Table Of ContentDISCRETE COMPACTNESS FOR P-VERSION OF TETRAHEDRAL
EDGE ELEMENTS
R. HIPTMAIR∗
Report 2008-31,Seminar for Applied Mathematics, ETH Zurich
Abstract. We consider the first family of H(curl,Ω)-conforming Ned´el´ec finite elements on
tetrahedral meshes. Spectral approximation (p-version) is achieved by keeping the mesh fixed and
9 raisingthepolynomialdegreepuniformlyinallmeshcells.Weprovethattheassociatedsubspacesof
0 discretelyweaklydivergencefreepiecewisepolynomialvectorfieldsenjoyalongconjectureddiscrete
0 compactness property as p → ∞. This permits us to conclude asymptotic spectral correctness of
2 spectralGalerkinfiniteelementapproximationsofMaxwelleigenvalueproblems.
n
a Keywords.Edgeelements,Maxwelleigenvalueproblem,discretecompactness,Poincar´elifting,
J projectionbasedinterpolation
7
AMS subject classifications. 65N30, 65N25,78M10
]
A 1. Introduction. Identifying spectrally correct conforming Galerkin approxi-
N mations of the Maxwell eigenvalue problem [15]: seek1 u H(curl,Ω) and ω > 0,
∈
such that
.
h
at µ−1curlu,curlv L2(Ω) =ω2(ǫu,v)L2(Ω) ∀v∈H(curl,Ω), (1.1)
m
(cid:0) (cid:1)
[ (ǫ,µ (L∞(Ω))3,3 uniformly positive definite material tensors) has turned out to
be a h∈ighly inspiring challenge in numerical analysis. Obviously, eigenfunctions of
v1 (1.1) belong to H(curl,Ω)∩H0(divǫ0,Ω) and the compact embedding L2(Ω) ֒→
1 H(curl,Ω) H(divǫ0,Ω) [30] relates (1.1) to an eigenvalue problem for a com-
∩
6 pact selfadjoint operator. However, asymptotically dense families of finite elements
7 in H(curl,Ω) H(divǫ0,Ω) are not known in general.
0 Letusassu∩methatamerelyH(curl,Ω)-conformingfamily 1( ) offinite
. Wp M p∈N
1 dimensional trial and test spaces 1( ) H(curl,Ω) for (1.1) is employed for
0 Wp M ⊂ (cid:0) (cid:1)
the Galerkin discretization of (1.1). The corresponding discrete eigenfunctions u
9 p ∈
1( ), if they exist, will satisfy
0 Wp M
:
v u 1( ):= w 1( ):(ǫw ,v ) =0 v Ker(curl) 1( ) .
i p ∈Xp M { p ∈Wp M p p L2(Ω) ∀ p ∈ ∩Wp M }
X (1.2)
r
a We cannot expect Xp1(M) ⊂ H(curl,Ω) ∩ H0(divǫ0,Ω) and, thus, a standard
Galerkin approximation of (1.1) boils down to an outer approximation of the eigen-
value problem. Good approximation properties of the finite element space no longer
automatically translate into convergence of eigenvalues and eigenfunctions. As inves-
tigated Caorsi, Fernandes and Rafetto in [12], an array of other requirements has
to be met by the finite element spaces, the most prominent of which is the discrete
compactness property [3].
Definition 1.1. The discrete compactness property holds for an asymptoti-
cally dense family 1( ) of finite dimensional subspaces of H(curl,Ω), if any
Wp M p∈N
(cid:0) (cid:1)
∗SAM,ETHZu¨rich,CH-8092Zu¨rich,[email protected]
1We use the customary notations for Sobolev spaces like Hs(Ω), H(curl,Ω), H(div,Ω), etc.,
and write H(curl0,Ω), H(curl0,Ω), etc., for the kernels of differential operators. The reader is
referredto[24,§I.2]and[29,Sect. 2.4]formoreinformation.
1
2 R.Hiptmair
bounded sequence in 1( ) H(curl,Ω) contains a subsequence that converges in
Xp M ⊂
L2(Ω).
