k∂i (v −
I` v )k2K
=
d
X
k∂i v − I`−1 (∂i v )k2K
i=1
k(∂i v − Qi ) − I`−1 (∂i v − Qi )k2K
i=1
≤ Cde k∇v − Qk2K
Independence on element shape by using an idea of
Dekel/Leviatan ’04.
A. Veeser
FE tree approximation