Stable broken $H^1$ and $\bf H(\mathrm{div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions

0.7 MB·

by  Alexandre Ern

Stable broken H1 and H(div) polynomial extensions for
polynomial-degree-robust potential and flux reconstruction in three
space dimensions∗
Alexandre Ern† Martin Vohral´ık‡
January 10, 2017
7
1
0
Abstract
2
Westudyextensionsofpiecewisepolynomialdataprescribedonfacesandpossiblyinelementsofapatch
n
a ofsimplicessharingavertex. IntheH1 setting, welookforfunctionswhosejumpsacrossthefacesare
J prescribed, whereas in the H(div) setting, the normal component jumps and the piecewise divergence
9 are prescribed. We show stability in the sense that the minimizers over piecewise polynomial spaces
of the same degree as the data are subordinate in the broken energy norm to the minimizers over the

] whole broken H1 and H(div) spaces. Our proofs are constructive and yield constants independent of
A
thepolynomialdegree. Oneparticularapplicationoftheseresultsisinaposteriorierroranalysis,where
N the present results justify polynomial-degree-robust efficiency of potential and flux reconstructions.
.
h
t Key words: polynomial extension operator, broken Sobolev space, potential reconstruction, flux recon-
a
struction, aposteriorierrorestimate, robustness, polynomialdegree, bestapproximation, patchofelements
m
[

1 Introduction
1
v
1 Braesset al.[1,Theorem1]showedthatequilibratedfluxaposteriorierrorestimatesleadtolocalefficiency
6 andpolynomial-degreerobustness(inshort,p-robustness). Thismeansthattheestimatorsupper-bounding
1
the error also give local lower bounds for the error, up to a generic conctant independent of the polynomial
2
degree of the approximate solution. These results apply to conforming finite element methods in two space
0
dimensions. They are based on flux reconstructions obtained by solving, via the mixed finite element
.
1
method, a homogeneous Neumann, hat-function-weighted residual problem on each vertex-centered patch
0
ofthemesh. Theproofofthep-robustnessin[1]reliesontwokeycomponents: p-robuststabilityoftheright
7
1 inverse of the divergence operator shown in Costabel and McIntosh [9, Corollary 3.4] and p-robust stability
: oftherightinverseofthenormaltraceshowninDemkowiczetal.[12,Theorem7.1]. Inourcontribution[19,
v
Theorem3.17],weextendedp-robustnessofaposteriorierrorestimatestoanynumericalschemesatisfyinga
i
X couple of clearly identified assumptions, including nonconforming, discontinuous Galerkin, and mixed finite
r elements, still in two space dimensions, while proceeding through similar stability arguments. A second
a
type of local problem appears here, where one is led to solve a homogeneous Dirichlet, conforming finite
elementproblemoneachvertex-centeredelementpatch,withahat-function-weighteddiscontinuousdatum,
yielding a potential reconstruction.
The present work extends the results of [1] on flux reconstruction to three space dimensions and refor-
mulatesthemethodologyof[19]forpotentialreconstructionsothatitcanbeappliedinthesamewayintwo
and three space dimensions. In doing so, we adopt a different viewpoint leading to a larger abstract setting
not necessarily linked to a posteriori error analysis. The two main results of this paper are Theorems 2.2
and 2.3. They concern a setting where one considers a shape-regular patch of simplicial mesh elements
sharing a given vertex, say a, together with a p-degree polynomial r associated with each face F of the
F
∗ThisprojecthasreceivedfundingfromtheEuropeanResearchCouncil(ERC)undertheEuropeanUnionsHorizon2020
researchandinnovationprogram(grantagreementNo647134GATIPOR).
†Universit´eParis-Est,CERMICS(ENPC),77455Marne-la-Vall´ee2,France([email protected]).
‡INRIAParis,2rueSimoneIff,75589Paris,France([email protected]).
1
patch (H1 potential reconstruction setting) or p-degree polynomials r and r associated with each face
F K
F and element K of the patch respectively (H(div) flux reconstruction setting). These data, satisfying
appropriate compatibility conditions, are to be extended to functions defined over the patch, such that
the jumps across the interior faces are prescribed by r (H1 setting) or such that the normal component
F
jumps and boundary values are prescribed by r and the piecewise divergence is prescribed by r (H(div)
F K
setting). Crucially, we prove that the extension into piecewise polynomials of degree p that minimizes the
broken energy norm is, up to a constant only depending on the patch shape regularity, as good as the
extension into the whole broken H1 space with the same jump constraints. Similarly, our broken p-degree
Raviart–Thomas–N´ed´elec extension is stable with respect to the broken H(div) one.
Section 3 reformulates equivalently the above theorems as Corollaries 3.1 and 3.3 to show that best-
approximation of discontinuous or normal-trace discontinuous piecewise polynomial data by H1(ω )- or
0 a
H (div,ω )-conforming piecewise polynomials (i.e., by continuous or normal-trace continuous piecewise
0 a
polynomials) is, up to a p-independent constant, as good as by all H1(ω ) or H (div,ω ) Sobolev func-
0 a 0 a
tions. This section also sheds more light on the continuous level, uncovering that three different equivalent
formulationsofourresultscanbedevisedusingtheequivalenceprincipleofprimalanddualenergies. This,
in particular, allows us to make a link with the previously obtained results in [1, 19] and to describe the
application of our results to a posteriori error analysis in Section 4. In particular, p-robust local efficiency
for flux reconstructions is stated in formula (4.9) and p-robust local efficiency for potential reconstructions
is stated in formula (4.7).