The same notion applies in the case of homogeneous Dirichlet boundary condi-
tions, when (1.1) is considered in H (curl,Ω). In this case the eigenfunctions will
0
belong to H0(curl,Ω) H(divǫ0,Ω) and zero tangential trace on ∂Ω has to be im-
∩
posed on trial and test functions.
The discrete compactnessproperty of 1( ) is key to establishing spectralcor-
Wp M
rectnessandasymptoticoptimalityofGalerkinapproximationsof (1.1),see[11,12,15]
fordetails.Smallwonder,thatsubstantialefforthasbeenspentonprovingthis prop-
ertyforvariousasymptoticallydense familiesofH(curl,Ω)/H (curl,Ω)-conforming
0
finite elements. For the h-version of Ned´el´ec’s edge elements Kikuchi [31–33] accom-
plishedthefirstproof,whichwaslatergeneralizedin[7,23,36],see[35,Sect.7.3.2],[29,
Sect. 4], and [15] for a survey. Conversely, spectral edge element schemes in 3D have
longdefiedallattemptstoprovetheirdiscretecompactnessproperty,thoughtheyper-
form well for Maxwell eigenvalue problems [15,17,39]. Partial success was reported
for edge elements in 2D: In [9] the analysis of the discrete compactness property for
triangularhp finite elements has been tackled, but the proofof the main resultrelied
on a conjectured L2 estimate, which had only been demonstrated numerically. The
first fully rigorous analysis of 2D hp edge elements on rectangles was devised in [8].
In[29,Remark15]aninterpolationestimatewasidentifiedascrucialmissingstep
in the analysis. Since then, two major advances have paved the way for closing the
gap:
1. In [16] M. Costabel and M. McIntosh discovered a construction of H1(Ω)-
stablevectorpotentialsby meansofasmoothedPoincar´emapping.This will
be reviewed in Sect. 2 of the present paper.
2. In the breakthrough paper [21] L. Demkowicz and A. Buffa achieved a com-
prehensive analysis of commuting projection based interpolation operators.
To maintain the article self-contained, their approach will be explained in
Sect. 4 and their interpolation error estimates will be presented in Sect. 5.
In addition, we exploit the possibility to construct high order versions of Ned´el´ec’s
firstfamilyofedgeelements[38]byusingCartan’sPoincar´emap[4,27–29],seeSect.3
for details. Another important tool are stable polynomial preserving extension oper-
ators developed, for example, in [1,5,22,37]. In addition, we heavily rely on spectral
polynomial approximation estimates, see [6,37,40].
Thus, standing on the shoulders of giants and combining all these profound the-
oriesofnumericalanalysis,this articlemanagesto givethe firstprooffor the discrete
compactness property of the p-version for the first family of Ned´el´ec’s edge elements
on tetrahedral meshes of Lipschitz polyhedra Ω, consult Sect. 6 for the proof.
Theorem 1.2. The sequence 1( ) of trial spaces generated by the p-
Wp M p∈N
version of the first family of Ned´el´ec’s H(curl,Ω)- or H (curl,Ω)-conforming finite
(cid:0) (cid:1) 0
elements on a fixed tetrahedral mesh of a bounded Lipschitz polyhedron Ω R3
M ⊂
satisfies the discrete compactness property.
TheideaoftheproofistoinspecttheL2(Ω)-orthogonalHelmholtzdecomposition
[24, I.3]
§
w =w w0 , w0 Ker(curl), (1.3)
p p⊕L2 p p ∈
whose so-called solenoidal components w belong to H(curl,Ω) H (div0,Ω). The
e p ∩ 0
above mentioned compact embedding guarantees the existence of a subsequence of
(wp)p∈NthatconvergesinL2(Ω).Hence,eit“merely”takestoshowkwp−wpkL2(Ω) →
e e
Discretecompactness 3
0 for p in order to establish discrete compactness. Clever use of projection
→ ∞
operators that enjoy a commuting diagram property, converts this task to a uniform
interpolation estimate. The core of this paper is devoted to this seemingly humble
program.
