THE BUBBLE TRANSFORM, PART II
Abstract. The purpose of this note is to discuss a generalization of the bubble transform to differential forms. This transform was dicussed for ordinary
functions, or zero-forms, in [4]. The present note represents the beginnig of
a similar study for differential forms, where we build a transformation which
decomposes any function in HΛk (Ω) into a sum of local bubbles supported on
appropriate macroelements with respect to a given simplicial triangulation T
of the domain Ω. A key points are that the transform in bounded in HΛk (Ω),
and it should commute with the exterior derivative. Furthermore, the transform should preserve the piecewise polynomial structure of the standard finite
element spaces of k-forms. The manuscript [4] gives a detailed construction
and a rigorous analysis of the situation for zero-forms. In contrast, the present
note is more restricted. However, when Ω ⊂ R2 the construction of the transform given below is close to complete.
1. Introduction
The bubble transform for scalar functions, or zero forms, was presented in [4]. In
this note we will generalize this construction to the spaces HΛk (Ω), i.e., to spaces
of differential forms. Basically, we will adopt the notation used in that paper and in
[3]. Our goal is to extend the construction of the bubble transform to the complete
de Rham complex as presented in [1, 2]. More precisely, for all integers k ranging
from 0 to n we will construct a map
M
H̊Λk (Ωf )
B k = BTk : HΛk (Ω) →
f ∈∆(T )
dim f ≥k
P
such that u = f Bfk u, Bfk u ∈ H̊Λk (Ωf ). In addition, B k is a local operator, B k is
bounded as an operator from H̊Λk (Ω) to itself, and it preserves the piecewise polynomial spaces Pr Λk (T ) and Pr− Λk (T ). Furthermore, an additional key property is
that the transform should commute with the exterior derivative.
2. The bubble transform for zero-forms
We will first review the construction for zero forms given in [4]. Given a triangulation T of Ω, we define the map B 0 = BT0 by a recursion with respect to the
dimension of subsimplexes f ∈ ∆(T ). The map BT0 can be defined on the space
Notes by Ragnar Winther, February, 2013.
1
2
THE BUBBLE TRANSFORM
L2 (Ω). Assume that u ∈ L2 (Ω), and that we have defined Bg0 u for all g ∈ ∆j (T ),
0
j < m. We then define um = u − Bm−1
u, where
X
0
Bg0 u.
(2.1)
Bm−1
u=
g∈∆j (T )
j<m
For each f ∈ ∆m (T ) and m < n we define
0
Bf0 u = (Km
◦ A0f )um (λf (·)),
(2.2)
0
where the two operators A0f and Km
will be defined below, and where λf ∈ Rm are
the barycentric coordinates associated to f . Hence, if we regard λf as a map from
Ωf to Sm , then Bf0 u can be rewritten as
0
Bf0 u = (λ∗f ◦ Km
◦ A0f )um .
For f ∈ ∆n (T ) = T , we simply define Bf0 u = un |f .
For f = [x0 , x1 , . . . xm ] ∈ ∆m (T ) the average operator A0f : L2 Λ0 (Ωf ) →
L2 Λ0 (Sm ) is given by
Z
m
X
0
λj (xj − y)) dy.
Af v (λ) = − v(y +
Ωf
j=0
Note that if λ ∈ Sm (1) then
Pthe integrand is independent of y, and therefore
A0f v(λ) = v(x), where x =
j λj xj ∈ f . In fact, this average operator has a
natural analog for k-forms which we will introduce below.
0
, mapping the space of functions defined on Sm to itself, is
The operator Km
defined by
X
1 − |λ|
0
(2.3)
Km
w(λ) =
(−1)|I|
w(PI λ),
1 − |PI λ|
I∈Im
where we recall that Im are the set of increasing subsets of {0, 1, . . . , m}, and |I
denotes the cardinality of I. The key property is that Km preserves the trace
0
w on the rest of the boundary of Sm is zero.
on Sm (1), while the trace of Km
0
In particular, if w = (1 − |λ|) then Km
w = 0. Hence, linear functions which
0
0
0
vanishP
on Sm (1) are in the kernel of Km
. The transform
P 0 B is given 0as B =
0
0
Bn = f ∈∆(T ) Bf . By construction, we have u = f Bf u and supp Bf u ∈ Ωf .
Furthermore, it is established in [4] that Bf0 u ∈ H̊ 1 (Ωf ).