The proofs of Theorems 2.2 and 2.3 are respectively presented in Sections 5 and 6. In contrast to [1],
where the work with dual norms was essential, we design here a procedure only working in the (broken)
energy norms. The proofs are constructive and therefore indicate a possible practical reconstruction of
the potential and the flux avoiding the patchwise problem solves. These are replaced by a single explicit
run through the patch, with possibly a solve on each element. The key ingredients on a single element
are still the right inverse of the divergence [9, Corollary 3.4] and the right inverse of the normal trace [12,
Theorem 7.1] in the H(div) setting, but this becomes the right inverse of the trace shown in Demkowicz et
al. [10, Theorem 6.1] in the H1 setting. We combine these building blocks into a stability result on one
tetrahedron in Lemmas A.1 and A.3 in Appendix A. We find that they are of independent interest. Gluing
theelementalcontributionstogetheratthepatchlevelturnsouttobeanotherratherinvolvedingredientof
the proofs in three space dimensions, and we collect some auxiliary results for that purpose in Appendix B.
A firstdifficultyis that thetwo-dimensional argumentof turning arounda vertex can nolonger beinvoked.
To achieve a suitable enumeration of the mesh cells composing the patch in three dimensions, we rely on
the notion of shelling of polytopes, see Ziegler [21, Chap. 8], which we reformulate for the present purposes
in Lemma B.1. Another difficulty is that we need to devise suitable functional transformations between
different cells in the patch. This is done by introducing two- and three-coloring of some vertices lying on
the boundary of the patch, possibly on a submesh of the original patch; how to achieve such colorings is
described in Lemmas B.2 and B.3.
Forthesakeofclarityofourexposition,wefocusondiscussingindetailspatchescompletelysurrounding
thevertexa,correspondingtoan“interior”vertexwhenconsideringameshofsomecomputationaldomain.
Our technique, though, extends to the case where one considers a “boundary” vertex as well. Our main
results in this context are Theorems 2.4 and 2.5, whereas the reformulations as best-approximation results
on piecewise polynomial data can be found in Corollaries 3.7 and 3.8. The proofs of our main results
concerning boundary vertices are given in Section 7, with the description of the enumeration of boundary
patches presented in Appendix C. The aforementioned application to a posteriori error analysis (Section 4)
then also covers general inhomogeneous Dirichlet and Neumann boundary conditions.
Our results combine and extend those of [9, Corollary 3.4], [12, Theorem 7.1], and [10, Theorem 6.1].
In particular, we obtain stable H1 and H(div) polynomial extensions on an arbitrary tetrahedron in Lem-
masA.1andA.3andonanarbitrarypatchoftetrahedrainTheorems2.2and2.3. Inafurtherextension[17],
we were recently able to employ them to construct p-robust H(div) liftings over arbitrary domains, not
being patches of elements. A natural extension of [17] would be to obtain the same type of results in the
H1 setting. Another extension is to cover the H(curl) case, hinging on the single tetrahedron results of
Demkowicz et al. [11, Theorem 7.2]. Finally, we mention that application of the present results to the
construction of p-robust a posteriori error estimates for problems with arbitrarily jumping coefficients is
detailed in [8], to eigenvalue problems in [5, 4], to the Stokes problem in [7], to linear elasticity in [16], and
to the heat equation in [18].
2
2 Main results
This section presents our main results, once the setting and basic notation have been fixed.
2.1 Setting and basic notation
We call tetrahedron any non-degenerate (closed) simplex in R3, uniquely determined by four points in R3
not lying in a plane. Let a be a point in R3. We consider a patch of tetrahedra around a, say T , i.e., a
a
finite collection of tetrahedra having a as vertex, such that the intersection of any two distinct tetrahedra
in T is either a, or a common edge, or a common face. A generic tetrahedron in T is denoted by K and
a a
is also called an element or a cell. We let ω ⊂R3 denote the interior of the subset ∪ K. For the time
a K∈Ta
being, we focus on the case where ω contains an open ball around a. The main application we have in
a
mind is when a is the interior vertex of a simplicial mesh T of some computational domain Ω. The case
h
whereaisaboundaryvertexofthemeshentailssomeadditionaltechnicalitiesthatwedetailinSection2.4.
AllthefacesoftheelementsinthepatchT arecollectedinthesetF whichissplitintoF =Fi ∪Fb
a a a a a
with Fi collecting all the interior faces (containing the vertex a and shared by two distinct elements in T )
a a
and Fb collecting the faces located in ∂ω . For all faces F ∈ F , n denotes a unit normal vector to F
a a a F
whose orientation is arbitrary but fixed for all F ∈Fi and coinciding with the unit outward normal to ω
a a
for all F ∈Fb. We consider the jump operator [[·]] for all F ∈Fi, yielding the difference (evaluated along
a F a
n ) of the traces of the argument from the two elements that share the interior face F (the subscript F is
F
omitted if there is no ambiguity). We also need to consider edges. Let E collect all the edges in T sharing
a a
thevertexa; werefertotheseedgesasinterioredges. Then, foreache∈E ,thesetF collectsallthefaces
a e
in Fi sharing e, and the set T collects all the cells in T sharing e. For each e∈E , we fix one direction of
a e a a
rotation around e, and indicate by ι either equal to 1 or to −1 whether n complies with this direction
F,e F
or not for all F ∈F.