Remark 1.1. Generalizations of Thm. 1.2 to other families of tetrahedral edge
elements, and corresponding hp-finite element schemes are straightforward [8]. For
the sake of readability, these extensions will not be pursued in the present paper.
SincethePoncar´emapdoesnotfitatensorproductstructure,extendingtheresults
of this paper to 3D hexahedral edge elements will take some new ideas.
2. Poincar´e lifting. Let D R3 stand for a bounded domain that is star-
⊂
shaped with respect to a subdomain B D, that is,
⊂
a B, x D : ta+(1 t)x, 0<t<1 D . (2.1)
∀ ∈ ∈ { − }⊂
Definition 2.1. The Poincar´e lifting2 Ra : C0(Ω) C0(Ω), a B, is defined
7→ ∈
as
1
Ra(u)(x):= tu(x+t(x a))dt (x a), x D , (2.2)
− × − ∈
Z0
where designates the cross product of two vectors in R3.
×
This is a special case of the generalized path integral formula for differential
forms, which is instrumental in proving the exactness of closed forms on star-shaped
domains, the so-called “Poincar´elemma”, see [13, Sect. 2.13].
ThelinearmappingRaprovidesarightinverseofthecurl-operatorondivergence-
free vectorfields, see [25, Prop. 2.1] for the simple proof, and [13, Sect. 2.13] for a
general proof based on differential forms.
Lemma 2.2. If divu=0, then, for any a B, curlRau=u for all u C1(Ω).
∈ ∈
Unfortunately, the mapping Ra cannot be extended to a continuous mapping
L2(D) H1(D), cf. [25, Thm. 2.1]. As discovered in the breakthrough paper [16]
based o7→n earlier work of Bogovskiˇi [10], it takes a smoothed version to accomplish
this: we introduce the smoothed Poincar´e lifting 3
R(u):= Φ(a)Ra(u)da, (2.3)
ZB
where
Φ C∞(R3), suppΦ B , Φ(a)da=1. (2.4)
∈ ⊂
ZB
The substitution
1
y :=a+t(x a) , τ := , (2.5)
− 1 t
−
transforms the integral (2.4) into
∞
R(u)(x)= τ(1 τ)u(y) (x y)Φ(y+τ(y x))dτdy
− × − −
RZ3 Z1 (2.6)
= k(x,y x) u(y)dy ,
− ×
RZ3
2Boldsymbolswillgenerallybeusedtotagvector valuedfunctions andspacesofsuch.
3Thedependence ofRonΦisdroppedfromthenotation.
4 R.Hiptmair
that is, R is a convolution-type integral operator with kernel
∞
k(x,z)= τ(1+τ)Φ(x+τz)zdτ
Z1 (2.7)
z ∞ z z ∞ z
= ζΦ(x+ζ )dζ + ζ2Φ(x+ζ )dζ .
z 2 z z 3 z
| | Z1 | | | | Z1 | |
The kernel can be bounded by k(x,z) K(x)z −2, where K C∞(R3) depends
| | ≤ | | ∈
only on Φ and is locally uniformly bounded. As a consequence, (2.6) exists as an
improper integral.
The intricate but elementary analysis of [16, Sect. 3.3] further shows, that k be-
longs to the Ho¨rmander symbolclassS−1(R3), see [41,Ch. 7].Invokingthe theory of
1,0
pseudo-differential operators [41, Prop. 5.5] we obtain the following following conti-
nuity result, which is a special case of [16, Cor. 3.4]
Theorem 2.3. The mapping R can be extended to a continuous linear operator
L2(D) H1(D), which is still denoted by R. It satisfies
7→
curlRu=u u H(div0,D). (2.8)
∀ ∈
The smoothed Poincar´e lifting shares this continuity property with many other
mappings, see [29, Sect. 2.4]. Yet, it enjoys another essential feature, which is imme-
diate from its definition (2.2): R maps polynomials of degree p to other polynomials
of degree p+1. The next section will highlight the significance of this observation.
