ENERGY NORM A-POSTERIORI ERROR ESTIMATION FOR
HP-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR
ELLIPTIC PROBLEMS IN THREE DIMENSIONS
LIANG ZHU∗, STEFANO GIANI†, PAUL HOUSTON‡, AND DOMINIK SCH ¨OTZAU §
Abstract. We develop the energy norm a-posteriori error estimation forhp-version discontinu-ous Galerkin (DG) discretizations of elliptic boundary-value problems on 1-irregularly, isotropically refined affine hexahedral meshes in three dimensions. We derive a reliable and efficient indicator for the errors measured in terms of the natural energy norm. The ratio of the efficiency and reliability constants is independent of the local mesh sizes and weakly depending on the polynomial degrees. In our analysis we make use of anhp-version averaging operator in three dimensions, which we explic-itly construct and analyze. We use our error indicator in anhp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples. Our numerical results indicate that exponential rates of convergence are achieved for problems with smooth solutions, as well as for problems with isotropic corner singularities.
1. Introduction. In this paper we develop the energy norm a-posteriori error estimation forhp-adaptive discontinuous Galerkin (DG) discretizations of the follow-ing model diffusion equation in three dimensions:
−∆u=f(x) in Ω⊂R3,
u= 0 on Γ. (1.1)
Here, Ω is a bounded Lipschitz polyhedron inR3 with boundary Γ =∂Ω. The
right-hand sidef(x) is a given function inL2(Ω). The standard weak formulation of (1.1) is to findu∈H1
0(Ω) such that
A(u, v)≡
Z
Ω
∇u· ∇v dx= Z
Ω
f v dx ∀v∈H01(Ω). (1.2)
DG methods are ideally suited for realizinghp-adaptivity for second-order bound-ary-value problems, an advantage that has been noted early on in the recent devel-opment of these methods; see, for example, [6, 11, 17, 25, 26, 30] and the references therein. Indeed, working with discontinuous finite element spaces easily facilitates the use of variable polynomial degrees and local mesh refinement techniques on possibly irregularly refined meshes – the two key ingredients forhp-adaptive algorithms.
The development of energy-norm error estimation for hp-adaptive DG methods for elliptic boundary-value problems was initiated in [16] where a residual-basedhp -version error estimator was derived for regular meshes of triangular and quadrilateral elements on two-dimensional domains. It was verified numerically that the resulting hp-adaptive algorithm achieves exponential rates of convergence for problems with piecewise smooth data. In [21], a similar approach was presented for quasi-linear second-order problems in two dimensions. By using an underlying auxiliary mesh,
∗Mathematics Department, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
†School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK ([email protected]).
‡School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK ([email protected]).
§Mathematics Department, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
it was possible to also analyze the case of irregular meshes. Another technique to deal with irregular meshes was proposed in [32] where hp-version a-posteriori error estimates for two-dimensional convection-diffusion equations were derived that are robust in the P´eclet number of the problem.
In this paper, we extend the two-dimensional analysis presented in [16] to 1-irregularly, isotropically refined affine hexahedral meshes in three space dimensions. We propose an energy norm error estimator which gives rise to global upper and local lower bounds of the error measured in the natural DG energy norm. As in [16], the ratio of these error bounds is independent of the local mesh sizes and weakly depends on the local polynomial degrees. Crucial in our analysis is the use of an averaging operator which allows us to approximate a discontinuous finite element function by a continuous one. Operators of this type were originally introduced in [22] for the energy norm a-posteriori error analysis of DG methods for elliptic problems. The same operators have been employed in the papers [9, 15, 16, 21, 28, 32], both forh -andhp-version DG methods.
Here, we follow the approach of [32] and extend the analysis there to three space dimensions. By doing so, we also obtain an optimal L2-norm error bound for the averaging operator on irregular meshes which is of interest on its own. We use our estimators as error indicators in hp-adaptive computations and present a set of nu-merical experiments. We first test the resulting algorithms for problems with smooth solutions. Then we also show the performance of our method for a problem in the classical Fichera polyhedron, with a solution that has an isotropic singularity at the reentrant corner. In both cases, our numerical results indicate that exponential rates of convergence are achieved with respect to the number of degrees of freedom.
We emphasize that our analysis and techniques of proof are valid only for isotrop-ically refined elements. In light of the hp-version a-priori error analysis for elliptic boundary-value problems presented in [27], anisotropic refinement is essential for re-solving edge and edge-corner singularities with exponential rates of convergence. The extension of our results to anisotropic elements (and anisotropic polynomial spaces) remains an open question and is the subject of current research.
The outline of the rest of this article is as follows. In Section 2, we introduce thehp-adaptive DG discretization of the model problem stated in (1.1). In Section 3, we present our energy norm a-posteriori error estimate and discuss its reliability and local efficiency. The reliability proof shall be presented in Section 4. As an analysis tool, we use a new hp-version averaging operator that is analyzed in Section 5. In Section 6, we present a series of numerical tests that verify the theoretical results. Finally, in Section 7, we end with some concluding remarks.
2. Discontinuous Galerkin discretization of a diffusion problem. In this section, we introduce anhp-version interior penalty DG finite element method for the discretization of (1.1).