e
We define the broken H1-space on the patch T as
a
H1(T ):={v ∈L2(ω ); v| ∈H1(K), ∀K ∈T }, (2.1)
a a K a
and similarly the broken H(div)-space on the patch T as
a
H(div,T ):={v ∈L2(ω ); v| ∈H(div,K), ∀K ∈T }. (2.2)
a a K a
For any v ∈ H1(T ), we can consider its piecewise (broken) gradient ∇ v defined as (∇ v)| = ∇(v| ),
a T T K K
and similarly for any v ∈ H(div,T ), we can consider its piecewise (broken) divergence ∇ ·v defined as
a T
(∇ ·v)| =∇·(v| ), for all K ∈T . For any v ∈H1(T ), the jumps [[v]] across any face F ∈Fi are well
T K K a a F a
defined since the traces of v on F from the two cells sharing F are in L2(F); similarly, the traces v| are
Fb
a
well-defined. We note that any smooth enough function v ∈H1(T ) is such that
a
(cid:88)
ι [[v]] | =0 for all interior edges e∈E , (2.3)
F,e F e a
F∈Fe
sincetheorientedsumofthejumpsalongaclosedpatharoundaninterioredgeisalwayszero. Thedefinition
of traces is a bit more subtle when one considers a field v ∈ H(div,T ). Let r ∈ L2(F) for all F ∈ F .
a F a
Then we say that
v·n =r ∀F ∈Fb, (2.4a)
F F a
[[v]]·n =r ∀F ∈Fi (2.4b)
F F a
for a function v ∈H(div,T ) if and only if
a
(cid:88)
(∇ ·v,v) +(v,∇v) = (r ,v) ∀v ∈H1(ω ). (2.4c)
T ωa ωa F F a
F∈Fa
We will also need to prescribe the normal component of vector fields in a single cell K ∈ T with unit
a
outward normal n . Consider a non-empty subset FN ⊂ F where F collects the faces of K. Given
K K K K
functions r ∈L2(F) for all F ∈FN, we say that v·n | =r , ∀F ∈FN, for a function v ∈H(div,K) if
F K K F F K
(cid:88)
(∇·v,φ) +(v,∇φ) = (r ,φ) ∀φ∈H1(K) s.t. φ| =0∀F ∈F \FN. (2.5)
K K F F F K K
F∈FN
K
3
Let p ≥ 0 denote a polynomial degree. We use the notation P (K) for polynomials of order at most p
p
in the element K ∈ T and P (F) for polynomials of order at most p in the face F ∈ F . We denote by
a p a
P (T ) the space composed of all functions supported on the patch T whose restriction to any K ∈ T is
p a a a
in P (K). Similarly, P (F ) stands for the space composed of all functions supported on all faces from F
p p a a
whose restriction to any F ∈F is in P (F). Analogous notation is used for any subset of F . We denote
a p a
by r the restriction of r ∈P (T ) to K ∈T and similarly by r the restriction of r ∈P (F ) to F ∈F .
K p a a F p a a
Let RTN (K) be the Raviart–Thomas–N´ed´elec polynomial space of vector-valued functions of order p in
p
the element K ∈T , i.e., RTN (K):=[P (K)]3+P (K)x. Finally, RTN (T ) denotes the broken space
a p p p p a
composed of all functions whose restriction to any element K ∈T is in RTN (K).
a p
For an element K ∈ T , its shape-regularity parameter γ is defined to be the ratio of its diameter
a K
to the diameter of the largest inscribed ball, and the shape-regularity parameter of the patch T is then
a
defined to be γ :=max γ .
Ta K∈Ta K
Remark 2.1 (Orientation). The orientation of n is irrelevant in (2.3). Indeed, changing the orientation
F
of n changes the sign of the jumps (evaluated along n ) and at the same the sign of ι . Similarly, the
F F F,e
orientation of n is irrelevant also in the left-hand side of (2.4b).
F
2.2 Broken H1 polynomial extension
Our main result for broken scalar extensions is the following.
Theorem 2.2 (Stable broken H1 polynomial extension). Let p ≥ 1. Let the interface-based p-degree
polynomial r ∈P (Fi) satisfy the following compatibility conditions:
p a
r | =0 on all interior faces F ∈Fi, (2.6a)
F F∩∂ωa a
(cid:88)
ι r | =0 on all interior edges e∈E . (2.6b)
F,e F e a
F∈Fe
Then there exists a constant C >0 only depending on the patch shape-regularity parameter γ such that
st Ta
min (cid:107)∇ v (cid:107) ≤C min (cid:107)∇ v(cid:107) , (2.7)
vp∈Pp(Ta) T p ωa st v∈H1(Ta) T ωa
vp|F=0∀F∈Fab, v|F=0∀F∈Fab,
[[vp]]F=rF ∀F∈Fai [[v]]F=rF ∀F∈Fai
where the minimization sets are non-empty and both minimizers in (2.7) are unique.
The compatibility conditions (2.6) are natural since r is used to prescribe interface jumps. Indeed,
F
these jumps necessarily vanish on the points of the interfaces located on ∂ω since the considered functions
a
vanish on ∂ω ; moreover, (2.6b) follows from (2.3). The minimizers in (2.7) are respectively denoted by v∗
a p
and v∗, so that (2.7) becomes
(cid:107)∇ v∗(cid:107) ≤C (cid:107)∇ v∗(cid:107) . (2.8)
T p ωa st T ωa
Note also that since the minimization sets are non-empty and the left one is a subset of the right one by
definition, the inequality in the other direction, (cid:107)∇ v∗(cid:107) ≤(cid:107)∇ v∗(cid:107) , is trivial.
T ωa T p ωa
2.3 Broken H(div) polynomial extension
Our main result for broken vector extensions is the following.