≤
3. Tetrahedral edge elements. In [38] Ned´el´ec introduced a family of
H(curl,Ω)-conforming, that is, tangentially continuous, finite element spaces. On
a tetrahedraltriangulation ofΩ, the correspondingfinite elementspacesofdegree
M
p are given by
1( ):= v H(curl,Ω): v 1(T) T ,
Wp M { ∈ |T ∈Wp ∀ ∈M}
1(T):= v C∞(T): v(x)=p(x)+q(x) x, p,q P (R3), x T .
Wp { ∈ × ∈ p ∈ }
We wrote (R3) for the space of 3-variate polynomials of total degree p, p N ,
p 0
and the boPld symbol P (R3) for vectorfields with three components in≤ (R3∈). To
p p
P
emphasizethatpolynomialsonatetrahedronT arebeingconsidered,wemayuse the
notations (T)/P (T)insteadof (R3)/P(R3).Wealsoadopttheconventionthat
p p p
(R3)=P0 , if p<0. Another rePlevant polynomial space is
p
P { }
P (div0,R3):= q P (R3): divq=0 . (3.1)
p p
{ ∈ }
Deepinsightscanbegainedbyregardingedgeelementsasdiscrete1-forms.Thispro-
videsaveryelegantconstructionofhigherorderedgeelementspacesandimmediately
reveals their relationships with standard Lagrangian finite elements and H(div,Ω)-
conforming face elements (see below). In particular, the Poincar´e lifting becomes a
powerfultoolforbuildingdiscretedifferentialformsofhighpolynomialdegree.Thisis
exploredin [27,28],[29,Sect. 3.4],and [4,Sect. 1.4]in arbitrarydimension,using the
calculus of differentialforms.In this article we prefer to stick to the classicalcalculus
of vector analysis, because we are only concerned with 3D. We hope, that, thus, the
presentation will be more accessible to an audience of numerical analysts. Yet, the
differential forms background has inspired our notations: integer superscripts label
Discretecompactness 5
spaces and operators related to differential forms. For instance, 1( ) can be read
Wp M
as a space of discrete 1-forms.
According to [27, Sect. 3], for any T , a T, we can obtain the local space
∈ M ∈
as
Wp1(T)=Pp(R3)+Ra Pp(div0,R3) . (3.2)
Independence ofa is discussedin [27,Sect. 3].(cid:0)The represent(cid:1)ation(3.2) canbe estab-
lished by dimensional arguments: from the formula (2.2) for the Poincar´e lifting we
immediatelyseethatPp(R3)+Ra(Pp(R3))⊂Wp1(T).Inaddition,from[38,Lemma4]
and[27,Thm. 6,casel =1,n=3]we learnthat the dimensions ofboth spaces agree
and are equal to
dim 1(T)= 1(1+p)(3+p)(4+p). (3.3)
Wp 2
As a consequence, the two finite dimensional spaces must agree.
For the remainder of this section, which focuses on local spaces, we single out a
tetrahedronT .OnT wecanintroduceasmoothedPoincar´eliftingR according
T
∈M
to (2.3) with B =T and a suitable Φ C∞(T) complying with (2.4). An immediate
∈ 0
consequence of (3.2) is that
R P (div0,R3) 1(T). (3.4)
T p ⊂Wp
Weintroducethenotation(cid:0) (T)forthe(cid:1)setofallm-dimensionalfacetsofT,m=
m
F
0,1,2,3. Hence, (T) contains the vertices of T, (T) the edges, (T) the faces,
0 1 2
F F F
and (T) = T . Moreover, for some F (T), m = 1,2,3, (F) denotes the
3 m p
F { } ∈ F P
spaceofm-variatepolynomialsoftotaldegree pinalocalcoordinatesystemofthe
≤
facet F, and P (F) will designate corresponding tangential polynomial vectorfields.