2.1. Meshes and traces. Throughout, we assume that the computational do-main Ω can be partitioned into shape-regular and affine sequences of meshesT ={K}
of hexahedra K. Each element K∈ T is the image of the cubeKb = (−1,1)3 under an affine elemental mappingTK :Kb →K. As usual, we denote by hK the diameter
ofK. We store the elemental diameters in the mesh size vectorh={hK : K∈ T }.
For an elementK∈ T, we make use of the following sets of elemental faces: the set F(K) consists of the six elemental faces ofK. We further denote byFB(K) the
elemental faces ofKthat lie on Γ, and byFI(K) the set of interior faces; thereby, we
have thatF(K) =FB(K)∪ FI(K).
In order to be able to deal with irregular meshes, we also need to define the faces of a meshT. We refer toF as an interior mesh face ofT ifF =∂K∩∂K0 for two neighboring elements K, K0 ∈ T whose intersection has a positive surface measure. The set of all interior mesh faces is denoted byFI(T). Analogously, if the intersection
F =∂K∩Γ of the boundary of an elementK∈ T and Γ is of positive surface measure, we refer toF as a boundary mesh face ofT. The set of all boundary mesh faces ofT
is denoted byFB(T) and we setF(T) =FI(T)∪ FB(T). The diameter of a faceF
is denoted byhF.
We allow for 1-irregularly refined meshes T defined as follows. Let K be an element of T and F an elemental face in F(K). Then F may contain at most one hanging node located in the center ofF and at most one hanging node in the middle of each elemental edge ofF. That is, we have thatF is either a mesh face belonging to F(T) or F can be written as F = ∪4
i=1Fi, with four mesh faces Fi ∈ F(T),
i= 1, . . . ,4, of diameterhFi =hF/2, respectively.
Next, let us define the jumps and averages of piecewise smooth functions across faces of the mesh T. To that end, let the interior faceF ∈ FI(T) be shared by two
neighboring elementsKand Ke where the superscript e stands for “exterior”. For a piecewise smooth functionv, we denote by v|F the trace onF taken from inside K,
and by ve|
F the one taken from inside Ke. The average and jump of v across the
faceF are then defined as
{{v}}= 1
2(v|F+v e|
F), [[v]] =v|F nK+v
e|
F nKe.
Here, nK and nKe denote the unit outward normal vectors on the boundary of
el-ements K and Ke, respectively. Similarly, if q is piecewise smooth vector field, its average and (normal) jump acrossF are given by
{{q}}= 1
2 q|F+q e
|F
, [[q]] =q|F·nK+q
e
|F·nKe.
On a boundary faceF ∈ FB(T), we accordingly set{{q}}=q and [[v]] =vn, with n
denoting the unit outward normal vector on Γ. The other trace operators will not be used on boundary faces and are thereby left undefined.
2.2. Finite element spaces. We begin by introducing polynomial spaces on elements and faces. To that end, letK∈ T be an element. We set
Qp(K) ={v : K→R : v◦TK ∈ Qp(Kb)}, (2.1)
withQp(Kb) denoting the set of tensor product polynomials on the reference element
b
K of degree less than or equal topin each coordinate direction onKb. In addition, if F ∈ F(K) is a face ofK and Fb the corresponding one on the reference element Kb, we define
Qp(F) ={v : F →R : v◦TK|F ∈ Qp(Fb)}, (2.2)
whereQp(Fb) denotes the set of tensor product polynomials onFbof degree less than or equal topin each coordinate direction onFb.
To definehp-version finite element spaces, we assign a polynomial degreepK≥1
with each elementK of the meshT. We then introduce the degree vectorp={pK :
K∈ T }. We assume thatpis of bounded local variation, that is, there is a constant %≥1, independent of the meshT sequence under consideration, such that
%−1≤pK/pK0 ≤% (2.3)
for any pair of neighboring elements K, K0 ∈ T. For a mesh face F ∈ F(T), we introduce the face polynomial degreepF by
pF =
(
max{pK, pK0}, ifF =∂K∩∂K0∈ FI(T),
pK, ifF =∂K∩Γ∈ FB(T).
(2.4)
For a partition T of Ω and a polynomial degree vector p on T, we define the hp-version DG finite element space by
Sp(T) ={v∈L2(Ω) : v|K ∈ QpK(K), K ∈ T }. (2.5)
2.3. Interior penalty discretization. We now consider the following interior penalty DG discretization for the numerical approximation of the diffusion prob-lem (1.1): finduhp∈Sp(T) such that
Ahp(uhp, v) =
Z
Ω
f v dx ∀v∈Sp(T). (2.6)
The bilinear formAhp(u, v) is given by
Ahp(u, v) =
X
K∈T Z
K
∇u· ∇v dx− X
F∈F(T) Z
F
{{∇u}} ·[[v]] +{{∇v}} ·[[u]]ds
+ X
F∈F(T) γp2
F
hF
Z
F
[[u]]·[[v]]ds,
where the gradient operator ∇ is defined elementwise. The parameter γ > 0 is the interior penalty parameter. The method in (2.6) is a straightforward extension of the classical (symmetric) interior penalty method introduced in [4, 24] to the context of thehp-version finite element method; see also [5, 17, 30] and the references therein.