Theorem 2.3 (Stable broken H(div) polynomial extension). Let p ≥ 0. Let the element- and face-based
p-degree polynomial r ∈P (T )×P (F ) satisfy the following compatibility condition:
p a p a
(cid:88) (cid:88)
(r ,1) − (r ,1) =0. (2.9)
K K F F
K∈Ta F∈Fa
Then there exists a constant C >0 only depending on the patch shape-regularity parameter γ such that
st Ta
min (cid:107)v (cid:107) ≤C min (cid:107)v(cid:107) , (2.10)
vp∈RTNp(Ta) p ωa st v∈H(div,Ta) ωa
vp·nF=rF ∀F∈Fab v·nF=rF ∀F∈Fab
[[vp]]·nF=rF ∀F∈Fai [[v]]·nF=rF ∀F∈Fai
∇T·vp|K=rK ∀K∈Ta ∇T·v|K=rK ∀K∈Ta
where the minimization sets are non-empty and both minimizers in (2.10) are unique.
4
Thecompatibilitycondition(2.9)isagainnaturalhere,sinceitfollowsfrom(2.4c)withthetestfunction
equal to 1 in ω . The minimizers in (2.10) are respectively denoted by v∗ and v∗, so that (2.10) becomes
a p
(cid:107)v∗(cid:107) ≤C (cid:107)v∗(cid:107) . (2.11)
p ωa st ωa
Since the minimization sets are non-empty and the left one is a subset of the right one by definition, the
inequality in the other direction, (cid:107)v∗(cid:107) ≤(cid:107)v∗(cid:107) , again is trivial.
ωa p ωa
2.4 Boundary vertices
We consider in this section the case where the patch domain ω does not contain an open ball around the
a
point a; typically, a is a mesh vertex lying on the boundary of some computational domain Ω. In this case,
the patch domain ω only contains an open ball around a minus some sector with solid angle θ ∈(0,4π).
a a
The set F collecting all the faces of T is now divided into four disjoint subsets: the set Fi collecting
a a a
(as before) the interior faces containing the vertex a and shared by two distinct elements in T , the set
a
Fb collecting the faces that are subsets of ∂ω that do not contain a, and the sets FD and FN collecting
a a a a
the faces that are subsets of ∂ω that contain a. Note that the faces in Fb, FD, and FN altogether cover
a a a a
∂ω . Faces in these three sets are assigned a unit normal vector n pointing outward ω . The distinction
a F a
between the two sets FD and FN is introduced so as to handle different types of boundary conditions in
a a
the context of a posteriori error estimates. We remark that Fi can be empty (if T consists of a single
a a
tetrahedron), that Fb is always non-empty, and that either FD or FN can be empty, but not both at the
a a a
same time. Let us set ∂ωb =∪ F. Finally, the set E collects all the edges in T sharing the vertex a
a F∈Fb a a
a
(note that some of these edges are now located on ∂ω ) and, for each edge e∈E , F collects all the faces
a a e
in F sharing e (note that F is now a subset of Fi ∪FD∪FN).
a e a a a
Our main results for boundary vertices are the following.
Theorem 2.4 (Stable broken H1 polynomial extension). Let p ≥ 1 and let Assumption C.1 hold. Let
r ∈P (Fi ∪FD) satisfy the following compatibility conditions:
p a a
r | =0 ∀F ∈Fi ∪FD, (2.12a)
F F∩∂ωb a a
a
(cid:88)
ι r | =0 ∀e∈E such that F ∩FN =∅. (2.12b)
F,e F e a e a
F∈Fe
Then there exists a constant C >0 only depending on the patch shape-regularity parameter γ such that
st Ta
min (cid:107)∇ v (cid:107) ≤C min (cid:107)∇ v(cid:107) , (2.13)
vp∈Pp(Ta) T p ωa st v∈H1(Ta) T ωa
vp|F=0∀F∈Fab, v|F=0∀F∈Fab,
vp|F=rF ∀F∈FaD, v|F=rF ∀F∈FaD,
[[vp]]F=rF ∀F∈Fai [[v]]F=rF ∀F∈Fai
where the minimization sets are non-empty and both minimizers in (2.13) are unique.
Theorem 2.5 (StablebrokenH(div)polynomialextension). Letp≥0. Letr ∈P (T )×P (Fi ∪Fb∪FN)
p a p a a a
satisfy the following compatibility condition:
(cid:88) (cid:88)
(r ,1) − (r ,1) =0 if FD =∅. (2.14)
K K F F a
K∈Ta F∈Fai∪Fab∪FaN
Then there exists a constant C >0 only depending on the patch shape-regularity parameter γ such that
st Ta
min (cid:107)v (cid:107) ≤C min (cid:107)v(cid:107) , (2.15)
vp∈RTNp(Ta) p ωa st v∈H(div,Ta) ωa
vp·nF=rF ∀F∈Fab v·nF=rF ∀F∈Fab
vp·nF=rF ∀F∈FaN v·nF=rF ∀F∈FaN
[[vp]]·nF=rF ∀F∈Fai [[v]]·nF=rF ∀F∈Fai
∇T·vp|K=rK ∀K∈Ta ∇T·v|K=rK ∀K∈Ta
where the minimization sets are non-empty and both minimizers in (2.15) are unique.
5
3 Equivalent reformulations
We reformulate in this section Theorems 2.2 and 2.3 in an equivalent way as best-approximation results of
discontinuous piecewise polynomial data. This will in particular allow for a straightforward application to
a posteriori error analysis in the forthcoming section. For further insight, as well as to make a link with
previouscontributionsonthesubject, wealsogiveequivalentreformulationsoftheright-handsidesin(2.7)
and (2.10).
3.1 Reformulation as best-approximation results
Let us set
H1(ω ):={v ∈H1(ω ); v| =0}, (3.1a)
0 a a ∂ωa
H (div,ω ):={v ∈H(div,ω ); v·n =0}. (3.1b)
0 a a ∂ωa
The result of Theorem 2.2 can be rephrased as follows.