p
Further, we write
1(e)= 1(T) t , t the unit tangent vector of e, e (T), (3.5)
Wp Wp · e e ∈F1
1(f)= 1(T) n , n the unit normal vector of f, f (T), (3.6)
Wp Wp × f f ∈F2
forthetangentialtracesoflocaledgeelementvectorfieldsontoedgesandfaces.Simple
vector analytic manipulations permit us to deduce from (3.2) that
1(e)= (e), e (T), (3.7)
Wp Pp ∈F1
1(f)=P (f)+R2D( (f)), a f , f (T), (3.8)
Wp p a Pp ∈ ∈F2
where the projection R2aD of the Poincar´elifting in the plane reads
1
R2aD(u)(x):= tu(a+t(x a)](x a)dt, a R2 . (3.9)
− − ∈
Z0
It satisfies div R2D(u)= u for all u C∞(R2). We point out that, along with (3.2),
Γ a
∈
the formulas(3.7)and(3.8) arespecialversionsofthe generalrepresentationformula
for discrete 1-forms, see [27, Formula (16)]. Special facet tangential trace spaces will
also be needed:
◦
1(e):= u 1(e): udl=0 , e (T), (3.10)
Wp { ∈Wp } ∈F1
Ze
◦
1(f):= u 1(f): u n 0 e (T), e ∂f , f (T), (3.11)
Wp { ∈Wp · e,f ≡ ∀ ∈F1 ⊂ } ∈F2
◦
1(T):= u 1(T): u n 0 f (T) . (3.12)
Wp { ∈Wp × f ≡ ∀ ∈F2 }
6 R.Hiptmair
Here n represents an exterior face unit normal of T, n the in plane normal of a
f e,f
face w.r.t. an edge e ∂f.
⊂
According to [38, Sect. 1.2] and [27, Sect. 4], the local degrees of freedom for
1(T)aregivenbythe firstp 2vectorialmomentsonthe cells of ,the firstp 1
Wp − M −
vectorial moments of the tangential components on the faces of and the first p
M
tangential moments along the edges of T, see (3.14) for concrete formulas. Then the
set dof1(T) can be partitioned as
p
dof1(T)= ldf1(e) ldf1(f) ldf1(T), (3.13)
p p ∪ p ∪ p
e∈F[1(T) f∈F[2(T)
where the functionals in ldf1(e), ldf1(f), and ldf1(T) are supported on an edge, face,
p p p
and T, respectively, and read
κ ldf1(e) κ(u)= pξ t dl for e (T), suitable p (e),
∈ p ⇒ e · e ∈F1 ∈Pp
κ ldf1(f) κ(u)= p (ξ n)dS for f (T), suitable p P (f),
∈ p ⇒ Rf · × ∈F2 ∈ p−1
κ ldf1(T) κ(u):= p ξdx for suitable p P (T).
∈ p ⇒ RT · ∈ p−2
(3.14)
R
These functionals are unisolvent on 1(T) and locally fix the tangential trace of
Wp
u 1(T). There is a splitting of 1(T) dual to (3.13): Defining
∈Wp Wp
1(F):= v 1(T): κ(v)=0 κ dof1(T) ldf1(F) (3.15)
Yp { ∈W ∀ ∈ p \ p }
for F (T), m=1,2,3, we find the direct sum decomposition
m
∈F
3
1(T)= 1(F). (3.16)
Wp Yp
mX=1F∈XFm(T)
In addition, note that the tangential trace of u 1(F) vanishes on all facets = F,
∈ Xp 6
whosedimensionissmallerorequalthedimensionofF.Bytheunisolvenceofdof1(T),
p
there are bijective linear extension operators
E1 : 1(e) 1(e), e (T), (3.17)
e,p W◦p 7→Yp ∈F1
E1 : 1(f) 1(f), f (T). (3.18)
f,p Wp 7→Yp ∈F2
Thecurlconnectstheedgeelementspaces 1( )andtheso-calledfaceelement
Wp M
spaces of discrete 2-forms [38, Sect. 1.3]
2( ):= v H(div,Ω): v 2(T) T ,
Wp M { ∈ |T ∈Wp ∀ ∈M}
2(T):= v C∞(T): v(x)=p(x)+q(x)x, p P (T), q (T) .