3. The construction for k-forms
The construction of the transform B k , for 1 ≤ k ≤ n, will mainly follow the
k
pattern above. We will define Bm
for k ≤ m ≤ n on the form
X
k
Bm
v=
Bgk v.
g∈∆j (T )
j≤m
k
Bnk .
In particular, B =
For dim f = m, where k < m < n, we will define Bfk on
the form
k
k
Bfk v = (λ∗f ◦ Km
◦ Akf )vm , vm = v − Bm−1
v.
THE BUBBLE TRANSFORM
3
k
0
Here the operators Akf and Km
are generalizations of the operators A0f and Km
introduced above. For dim f = k we have to use a special definition of Bfk , while
for f ∈ ∆n (T ), and k < n, we take Bfk v = vn |f .
3.1. The operator Akf . We start by generalizing the average operator A0f to kforms. So let f = [x0 , x1 , . . . , xm ] ∈ ∆m (T ) be fixed, and consider the map Gm :
Sm × Ωf → Ωf given by
Gm (λ, y) = y +
m
X
λj (xj − y) =
j=0
m
X
λj xj + (1 − |λ|)y,
λ ∈ Sm , y ∈ Ωf .
j=0
The operator Akf is defined by
Akf v
Z
= − Gm (·, y)∗ v dy,
Ωf
and maps k-forms defined on Ωf to k-forms defined on Sm for f ∈ ∆m (T ). Note
that since pull-backs commute with the exterior derivative, so do the operators
Akf . Furthermore, the operator Akf preserves the finite element spaces Pr Λk (T )
and Pr− Λk (T ). In fact, we have the following result which shows that the operator
Akf maps piecewise polynomial forms to polynomial forms.
Lemma 3.1. Let f ∈ ∆m (T ). Then the operator Akf maps Pr Λk (Tf ) to Pr Λk (Sm ),
and Pr− Λk (Tf ) to Pr− Λk (Sm )
Proof. Assume that v ∈ Pr Λk (Tf ). Fix an element y ∈ T ∈ Tf . Then all the points
Gm (λ, y) belongs to same simplex T of Tf . Therefore, since Gm (λ, y) is linear in λ,
λ ∈ Sm , it follows that Gm (·, y)∗ v ∈ Pr Λ(Sm ). By taking the average we conclude
that
Z
Akf v = − Gm (·, y) dy ∈ Pr Λ(Sm ).
Ωf
This establishes the first part of the lemma.
To show the second part we assume that v ∈ Pr− Λk (Tf ). To show that Akf v ∈
Pr− Λk (Sm ) it is enough to show that Gm (·, y)∗ v ∈ Pr− Λk (Sm ) for all fixed y ∈ Ωf .
Furthermore, Gm (·, y)∗ v ∈ Pr− Λk (Sm ) if and only if (Gm (·, y)∗ v)yλ ∈ Pr Λk−1 (Sm ).
However, the identity
DGm (λ, y)λ = Gm (λ, y) − y,
where DGm ∈ Rn×m is derivative of Gm with respect to λ, implies that
(3.1)
(Gm (·, y)∗ v)yλ = Gm (·, y)∗ (vy(x − y)).
Furthermore, since v ∈ Pr− Λk (Tf ) we have that vy(x−y) ∈ Pr Λk (Tf ) for all y ∈ Ωf .
By the first opart of the proof we therefore can conclude that Gm (·, y)∗ (vy(x−y)) ∈
Pr Λk−1 (Sm ). By (3.1) we further obtain that (Gm (·, y)∗ v)yλ ∈ Pr Λk−1 (Sm ) for
each y, and this completes the proof.
4
THE BUBBLE TRANSFORM
3.2. Compute dB 0 . To derive how to define the bubble transform for k > 0,
such that the transform commutes with exterior derivative, we will first compute
dB00 . This term is very special part of dB 0 since the transform B 1 for one-forms
are only allowed to utilize components with support in macroelements Ωf , f ∈
∆(T ) for dim f ≥ 1. Therefore, to obtain a commuting transform, not only do we
need express dB00 u in terms of du, but we also need to be able to decompose this
expression as a sum of components with support in Ωf , f ∈ ∆(T ) with dim f ≥ 1.
In fact, we will show that dB00 u can be decomposed as a sum of components with
support in Ωf , f ∈ ∆1 (T ).
