Mesh refinement with tree structure for PDEs
Adaptive tree approximation
with finite elements
Andreas Veeser
Università degli Studi di Milano (Italy)
July 2015 / Cimpa School / Mumbai
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
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
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Literature
Two-dimensional case: Section 1.3 and Chapter 6 in
Nochetto/Veeser, Primer of Adaptive FEM, in: Lecture
Notes in Math 2040, Springer 2012
Multidimensional case: Chapter 4 in
Nochetto/Siebert/Veeser, Theory of Adaptive FEM: An
introduction, in: Multiscale, Nonlinear and Adaptive
Approximation, DeVore/Kunoth (Eds.), Springer, 2009
Here we will consider only the 2d case.
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Trianglar mesh refinement and PDEs
Assume we discretize a PDE with a triangular mesh. Then the
following properties are often exploited:
Shape regularity: The minimum angle of all triangles is
uniformly bounded away from zero.
Conformity (edge-to-edge): Triangles meet only in vertices
or edges.
Notice:
Shape regularity puts constraints on iterated subdivisions
of triangles.
Conformity or limited non-conformity may propagate
refinement (!).
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Outline
1
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Definition
Given a triangle with a labeling (0, 1, 1) of its edges, we
prescribe
where the new vertex is the midpoint of the edge i.
This associates a tree with any triangle of an initial triangulation
M0 .
The generation g(K ) of a triangle K is its lowest edge label.
Incrementing the generation halves the area.
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Shape regularity, diameter and generation
The minimum angles of the triangles in a tree are bounded
away from 0 in terms of the root triangle:
Thus, we have diam K ≈ |K |1/2 for all triangle in a tree, whence
diam K ≈ 2−g(K )/2 .
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
A few bisections in a triangulation . . .
The labeled bisections
correspond to . . .
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
. . . and its trees
to the trees (or forest)
where
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Outline
1
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Refinement propagation due to conformity – 1
The bisection of one triangle may require the bisection of a
triangle with lower generation:
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Refinement propagation due to conformity – 2
. . . with arbitrary (!) lower generation (longest edge has always
lowest label):
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Recursive bisection – 1
Let K ∈ M be a triangle of a conforming mesh and denote by
E its edge with the lowest label.
If E is a boundary edge, then we set
FM (K ) := ∅,
else there exists a unique K 0 ∈ M with E = K ∩ K 0 and we set
FM (K ) := K 0 .
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Recursive bisection – 2
To construct the smallest conforming refinement of M in which
K is bisected, we consider
rec-bisect(M, K )
if FM (K ) = ∅ then
M =bisect(M, K )
else
2 (K ) 6= K then
if FM
rec-bisect(FM (K )) % FM (K ) changes
M =bisect(M, K , FM (K ))
return(M)
Does it terminate and is the output conforming?
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
A sufficient condition for termination, . . .
If M is conforming and
each interelement edge receives coinciding labels,
then rec-bisect(M, K ) has at most g(K ) recursive calls.
In fact, the edge labeling of K is (i, i + 1, i + 1), where
E = K ∩ FM (K ) has label i also from FM (K ). Another call of
rec-bisect with FM (K ) happens only if M is not changed
and FM (K ) has the edge labeling (i − 1, i, i). therefore
g(FM (K )) = g(K ) − 1.
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
for conservation of uniqueness of edge labeling, . . .
In the aforementioned case all intermediate meshes have a
unique edge labeling
because, apart from the case FM (K ) = ∅, only compatible
bisections
are performed.
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
and for conforming output
All intermediate mesh and the final one are conforming.
It remains to consider the situation after the return of a
recursive call. Since rec-bisect(M, K ) bisects K before
returning, we obtain
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Assumption for initial labeling
Let M0 be a conforming initial triangulation and assume that
the edge labeling (0, 1, 1) for any triangle is such that
each interelement edge of M0 has coinciding labels.
Then iterative applications of rec-bisect are well-defined.
Non-constructive existence by Peterson’s theorem about
matching in cubic graphs. In alternative,
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Two useful properties of rec-bisect
Let M be some mesh arising from a suitable M0 and iterative
applications of rec-bisect. If
K ∈ M and K 0 ∈ rec-bisect(M, K ) \ M,
then
g(K 0 ) ≤ g(K ) + 1
and
dist(K 0 , K ) ≤ Cdist 2−g(K
A. Veeser
0 )/2
FE tree approximation
Single bisections
Maintaining conformity
Cost of conformity
Mesh refinement with tree structure for PDEs
Proof of dist bound
Let K0 = K , K1 , . . . , Km be the triangles in the recursive calls of
rec-bisect and take j such that K 0 ⊂ Kj . Wlog j > 1. Then
0
dist(K , K ) ≤ diam(Kj−1 ) + dist(Kj−1 , K0 ) ≤
j−1
X
diam(Kk )
k =0
4
j−1
X
−g(Kk )/2
2
−g(Kj−1 )/2
=2
k =0
j−1
X
2−k /2 ≈ 2−g(K
k =0
A. Veeser
FE tree approximation
0 )/2
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
refine
Given an ‘admissible’ initial mesh M0 , let Mbisect (M0 ) denote
the set of conforming, shape-regular meshes that can be
generated by successive applications rec-bisect, starting
from M0 .
Given M ∈ Mbisect (M0 ) and a set M? ⊂ M of triangles to be
bisected, we define
refine(M, M? )
while M ∩ M? 6= ∅
pick K ∈ M ∩ M?
M = rec-bisect(M, K )
return(M)
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Outline
1
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Bound by total number of requested bisections
Let M0 be an initial triangulation with coinciding edge labels.