Wp { ∈ ∈ p ∈Pp }
An alternative representation of the local face element space is [27, Formula (16) for
l=2, n=3]
Wp2(T)=Pp(T)+Da(Pp(T)), (3.19)
where the appropriate version of the Poincar´e lifting reads
1
(Dau)(x):= t2u(a+t(x a))(x a)dt, a T . (3.20)
− − ∈
Z0
Discretecompactness 7
Like(3.2)thisisaspecialincarnationofthegeneralformula(16)in[27].Again,dimen-
sionalargumentsbased on[38,Sect. 1.3]and[27,Thm. 6]confirmthe representation
(3.20). We remark that divDau=u, see [25, Prop. 1.2].
The normal trace space of 2(T) onto a face is
Wp
2(f):= 2(T) n = (f), f (T), (3.21)
Wp Wp · f Pp ∈F2
and as relevant space “with zero trace” we are going to need
◦
2(f):= u 2(f): udS =0 , f (T), (3.22)
Wp { ∈Wp } ∈F2
Zf
◦
2(T):= u 2(T): u n =0 . (3.23)
Wp { ∈Wp · ∂T }
The connection between the local spaces 1(T), 2(T) and full polynomial
Wp Wp
spaces is established through a local discrete DeRham exact sequence: To elucidate
therelationshipbetweendifferentialoperatorsandvarioustracesontofacesandedges,
we also include those in the statement of the following theorem. There n stands for
f
anexteriorfaceunitnormalofT,n forthe inplanenormalofafacew.r.t.anedge
e,f
e ∂f, and d is the differentiation w.r.t. arclength on an edge.
⊂ dl
Theorem 3.1. For f (T), e (T), e ∂f, all the sequences in
2 1
∈F ∈F ⊂
const Id (T) grad 1(T) curl 2(T) div (T) Id 0
−−−−→ Pp+1 −−−−→ Wp −−−−→ Wp −−−−→ Pp −−−−→ { }
.|f .×nf|f .·nf|f
const Id (f) curlΓ 1(f) divΓ (f) Id 0
−−−−→ Pp+y1 −−−−→ Wpy −−−−→ Ppy −−−−→ { }
.|e .·ne,f|e
const Id (e) ddl (e) Id 0
p+y1 py
−−−−→ P −−−−→ P −−−−→ { }
are exact and the diagram commutes.
Proof. The assertion about the top exact sequence is an immediate consequence
of representations (3.2) and (3.19) and the relationships
curlRa(u)=u u Pp(div0,T), divDa(u)=u u p(T).
∀ ∈ ∀ ∈P
For further discussions and the proof of the other exact sequence properties see [27,
Sect. 5 for n=3].
4. Projection based interpolation. Thedegreesoffreedomintroducedabove
definelocalfiniteelementprojectorsonto 1(T).Inconjunctionwithsuitablydefined
Wp
interpolation operators for degree p Lagrangian finite elements, they possess a very
desirable commuting diagramproperty [27, Thm. 13], which will be explained below.
However, they do not enjoy favorable continuity properties with increasing p. Thus,
L. Demkowicz [19–21], taking the cue from the theory of p-version Lagrangian finite
elements, invented an alternative in the form of local projection based interpolation.
4.1. Projections, liftings, and extensions. Again, consider a single tetra-
hedron T and fix the polynomial degree p N. Following the developments
∈ M ∈
of [29, Sect. 3.5], projection based interpolation requires building blocks in the form
8 R.Hiptmair
of local orthogonal projections Pl and liftings Ll4. Some operators will depend on a
∗ ∗
regularity parameter 0<ǫ< 1, which is considered fixed below and will be specified
2
in Sect. 5. To begin with, we define for every e (T)
1
∈F
d ◦ ◦
P1 :H−1+ǫ(e) (e)= 1(e) (4.1)
e,p 7→ dlPp+1 Wp
◦
as the H−1+ǫ(e)-orthogonal projection. Here, (F) denotes the space of degree p
p
P
polynomials on a facet F that vanish on ∂F.