We recall that
B00 u =
X
Bg0 u =
g∈∆0 (T )
X
(λ∗g ◦ K00 ◦ A0g )u.
g∈∆0 (T )
The operator K00 maps functions w = w(λ), defined on S0 = [0, 1], to
K00 w(λ) = w(λ) − (1 − λ)w(0).
Hence, dB00 u can be written as a sum of two terms
X
X
(3.2)
dB00 u =
d(λ∗g ◦ A0g )u +
(A0g u)(0)dλg .
g∈∆0 (T )
g∈∆0 (T )
We now show how each of the two terms can be expressed in terms of du. For the
first term, fix g ∈ ∆0 (T ) and consider d(λ∗g ◦ A0g )u. It folloows from the commuting
property of pull-backs, and the corresponding property of the average operators
that
d(λ∗g ◦ A0g )u = (λ∗g ◦ A1g )du.
Hence, if x ∈ Ωg and t is a vector in Rn (the tangent space of Ωg ) then it follows
from the commuting property of the operators Akf and the definition of pull-backs,
that
(d(λ∗g ◦ A0g )u)x (t) = (A1g du)λg (x) dλg (t).
Note that in the present case A1g du is a one form on S0 = [0, 1]. The one-form dλg
has support in Ωg . Furthermore,
X
(3.3)
dλg =
Ef1 volf .
x1 ∈∆0 (T )
f =[x1 ,g]∈∆1 (T )
Here, Ef1 volf are the Whitney 1-forms associeted f ∈ ∆1 (T ), scaled such that
R
E volf = 1 (cf. [3, Section 2]). Furthermore, Ef volf is has local support on the
f f
macroelement Ωf . We therefore can conclude that
X
X
X
d(λ∗g ◦ A0g )u =
(A1g (du))λg (·) Ef1 volf
g∈∆0 (T )
g∈∆0 (T )
f ∈∆1 (T )
f =[g,x1 ]
(3.4)
=
X
X
f ∈∆1 (T )
g∈∆0 (f )
sign(g, f )(A1g du)λg (·) Ef1 volf .
Here, sign(g, f ) = 1 if f is of the form [x1 , g] and sign(g, f ) = −1 if f is of the form
[g, x1 ].
THE BUBBLE TRANSFORM
5
Next, we need to show that the second term of the expression for dB00 u given in
(3.2) can be expressed in terms of du. If we adopt the notation of [3, Section 3]
this term can be written as
X Z
(
u ∧ volΩg )dEg0 volg .
g∈∆0 (T )
Ωg
However, using the double complex structure, it was established in Lemma 3.3 of
that paper that this term can be rewritten as
X Z
X Z
0
(3.5)
(
u ∧ volΩg )dEg volg =
(
du ∧ zf1 )Ef1 volf ,
g∈∆0 (T )
Ωg
f ∈∆1 (T )
Ωef
where zf1 ∈ P̊1− (Tfe ) satisfies
dzf1 = volΩg1 − volΩg0 ,
f = [g0 , g1 ].
Furthermore, Ωef is the extended macroelement associated f ∈ ∆1 (T ), i.e., Ωef =
Ωg0 ∩ Ωg1 , and Tfe is the restriction of T to Ωef .
To sum up the discussion above, we can conclude from (3.2), (3.4), and (3.5)
that
Z
X
dB00 u =
d(λ∗g ◦ A0g )u + (
u ∧ volΩg )dEg0 volg
Ωg
g∈∆0 (T )
(3.6)
=
Z
(
X
f ∈∆1 (T )
=
X
Ωef
du ∧ zf1 ) +
1
Bf,0
du
X
sign(g, f )(A1g du)λg (·) Ef1 volf
g∈∆0 (f )
=
B01 du,
f ∈∆1 (T )
1
where Bf,0
and B01 are implicitly defined from this equation, i.e.,
Z
X
1
Bf,0
v= (
v∧zf1 )+
sign(g, f )(A1g v)λg (·) Ef1 volf and B01 v =
Ωef
X
1
Bf,0
v.
f ∈∆1 (T )
g∈∆0 (f )
Here v is a 1-form.
3.3. The structure of the operator B11 . The operator B01 has the property that
B01 v can be decomposed as a sum of 1-forms with support in Ωf , f ∈ ∆1 (T ).