Corollary 3.1 (H1 best-approximation). Let the assumptions of Theorem 2.2 hold. Consider any τ ∈
p
P (T ) so that τ | =0 ∀F ∈Fb, and [[τ ]] =r ∀F ∈Fi. Then the following holds:
p a p F a p F F a
min (cid:107)∇ (τ −v )(cid:107) ≤C min (cid:107)∇ (τ −v)(cid:107) . (3.2)
vp∈Pp(Ta)∩H01(ωa) T p p ωa stv∈H01(ωa) T p ωa
Proof. Direct consequence of (2.7) upon shifting the minimization sets by τ . Note that the existence of τ
p p
follows from the non-emptiness of the discrete minimization set in (2.7).
Remark 3.2 (Minimizers). The unique minimizers in (3.2) are respectively sa ∈ P (T )∩H1(ω ) such
h p a 0 a
that
(∇sa,∇v ) =(∇ τ ,∇v ) ∀v ∈P (T )∩H1(ω ), (3.3)
h p ωa T p p ωa p p a 0 a
and sa ∈H1(ω ) such that
0 a
(∇sa,∇v) =(∇ τ ,∇v) ∀v ∈H1(ω ). (3.4)
ωa T p ωa 0 a
The minimizers in (2.7) are such that v∗ =τ −sa and v∗ =τ −sa.
p p h p
Similarly, in the H(div)-setting, Theorem 2.3 can be reformulated as follows.
Corollary 3.3 (H(div) best-approximation). Let the assumptions of Theorem 2.3 hold. Consider any
τ ∈RTN (T ) so that τ ·n =r ∀F ∈Fb and [[τ ]]·n =r ∀F ∈Fi. Then the following holds:
p p a p F F a p F F a
min (cid:107)τ +v (cid:107) ≤C min (cid:107)τ +v(cid:107) . (3.5)
vp∈RTNp(Ta)∩H0(div,ωa) p p ωa st v∈H0(div,ωa) p ωa
∇·vp|K=rK−∇T·τp|K ∀K∈Ta ∇·v|K=rK−∇T·τp|K ∀K∈Ta
Proof. Direct consequence of (2.10) upon shifting the minimization sets by τ , the existence of τ following
p p
from the non-emptiness of the discrete minimization set in (2.10).
Remark 3.4 (Minimizers). Theunique minimizersin (3.5) arerespectively σa ∈RTN (T )∩H (div,ω )
h p a 0 a
with ∇·σa| =r −∇ ·τ | for all K ∈T such that
h K K T p K a
(σa,v ) =−(τ ,v ) ∀v ∈RTN (T )∩H (div,ω ), ∇·v =0, (3.6)
h p ωa p p ωa p p a 0 a p
and σa ∈H (div,ω ) with ∇·σa| =r −∇ ·τ | for all K ∈T such that
0 a K K T p K a
(σa,v) =−(τ ,v) ∀v ∈H (div,ω ), ∇·v =0. (3.7)
ωa p ωa 0 a
The minimizers in (2.10) are such that v∗ =τ +σa and v∗ =τ +σa.
p p h p
6
3.2 Equivalent reformulations at the continuous level
We summarize here additional equivalence results on the continuous-level minimizations appearing in the
right-hand sides of (3.2) and (3.5). Let us first set
H1(ω ):={v ∈H1(ω ); (v,1) =0},
∗ a a ωa
and let us define the following subspace of H(curl,ω ):
a
H (curl,ω ):={v ∈H(curl,ω ); (v,∇φ) =0, ∀φ∈H1(ω )}.
∗ a a ωa ∗ a
WefirstshowthattheH1(ω )-minimizationofCorollary3.1isequivalenttoevaluatingadualH(curl)-
0 a
norm of a suitable linear form defined from the data r , and consequently to evaluating the energy norm
F
of its H (curl,ω )-lifting.
∗ a
Corollary 3.5 (H(curl) form of the H1-minimization). Let the assumptions of Corollary 3.1 hold. Let
ra ∈H (curl,ω ) solve
∗ a
(cid:88)
(∇×ra,∇×v) =− (r n ,∇×v) ∀v ∈H (curl,ω ). (3.8)
ωa F F F ∗ a
F∈Fi
a
Then
 
 (cid:88) 
min (cid:107)∇ (τ −v)(cid:107) =(cid:107)∇×ra(cid:107) = max (r n ,∇×v) . (3.9)
v∈H01(ωa) T p ωa ωa v(cid:107)∈∇H×(vc(cid:107)uωral,ω=a1)F∈Fai F F F
Proof. Since (∇sa−∇ τ ,∇v) =0 for all v ∈H1(ω ), a distributional argument implies that the vector
T p ωa 0 a
field ∇sa−∇ τ is divergence-free in ω . The boundary ∂ω being connected, we infer that there is ra ∈
T p a a
H(curl,ω ) such that ∇sa−∇ τ =∇×ra, and without loss of generality, we can take ra ∈H (curl,ω )
a T p ∗ a
since H(curl,ω ) = H (curl,ω )⊕∇H1(ω ) (the sum being L2-orthogonal) and fields in ∇H1(ω ) are
a ∗ a ∗ a ∗ a
curl-free. We now observe that we have
(∇×ra,∇×v) =−(∇ τ ,∇×v) ∀v ∈H (curl,ω ). (3.10)
ωa T p ωa ∗ a
Indeed, we have, for any H (curl,ω ),
∗ a
(∇×ra,∇×v) =(∇sa−∇ τ ,∇×v) =−(∇ τ ,∇×v) ,
ωa T p ωa T p ωa
sincesa ∈H1(ω ). Now,elementwiseGreenformulausingthedefinitionofτ showsthat(∇ τ ,∇×v) =
(cid:80) 0 a p T p ωa
(r n ,∇×v) , so that (3.8) follows. Finally,
F∈Fi F F F
a
min (cid:107)∇ (τ −v)(cid:107) =(cid:107)∇ (τ −sa)(cid:107) =(cid:107)∇×ra(cid:107) = max (∇ τ ,∇×v) ,
v∈H01(ωa) T p ωa T p ωa ωa v∈H(curl,ωa) T p ωa
(cid:107)∇×v(cid:107)ωa=1
using (3.10) and noting that any function v ∈ ∇H1(ω ) is automatically excluded from the maximization
∗ a
set by the constraint (cid:107)∇×v(cid:107) =1.