Similarly, for every face f (T) introduce
2
∈F
◦ ◦
P1f,p :H−12+ǫ(f)7→curlΓPp+1(f)={v∈W1p(f): divΓv=0}, (4.2)
◦ ◦
P2f,p :H−21+ǫ(f)7→divΓW1p(f)=W2p(f), (4.3)
as the corresponding H−12+ǫ(f)-orthogonalprojections. Eventually, let
◦ ◦
P1 :L2(T) grad (T)= v 1(T): curlv=0 , (4.4)
T,p 7→ P◦ p+1 { ∈◦Wp }
P2 :L2(T) curl 1(T)= v 2(T): divv=0 , (4.5)
T,p 7→ Wp { ∈Wp }
◦
P3 :L2(T) div 2(T)= v (T): v(x)dx=0 , (4.6)
T,p 7→ Wp { ∈Pp }
ZT
stand for the respective L2(T)-orthogonalprojections.
The lifting operators
◦ ◦
L1 : 1(e) (e), e (T), (4.7)
e,p Wp ◦7→Pp+1 ∈F1 ◦
L1 : v 1(f): div v=0 (f), f (T), (4.8)
f,p { ∈W◦ p Γ }7→P◦p+1 ∈F2
L1 : v 1(T): curlv=0 (T), (4.9)
T,p { ∈Wp }7→Pp+1
are uniquely defined by requiring
d ◦
L1 u=u u 1(e), (4.10)
dl e,p ∀ ∈Wp
◦
curl L1 u=u u 1(f): div v=0 , (4.11)
Γ f,p ∀ ∈{Wp ◦ Γ }
gradL1 u=u u v 1(T): curlv=0 . (4.12)
T,p ∀ ∈{ ∈Wp }
Another class of liftings provides right inverses for curl and div : Pick a face f
Γ
∈
(T), and, without loss of generality, assume the vertex opposite to the edge e to
2
F
coincide with 0. Then define
◦ ◦
div 1(f) 1(f) e
L2 : ΓWp 7→ Wp (4.13)
f,p (cid:26) u 7→ R20Du−curlΓE0ee,pL1ee,p(R20Du·nee,f).
This is a valid definition, since, by virtue of definition (3.9), the normal components
of R2Du will vanish on ∂f e. Moreover, div R2Du = u ensures that the normal
0 \ Γ 0
component of R2Du has zero averageon e. We infer
0
e
curlΓE0ee,pL1ee,p (R20Du·nee,f)|ee ·nee,f e|ee=
(cid:16) (cid:0) (cid:1) d (cid:17)
dlL1ee,p (R20Du)·nee,f |ee=R20Du·nee,f on e,
(cid:0) (cid:17)
4The parameter l in the notations for the extension operators El∗, the projections Pl∗, aned the
liftings Ll∗ refers to the degree of the discrete differential form they operate on. This is explained
moreclearlyin[29,Sect.3.5].
Discretecompactness 9
and see that the zero trace condition on ∂f is satisfied. The same idea underlies the
definition of
◦ ◦
curl 1(T) 1(T)
L2T,p :( Wup 7→7→ WR0pu−gradE0fe,pL1fe,p ((R0u)×nfe)|fe , (4.14)
(cid:0) (cid:1)
where f is the face opposite to vertex 0, and the definition of
◦ ◦
e div 2(T) 2(T)
L3T,p :( Wup 7→7→ DW0pu−curlE1fe,pL2fe,p((D0u·nfe)|fe). (4.15)
The relationshipsbetweenthe variousfacetfunctionspaceswithvanishing tracescan
be summarized in the following exact sequences:
0 Id ◦ (T) grad ◦ 1(T) curl ◦ 2(T) div (T) Id 0 ,
{ } −−−−→ Pp+1 −−L−1−→ Wp −−L−2−→ Wp −−L−3−→ Pp −−−−→ { }
T,p T,p T,p
0 Id ◦ (f) curlΓ ◦ 1(f) divΓ (f) Id 0 ,
{ } −−−−→ Pp+1 −−L−1−→ Wp −−L−2−→ Pp −−−−→ { }
f,p f,p
Id ◦ d Id
0 (e) dl (e) 0 ,
p+1 p
{ } −−−−→ P −−L−1−→ P −−−−→ { }
e,p
(4.16)
where (F)designatesdegreeppolynomialspacesonF withvanishingmean.These
p
P
relationships and the lifting mappings are studied in [29, Sect. 3.4].