Furthermore, it satisfies the commuting relation (3.6). The next step in our construction is to add more terms to B01 , to obtain an operator B11 , which still can be
decomposed as a sum of 1-forms with support in Ωf , f ∈ ∆1 (T ), but where we also
have dB10 = B11 du. In addition, v − B11 v should have zero trace on all f ∈ ∆1 (T ),
at least if v is a smooth 1-form.
The operator B11 will be of the form
X
B11 v = B01 v +
(λ∗f ◦ K11 ◦ A1f )v1 ,
v1 = v − B01 v.
f ∈∆1 (T )
K11
Here
is a trace preserving cut-off operator mapping the set of 1-forms defined
on S1 ⊂ R2 to itself. So to complete the definition of B11 it only remains to specify
6
THE BUBBLE TRANSFORM
K11 . Note that the operator K10 is a corresponding map on zero forms defined on
S1 . In fact, if K11 is constructed such that the commuting relation
dK10 w = K11 dw
(3.7)
holds, then we will also obtain the desired relaton
dB10 u = B11 du.
(3.8)
To see this we just use (3.6) and (3.7) to obtain
X
dB10 u = dB00 u + d
(λ∗f ◦ K10 ◦ A0f )(u − B00 u)
f ∈∆1 (T )
=
B01 du
X
+
(λ∗f ◦ K11 ◦ A1f )(du − B01 du)
f ∈∆1 (T )
= B11 du.
Note also, that the operator K10 is only applied to functions satisfying w(1, 0) =
w(0, 1) = 0 in a weak sense, cf. [4, Section 4]. Therefore, the real challenge is to
construct the proper operator K11 satisfying (3.7) for functions w satisfying these
constraints.
3.4. Compute dK10 . If w = w(λ) = w(λ0 , λ1 ) is a zero form defined on S1 , then
K10 w is given by
(K10 w)(λ) = w(λ) −
1 − |λ|
1 − |λ|
w(0, λ1 ) −
w(0, λ1 ) + (1 − |λ|)w(0),
1 − λ1
1 − λ1
cf. (2.3). Furthermore, if w = (1 − |λ|) then K10 w = 0. Hence, for any w, which is
smooth away from Sm (1), we have that
K10 w = K10 (w − Π0 w).
Here Π0 w denotes the interpolant of w into the space of linear functions on S1 ,
which are zero on S1 (1), i.e., Π0 w = (1 − |λ|)w(0). Hence, it is enough to consider
functions w such that w(0) = 0. For such w we now have
dK10 w = dw − d
1 − |λ| ∗ 1 − |λ| ∗ P0 w − d
P w ,
1 − λ1
1 − λ1 1
where the maps P0 and P1 are defined by
P0 (λ0 , λ1 ) = (0, λ1 )
and P1 (λ0 , λ1 ) = (λ0 , 0).
Introduce the Poincaré operator R1 , mapping one forms to zero forms, by
Z 1
(R1 v)λ =
(vtλ yλ) dt,
0
1
such that R dw = w − w(0) = w if w(0) = 0. Hence, the expression for dK10 w, for
w(0) = 0, can be rewritten as
dK10 w = dw − d
1 − |λ| ∗ 1 1 − |λ| ∗ 1 P0 R dw − d
P R dw .
1 − λ1
1 − λ1 1
THE BUBBLE TRANSFORM
7
3.5. The operator K11 . For any one form v, satisfying R1 v = 0 at the points (1, 0)
and (0, 1), we now define
1 − |λ| ∗ 1 1 − |λ| ∗ 1 P0 R v − d
P R v
1 − λ1
1 − λ1 1
1 − |λ| ∗
1 − |λ| ∗
P v−
P v
=v−
1 − λ1 0
1 − λ1 1
1 − |λ|
1 − |λ|
− d(
) ∧ P0∗ R1 v − d(
) ∧ P1∗ R1 v,
1 − λ1
1 − λ1
K11 v = v − d
(3.9)
where the last identity follows since Pi∗ dR1 v = Pi∗ v. If w is a zero form on S1 , which
vanish at the three corners of S1 , then by construction we have dK10 w = K11 dw.