ωa
Let us now show that the constrained H (div,ω )-minimization of Corollary 3.3 is equivalent to eval-
0 a
uating a dual H1-norm of a suitable linear form defined from the data r and r and consequently to
K F
evaluating the energy norm of its H1(ω )-lifting.
∗ a
Corollary 3.6 (H1 form of the H(div)-minimization). Let the assumptions of Corollary 3.3 hold. Let
ra ∈H1(ω ) solve
∗ a
(cid:88) (cid:88)
(∇ra,∇v) = (r ,v) − (r ,v) ∀v ∈H1(ω ). (3.11)
ωa K K F F ∗ a
K∈Ta F∈Fa
Then
(cid:40) (cid:41)
(cid:88) (cid:88)
min (cid:107)τ +v(cid:107) =(cid:107)∇ra(cid:107) = max (r ,v) − (r ,v) . (3.12)
∇·v|K=vr∈KH−0∇(Tdi·vτ,pω|Ka)∀K∈Ta p ωa ωa (cid:107)v∇∈Hv(cid:107)1ω(aω=a)1 K∈Ta K K F∈Fa F F
7
Proof. Elementwise Green formula combined with the definition of τ gives
p
(cid:88) (cid:88) (cid:88) (cid:88)
(r ,v) − (r ,v) = (r ,v) − (τ ·n ,v)
K K F F K K p K ∂K
K∈Ta F∈Fa K∈Ta K∈Ta
(cid:88) (cid:88)
= (r −∇ ·τ ,v) − (τ ,∇v) ,
K T p K p K
K∈Ta K∈Ta
and we immediately see that (3.11) is the primal formulation of (3.7). As both formulations are equivalent,
σa = −∇ra−τ , cf. [19, Remark 3.15]. The equality (3.12) follows immediately from (3.11), writing the
p
maximum first for all v ∈H1(ω ) with (cid:107)∇v(cid:107) =1 and then noting that any function v constant on ω is
∗ a ωa a
automatically excluded from the maximization set by the constraint (cid:107)∇v(cid:107) =1.
ωa
Corollaries3.5and3.6allowustodrawinsightfullinkswiththeliterature. Ontheonehand,Corollary3.5
explains how the right-hand side in (3.2) links to the continuous minimization used in [19, Lemma 3.13].
Therein, in two space dimensions, the field (cid:60)⊥(∇ τ ) has been employed in the definition of the function
T p
r by formulas (3.19) and (3.32), where (cid:60)⊥ =(cid:0)0−1(cid:1) is the rotation by π; then (cid:107)∇r (cid:107) of [19] equals the
a 1 0 2 a ωa
present min (cid:107)∇ (τ −v)(cid:107) , and, in particular, we have
v∈H01(ωa) T p ωa
min (cid:107)∇ (τ −v)(cid:107) = max (cid:110)−(cid:0)(cid:60)⊥(∇ τ ),∇v(cid:1) (cid:111).
v∈H01(ωa) T p ωa v∈H1(ωa) T p ωa
(cid:107)∇v(cid:107)ωa=1
On the other hand, the maximization form in Corollary 3.6 has been used previously in [1, Theorem 7]
and [19, Lemma 3.12 and Corollary 3.16].
3.3 Boundary vertices
In this section, we reformulate Theorems 2.4 and 2.5 as best-approximation results on discontinuous piece-
wise polynomial data. The proofs are omitted since they are similar to the previous ones. In view of
application to a posteriori error analysis of model problems with non-homogeneous boundary conditions, it
is convenient to introduce some additional boundary data denoted by uD and σN in the H1 and H(div)
a a
settings, respectively.
Corollary 3.7 (H1 best-approximation). Let the assumptions of Theorem 2.4 hold. Let uD ∈ P (FD)∩
a p a
C0(∂ωD) with ∂ωD = ∪ F. Consider any τ ∈ P (T ) so that τ | = 0 ∀F ∈ Fb, τ | −uD| = r
a a F∈FD p p a p F a p F a F F
a
∀F ∈FD, and [[τ ]] =r ∀F ∈Fi. Then the following holds:
a p F F a
min (cid:107)∇ (τ −v )(cid:107) ≤C min (cid:107)∇ (τ −v )(cid:107) (3.13)
vp∈Pp(Ta)∩H1(ωa) T p p ωa st v∈H1(ωa) T p p ωa
vp|F=0∀F∈Fab v|F=0∀F∈Fab
vp|F=uDa|F ∀F∈FaD v|F=uDa|F ∀F∈FaD
Corollary 3.8 (H(div)best-approximation). Let the assumptions of Theorem 2.5 hold. Let σN ∈P (FN).