Finally we need polynomial extension operators
◦
E0 : (e) (T), (4.17)
e,p P◦p+1 7→Pp+1
E0 : (f) (T) (4.18)
f,p Pp+1 7→Pp+1
that satisfy
E0 u =0 e′ (T) e , (4.19)
e,p |e′ ∀ ∈F1 \{ }
E0 u =0 f′ (T) f . (4.20)
f,p |f′ ∀ ∈F2 \{ }
Such extension operatorscan be constructed relying on a representationof a polyno-
mial on F, F (T), m = 1,2, as a homogeneous polynomial in the barycentric
m
∈ F
coordinatesofF,see[29,Lemma 3.4].Asanalternative,one mayusethe polynomial
preserving extension operators proposed in [22,37] and [1]. We stress that continuity
properties of the extensions El , l =0,1, F (T), are immaterial.
F ∈Fm
4.2. Interpolation operators. Now, we are in a position to define the projec-
tionbasedinterpolationoperatorslocallyonagenerictetrahedronT withverticesa ,
i
i=1,2,3,4.
First, we devise a suitable projection (depending on the regularity parameter
0<ǫ< 1, which is usually suppressed to keep notations manageable)
2
Π0 (=Π0 (ǫ)) : C∞(T) (T) (4.21)
T,p T,p 7→Pp+1
10 R.Hiptmair
for degree p Lagrangian H1(Ω)-conforming finite elements. For u C0(T) define (λ
i
∈
is the barycentric coordinate function belonging to vertex a of T)
i
4
u(0) :=u u(a )λ , (4.22)
i i
−
i=1
X
:=w(0)
d
u(1) :=u(0)| {z E}0 L1 P1 u(0) , (4.23)
− e,p e,p e,pds |e
e∈XF1(T)
:=w(1)
u(2) :=u(1)−| E0f,pL{1fz,pP1f,pcurlΓ(}u(1)|f), (4.24)
f∈XF1(T)
:=w(2)
Π0T,pu:=L1T,pP|1T,pgradu(2)+{zw(2)+w(1)+w}(0) . (4.25)
Observe that w(i) =0 for all F (T), 0 m<i 3. We point out that w(0) is
|F m
∈F ≤ ≤
the standard linear interpolant of u.
Lemma 4.1. The linear mapping Π0 , p N , is a projection onto Cp (T)
T,p ∈ 0 p+1
Proof. Assume u (T), which will carry over to all intermediate functions.
p+1
∈ P
Since u(0)(z )=0,i=1,...,4,weconclude fromtheprojectionpropertyofP1 that
i e,p
L1P1 d u(0) =u(0) for any edge e (T). As a consequence
e eds |e |e ∈F1
u(1) =u(0) E0 u(0) u(1) =0 e (T). (4.26)
− e,p |e ⇒ |e ∀ ∈F1
e∈XF1(T)
We infer L1 P1curl (u(1) )=u(1) on each face f (T), which implies
f,p f Γ |f |f ∈F2
u(2) =u(1) E0 (u(1) ) u(2) =0 f (T). (4.27)
− f,p |f ⇒ |f ∀ ∈F2
f∈XF1(T)
This means that L1 P1 gradu(2) =u(2) and finishes the proof.
T,p T,p
A similar stage by stage construction applies to edge elements and gives a pro-
jection
Π1 (=Π1 (ǫ)) : C∞(T) 1(T) : (4.28)
T,p T,p 7→W
for a directed edge e:=[a ,a ] we introduce the Whitney-1-form basis function
i j
b =λ gradλ λ gradλ . (4.29)
e i j j i
−