Moreover, given k ∈ N, let M?0 , . . . , M?k −1 and M1 , . . . , Mk
such that M?j ⊂ Mj and
Mj = refine(Mj−1 , M?j−1 )
for j = 0, . . . , k − 1.
Then
#Mk − #M0 4
kX
−1
#M?j
j=0
A. Veeser
FE tree approximation
Single bisections
Maintaining conformity
Cost of conformity
Mesh refinement with tree structure for PDEs
Approach
−1
Write R := Mk \ M0 and M?∪ := ∪kj=0
M?j and construct a
function
λ : R × M?∪ → R+
0
such that
X
λ(K , K ? ) ≥ C1
X
and
K ? ∈M?∪
λ(K , K ? ) ≤ C2
K ∈R
Then
C1 #R ≤
X
X
λ(K , K ? ) =
K ∈R K ? ∈M?∪
X
X
λ(K , K ? ) ≤ C2 #M?∪ .
K ? ∈M?∪ K ∈R
A. Veeser
FE tree approximation
Single bisections
Maintaining conformity
Cost of conformity
Mesh refinement with tree structure for PDEs
Allocation function
Let a(`)
`≥−1
⊂ R+ be such that
P
`≥−1 a(`)
=A<∞
and B, C 0 > 0 constants to be chosen later.
Given K ∈ R and K ? ∈ M?∪ , we set
λ(K , K ? ) = a g(K ? ) − g(K )
whenever g(K ) ≤ g(K ? ) + 1 and dist(K , K ? ) ≤ BC 0 2−g(K )/2 , as
well as
λ(K , K ? ) = 0
otherwise.
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Upper bound
Fix K ? ∈ M?∪ and, for 0 ≤ g ≤ g(K ? ) + 1, consider
M(g) := {K ∈ R | dist(K , K ? ) ≤ BC 0 2−g(K )/2 and g(K ) = g}
Since all these triangles lie in a ball centered in the midpoint of
K ? and with radius 4 2−g , we have #M(g) 4 1
Therefore
g(K ? )+1
X
K ∈R
λ(K , K ? ) 4
X
a(g(K ? ) − g) 4
g=0
A. Veeser
X
`≥−1
FE tree approximation
a(`) 4 A.
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Lower bound – going back some way
Fix K0 ∈ R and define K0 , . . . , KJ with J ≤ k − 1 by
Kj arises from rec-bisect(·, Kj+1 ) and Kj+1 ∈ M?∪ .
We have
g(KJ ) = 0
g(Kj+1 ) ≥ g(Kj ) − 1
Let s be the smallest value such that g(Ks ) = g(K0 ) − 1. Then
K1 , . . . , Ks are candidates for contributing to the lower bound:
g(K0 ) ≤ g(Kj ) + 1
and Kj ∈ M?∪
A. Veeser
(j = 1, . . . , s).
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Lower bound – definition of C 0
Fix any j ∈ {1, . . . , s}. Similar to before, we derive
dist(K0 , Kj ) ≤
j
X
dist(Ki−1 , Ki )
i=1
j−1
X
0
−g(Ki )/2
≤C
2
+
j−1
X
i=1
,
i=0
which defines C 0 .
A. Veeser
FE tree approximation
diam(Ki )
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Lower bound – m(`, j)
To estimate further, we introduce, for ` ≥ 0 and j = 1, . . . , s,
m(`, j) := # K ∈ {K0 , . . . , Kj−1 } | g(K ) = g(K0 ) + `
Observe that m(`, ·) is increasing and
(
1, if ` = 0,
m(`, 1) =
0, if ` > 0.
Then
0 −g(K0 )/2
dist(K0 , Kj ) ≤ C 2
∞
X
m(`, j)2−`/2
`=0
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Lower bound – definition of B
In order to compare with m(`, j), we choose b(`) `≥0 ⊂ R+
such that
∞
X
B :=
b(`)2−`/2 < ∞
`=0
and
inf b(`) ≥ 1,
`≥0
inf a(`)b(`) ≥ cab > 0
`≥1
e.g., b(`) = 2`/3 if a(`) = (` + 2)−2 .
According to the relationship of m and b, we consider several
cases.
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Lower bound – case m(·, s) ≤ b
If
m(`, s) ≤ b(`)
for all ` ≥ 0,
then
dist(K0 , Ks ) ≤ BC 0 2−g(K0 )/2
and so
X
λ(K0 , K ? ) ≥ λ(K0 , Ks ) = a(−1) > 0
K ? ∈M?∪
A. Veeser
FE tree approximation
Mesh refinement with tree structure for PDEs
Single bisections
Maintaining conformity
Cost of conformity
Lower bound – case m(·, s) 6≤ b
Assume there is ` ≥ 0 with m(`, s) > b(`). Choose `∗ ≥ 0 and
j ∗ ∈ {1, . . . , s} such that
m(`∗ , j ∗ ) > b(`∗ )
and
∀`, j
m(`, j) > b(`) =⇒ j ≥ j ∗ .
Since m(`∗ , 1) ≤ 1 and b(`∗ ) ≥ 1, we have j ∗ ≥ 2. Hence, for all
i = 1, . . . , j ∗ − 1, we have m(·, i) ≤ b and so
dist(K0 , Ki ) ≤ BC 0 2−g(K0 )/2 .
Therefore
X
λ(K0 , K ? ) ≥ m(`∗ , j ∗ )a(`∗ ) ≥ b(`∗ )a(`∗ ) ≥ cab > 0.
K ? ∈M?∪
A. Veeser
FE tree approximation