a p a
Consider any τ ∈RTN (T ) so that τ ·n =r ∀F ∈Fb, τ ·n +σN =r ∀F ∈FN, and [[τ ]]·n =r
p p a p F F a p F a F a p F F
∀F ∈Fi. Then the following holds:
a
min (cid:107)τ +v (cid:107) ≤C min (cid:107)τ +v(cid:107) . (3.14)
vp∈RTNp(Ta)∩H(div,ωa) p p ωa st v∈H(div,ωa) p ωa
vp·nF=0∀F∈Fab v·nF=0∀F∈Fab
vp·nF=σaN|F ∀F∈FaN v·nF=σaN|F ∀F∈FaN
∇·vp|K=rK−∇T·τp|K ∀K∈Ta ∇·v|K=rK−∇T·τp|K ∀K∈Ta
4 Application to a posteriori error analysis
We show in this section the application of our results to a posteriori error analysis. For this purpose, let
Ω ⊂ R3 be a polyhedral Lipschitz domain (open, bounded, and connected set). Let T be a matching
h
tetrahedralmeshofΩ,shape-regularwithparameterγ >0thatboundstheratioofanyelementdiameter
Th
to the diameter of its largest inscribed ball. All faces of the mesh are collected in the set F , with faces
h
8
lying on the boundary of Ω forming two disjoint sets FN and FD covering two subdomains Γ and Γ of
h h N D
∂Ω. Consider the Laplace problem
−∆u=f in Ω, (4.1a)
u=u on Γ , (4.1b)
D D
−∇u·n =σ on Γ , (4.1c)
Ω N N
where, for simplicity, f ∈ P (T ), u ∈ P (FD) ∩ C0(Γ ), and σ ∈ P (FN), for a polynomial
p(cid:48)−1 h D p(cid:48) h D N p(cid:48)−1 h
degree p(cid:48) ≥1. If |Γ |=0, we need to additionally suppose the Neumann compatibility condition (f,1) =
D Ω
(σ ,1) . Weak solution of problem (4.1) is a function u∈H1(Ω) such that u| =u and such that
N ∂Ω ΓD D
(∇u,∇v) =(f,v) −(σ ,v) ∀v ∈H1(Ω) such that v| =0. (4.2)
Ω Ω N ΓN ΓD
For more general data f, u , and σ , data oscillation terms arise in the a posteriori error analysis, see [15]
D N
and references therein for details.
Let u ∈ P (T ) be an approximate solution to the problem (4.1); u can be primal-nonconforming in
h p(cid:48) h h
the sense that u (cid:54)∈ H1(Ω) and u | (cid:54)= u , as well as dual-nonconforming in the sense that −∇ u (cid:54)∈
h h ΓD D T h
H(div,Ω), ∇·(−∇ u )(cid:54)=f, and (−∇ u ·n )| (cid:54)=σ . The results of this paper have a direct application
T h T h Ω ΓN N
to a posteriori error analysis since they allow us to construct two central objects leading to guaranteed
reliability and p-robust local efficiency. The first is a so-called potential reconstruction s ∈ P (T )∩
h p(cid:48)+1 h
H1(Ω),equaltou onΓ . Thesecondoneisaso-calledequilibratedfluxreconstructionσ ∈RTN (T )∩
D D h p(cid:48) h
H(div,Ω), such that ∇·σ = f in Ω and σ ·n| = σ on Γ . When u (cid:54)∈ H1(Ω) or u | (cid:54)= u so that
h h ΓN N N h h ΓD D
the potential reconstruction s is necessary, we additionally need to suppose that Assumption C.1 below
h
on boundary patches enumeration is satisfied; this may request some conditions on the structure of the
boundary subsets Γ and Γ , like that they are locally connected (no checkerboard patters).
D N
Collect all the mesh vertices in the set V , and for any mesh vertex a ∈ V , let the patch T ⊂ T be
h h a h
givenbytheelementsinT havingaasvertex, whereasω ⊂ΩisthecorrespondingopensubdomainofΩ.
h a
Let ψ be the “hat” function associated with the vertex a: this is a continuous function, piecewise affine
a
withrespecttothemeshT ,whichtakesthevalue1atthevertexaand0attheothervertices. Itssupport
h
isthusthepatchsubdomainω . WealsosplitthevertexsetasV =Vi ∪Vb,whereVi containsallinterior
a h h h h
vertices and Vb all boundary vertices. The faces of the elements in the interior patches T (i.e., associated
h a
with an interior vertex a) are collected in F = Fi ∪Fb, in conformity with Section 2.1. For a boundary
a a a
vertex a∈Vb, the split is F =Fi ∪Fb∪FD∪FN, as in Section 2.4, where Fi collects the interior faces
h a a a a a a
containing the vertex a and shared by two distinct elements in T , Fb the patch boundary faces from ∂ω
a a a
but not sharing the point a, FD the Dirichlet boundary faces FD from ∂ω ∩Γ and sharing the point
a a a D
a, and FN the Neumann boundary faces from ∂ω ∩Γ and sharing the point a. To have a more unified
a a N
formalism between interior and boundary vertices, we conventionally define FD and FN to be empty sets
a a
for all a∈Vi.