For a more general one form we define
(3.10)
K11 v = K11 (v − Π1 v),
where Π1 is the canonical interpolant onto the space of Whitney one-forms on S1 ,
having vanishing trace on S1 (1). In fact, this is a two dimensional space, spanned
by (1 − λ0 )dλ1 + λ1 dλ0 and (1 − λ1 )dλ0 + λ0 dλ1 . Note that by construction,
R1 (v − Π1 v) = 0 at the points (1, 0) and (0, 1), such that the right hand side of
(3.10) is already defined. For all zero forms w, satisfying w(0, 1) = w(1, 0) = 0, we
will have dΠ0 w = Π1 dw. As a further consequence, we obtain
dK10 w = dK10 (w − Π0 w) = K11 (dw − dΠ0 w) = K11 (dw − Π1 dw) = K11 dw
for such functions w, which is (3.7).
3.6. K11 is a trace preseving cut-off. We also want show that K11 is a trace
preserving cut-off operator in the sence that it preserves the trace of S1 (1), while
the trace of K1 v is zero on the rest of the boundary of S1 . To see this we just
use the second expression for K11 v given in (3.9) as a sum of five terms. It easy
to see that all the terms except the first have zero trace on S1 (1). So K11 is trace
preserving. To see that the trace is zero on the rest of the boundary, consider the
subsimplex λ0 = 0 of S1 . Clearly, the trace of the two first terms cancels. The third
and fifth term vanish since trλ0 =0 P1∗ = 0 by orthogonality. Finally, the fourth term
vanish since (1 − |λ|)/(1 − λ1 ) is constant on λ0 = 0.
3.7. K11 is polynomial preserving. The final property we will establish for the
operator K11 is that it is polynomial preserving. More presicely, we will have
K11 (Pr Λ1 (S1 )) ⊂ Pr Λ1 (S1 )
and K11 (Pr− Λ1 (S1 )) ⊂ Pr− Λ1 (S1 ).
To show this assume first that v ∈ Pr Λ1 (S1 ). As a consequence of (3.10) we cam
also assume that R1 v is equal zero at the points (1, 0) and (0, 1). Consider now the
second term in the first expression for K11 v given in (3.9), i.e.,
d
1 − |λ| ∗ 1 P R v .
1 − λ1 0
We note that trλ0 =0 R1 v = (1 − λ1 )w(λ1 ), where w ∈ Pr . Hence,
1 − |λ| ∗ 1
P R v ∈ Pr+1 (S1 ),
1 − λ1 0
8
THE BUBBLE TRANSFORM
and therefore
1 − |λ| ∗ 1 P R v ∈ Pr Λ1 (S1 ).
1 − λ1 0
A slight modification of the argument shows that if v ∈ Pr− Λ1 (S1 ) then
1 − |λ| ∗ 1 d
P R v ∈ Pr−1 Λ1 (S1 ) ⊂ Pr− Λ1 (S1 ).
1 − λ1 0
Just observe that in this case trλ0 =0 R1 v ∈ Pr instead of in Pr+1 , and argue as
above. Of course, the third term can be handled in a similar manner.
d
4. The two dimensional case
The analysis is above is not complete since we only handle the case k = 1,
in addition to the case k = 0 which is already studied in [4]. Furthermore, for
one-forms we only define the contribution to the transform B 1 given by B11 , i.e.,
the contribution with support in the macroelements associated elements of ∆1 (T ).
However, if we stick to the two dimensional case, i.e. n = 2, this contains nearly
all the information we need to construct B 1 . We simply take
X
X
B1 v =
Bf1 v = B11 v +
(v − B11 v)|T .
f ∈∆j (T )
T ∈T
1≤j≤2
A key property here is that trf (v − B11 v) = 0 for all f ∈ ∆1 (T ), and in the smooth
case this follows from the similar property of K11 v derived in Section 3.6. However,
to make a more rigorous proof, we have to utilize weaker concepts of vanishing trace,
similar to the concepts used in [4, Section 4-5]. The definition of the transform B 2
in the two dimensional case is more straightforward, since the space HΛ2 is simply
equal to the space of two-forms in L2 . However, still we have to define this transform
such that it commutes properly with d. We drop further details here. In fact, this
activity represents ongoing research in the spring of 2013, partly joint with R. Falk
and S. Christiansen.
References
1. D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica 15 (2006), 1–155.
2. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus:
from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2,
281–354. MR 2594630 (2011f:58005)
3. R.S. Falk and R. Winther, Local bounded cochain projections, arXiv:1211.5893v1 (2012).
4. R. Winther, The bubble transform, part I, 2012.