h
We define the potential reconstruction following [6] and [19, Construction 3.8 and Remark 3.10] as
s :=(cid:80) sa, where sa is the discrete minimizer of Corollary 3.1 given by (3.3) for interior vertices and
h a∈Vh h h
similarlyasthediscreteminimizerofCorollary3.7forboundaryvertices. Wechoosethepolynomialdegree
p of our theory to be p:=p(cid:48)+1 and we set for all a∈V ,
h
τ :=ψ u in ω , (4.3a)
p a h a
r :=ψ [[u ]] on all F ∈Fi, (4.3b)
F a h F a
r :=0 on all F ∈Fb, (4.3c)
F a
r :=ψ (u −u ) on all F ∈FD, (4.3d)
F a h D a
uD :=ψ u on all F ∈FD. (4.3e)
a a D a
By construction, the polynomial data satisfy the compatibility conditions (2.6) and (2.12a). Similarly,
following[13],[2,1],and[19,Construction3.4andRemark3.7],wedefinetheequilibratedfluxreconstruction
as σ :=(cid:80) σa, where σa is the discrete minimizer of Corollary 3.3 given by (3.6) for interior vertices
h a∈Vh h h
and similarly the discrete minimizer of Corollary 3.8 for boundary vertices, with the polynomial degree
9
p:=p(cid:48). Here we set, for all a∈V ,
h
τ :=ψ ∇ u in ω , (4.4a)
p a T h a
r :=ψ (f +∆ u ) in all K ∈T , (4.4b)
K a T h a
r :=ψ [[∇ u ]]·n on all F ∈Fi, (4.4c)
F a T h F a
r :=0 on all F ∈Fb, (4.4d)
F a
r :=ψ (∇ u ·n +σ ) on all F ∈FN, (4.4e)
F a T h F N a
σN :=ψ σ on all F ∈FN. (4.4f)
a a N a
Here, for all interior vertices and for those boundary vertices which are only shared by Neumann faces (i.e.,
FD =∅), the hat-function orthogonality
a
(∇ u ,∇ψ ) =(f,ψ ) −(σ ,ψ ) (4.5)
T h a ωa a ωa N a ∂ωa∩ΓN
is a necessary condition for the data compatibility conditions (2.9) and (2.14) to hold. This relation does
not hold, for example, for certain discontinuous Galerkin methods; a simple use of the discrete gradient
∇ u from [14, Section 4.3] in place of the broken gradient ∇ u allows to fix this, see [19, 15].
d h T h
The above potential and flux reconstructions lead to the guaranteed a posteriori error estimate
(cid:107)∇ (u−u )(cid:107)2 ≤ (cid:88) (cid:0)(cid:107)∇ u +σ (cid:107)2 +(cid:107)∇ (u −s )(cid:107)2 (cid:1), (4.6)
T h Ω T h h K T h h K
K∈Th
see [19, Theorem 3.3] or [15, Theorem 3.3] and the references therein. The crucial use of our results is in
the proof of the p-robust local efficiency. Corollary 3.1 for interior vertices and Corollary 3.7 for boundary
vertices immediately give
(cid:107)∇ (ψ u −sa)(cid:107) ≤C min (cid:107)∇ (ψ u −v)(cid:107) ≤C inf (cid:107)∇ (ψ (u −v))(cid:107) .
T a h h ωa st v∈H1(ωa) T a h ωa st v∈H1(ωa) T a h ωa
v|F=0∀F∈Fab v|F=uD ∀F∈FaD
v|F=ψauD ∀F∈FaD
Indeed, the right inequality follows immediately as any function v ∈H1(ω ), equal to u on the faces from
a D
FD, belongs to the minimization set of the middle term above when multiplied by the hat function ψ .
a a
This means that the discrete fully computable estimator (cid:107)∇ (ψ u −sa)(cid:107) is a local lower bound for
T a h h ωa
a ψ -weighted distance to the H1(ω ) space (or an affine subspace if FD (cid:54)= ∅). We now make the weak
a a a
solution u of (4.1) appear in the bound. For a ∈ Vi, let u˜ := u−c , where the constant c is chosen so
h a a
that (u˜,1) = (u ,1) . For a boundary vertex a ∈ Vb such that FD = ∅, let u˜ := u−c , where the
ωa h ωa (cid:80) (cid:80) h a a
constant c is chosen so that (u˜,1) = (u ,1) . In the other situations, we let u˜ := u.
a F∈FN F F∈FN h F
a a
Then, proceeding as in [19, Lemma 3.13 and Section 4.3] while relying on the broken Poincar´e–Friedrichs
inequality [3], we obtain
(cid:107)∇ (ψ (u −u˜))(cid:107) ≤(cid:107)∇ψ (cid:107) (cid:107)u −u˜(cid:107) +(cid:107)ψ (cid:107) (cid:107)∇ (u −u˜)(cid:107)
T a h ωa a ∞,ωa h ωa a ∞,ωa T h ωa
≤(1+C h (cid:107)∇ψ (cid:107) )(cid:107)∇ (u −u)(cid:107)
bPF,ωa ωa a ∞,ωa T h ωa
(cid:40) (cid:41)1/2
(cid:88) (cid:88)
+C h (cid:107)∇ψ (cid:107) h−1(cid:107)Π0[[u ]](cid:107)2 + h−1(cid:107)Π0(u −u )(cid:107)2 .
bPF,ωa ωa a ∞,ωa F F h F F F h D F
F∈Fi F∈FD
a a
In particular, if the mean values of the jumps in u are zero, i.e., ([[u ]],1) = 0 for all the faces F ∈ Fi
h h F h
and (u ,1) =(u ,1) for all the Dirichlet faces F ∈FD, we infer that
h F D F h
(cid:107)∇ (ψ u −sa)(cid:107) ≤C C (cid:107)∇ (u −u)(cid:107) , (4.7)
T a h h ωa st cont,bPF T h ωa
wheretheconstantC :=max {1+C h (cid:107)∇ψ (cid:107) }onlydependsontheshape-regularity
cont,bPF a∈Vh bPF,ωa ωa a ∞,ωa
parameter γ .
Th
Let
H1(ω ):={v ∈H1(ω ); (v,1) =0}, a∈Vi or a∈Vb and FD =∅, (4.8a)
∗ a a ωa h h a
H1(ω ):={v ∈H1(ω ); v =0 on all F ∈FD}, a∈Vb and FD (cid:54)=∅. (4.8b)
∗ a a a h a
10