Remark 2.1. The stability and well-posedness of the DG method (2.6) follow from the same arguments as those employed in [30, Proposition 3.8] used to analyze the scheme in two-dimensions: there is a threshold parameter γ0 > 0, independent of h and p, such that for γ ≥γ0 the formulation (2.6) possesses a unique solution uhp∈Sp(T).
3. Energy norm a-posteriori error estimates. In this section, we present and discuss our main results.
3.1. Energy norm and residuals. We measure the error in the following en-ergy norm associated with the DG formulation (2.6):
kuk2 E,T =
X
K∈T
k∇uk2
L2(K)+
X
F∈F(T) γp2
F
hF
k[[u]]k2
L2(F). (3.1)
To introduce our energy norm indicator, letuhp ∈Sp(T) be the DG
approxima-tion obtained by (2.6). Moreover, we denote by fhp a piecewise polynomial
approxi-mation inSp(T) of the right-hand sidef. For each elementK∈ T, we introduce the
following local error indicatorηK which is given by the sum of the three terms
η2K =η2R
K+η
2
FK+η
2
JK. (3.2)
The first termηRK is the interior residual defined by
η2R
K=p
−2
K h
2
Kkfhp+ ∆uhpk2L2(K).
The second termηFK is the face residual given by
η2FK= 1 2
X
F∈FI(K)
p−1F hFk[[∇uhp]]k2L2(F).
The last residual ηJK measures the jumps of the approximate solution uhp and is
defined as
ηJK2 =1 2
X
F∈FI(K)
γ2p3
F
hF
k[[uhp]]k2L2(F)+
X
F∈FB(K)
γ2p3
F
hF
k[[uhp]]k2L2(F).
We also introduce the local data approximation term
Θ2K=p−2K h2Kkf −fhpk2L2(K). (3.3)
Summing up the local error indicators, we introduce the global a-posteriori error estimator
η= X
K∈T ηK2
!12
. (3.4)
Similarly, we define the global data approximation term
Θ = X
K∈T Θ2K
!12
. (3.5)
3.2. Reliability. Our first theorem states that, up to a constant and to data approximation, the estimatorηin (3.4) gives rise to a reliable a-posteriori error bound. In this result and in the sequel, we shall use the symbols.and&to denote bounds that are valid up to positive constants independent ofhandp.
Theorem 3.1. Let u be the solution of (1.1) anduhp ∈Sp(T)its DG
approxi-mation obtained by (2.6) with γ ≥γ0. Let the error estimator η be defined by (3.4) and the data approximation error Θ by (3.5). Then we have the a-posteriori error bound
ku−uhpkE,T . η+ Θ.
The detailed proof of Theorem 3.1 will be given in Section 4. It is similar to the one given in [32] for two-dimensional convection-diffusion equations. Crucial in our proof, however, is the use of a three-dimensional averaging operator, whosehp-version approximation properties will be introduced in Theorem 4.1 and proven in Section 5.
Remark 3.2. As for the two-dimensional cases analyzed in [21, 32], the
penaliza-tion of the jump terms in the interior penalty formAhp(u, v)is of the orderp2Fh
−1
F on
each face, while the corresponding weight in the jump residualηJK is of the different
orderp3
Fh
−1
F . This suboptimality with respect to the powers of pF is due to the
possi-ble presence of hanging nodes in the underlying mesh T. Indeed, on meshes without irregular nodes, Theorem 3.1 holds true with the following (optimal) jump residual:
b ηJK2 =1
2 X
F∈FI(K)
γ2p2
F
hF
k[[uhp]]k2L2(F)+
X
F∈FB(K)
γ2p2
F
hF
k[[uhp]]k2L2(F);
see also Remark 4.3 below. The associated estimatorηbis then given by
b
η2= X
K∈T b
ηK2 with bηK2 =ηR2
K+η
2
FK+bη 2
JK. (3.6)
Our numerical experiments in Section 6 indicate that the indicators η and ηb yield almost identical results on 1-irregular meshes.
3.3. Efficiency. In our next result, we present a local lower bound for the error measured in the energy norm. As for many residual-based hp-version a-posteriori error estimates, reliability and efficiency bounds, which are uniform in p, are not readily available; cf. [16, 23] and the references therein. We thus restrict ourselves to stating a weaklyp-dependent local lower bound forηK defined in (3.2). We note that
our numerical results indicate that exponential rates of convergence are obtained for both smooth and non-smooth solutions; in this context, the p-suboptimality is less relevant.
For an elementK∈ T, we introduce the patch of neighboring elements as
wK ={K0∈ T : ∂K0∩∂K ∈ F(T)}. (3.7)
The local energy norm overwK is defined by
kuk2 E,wK=
X
K0∈wK
k∇uk2
L2(K0)+ X
F∈F(K) γp2
F
hF
k[[u]]k2
L2(F). (3.8)
Similarly, we set
ΘwK=
X
K0∈w
K
Θ2K0 !1/2
. (3.9)
With this notation the following result holds.
Theorem 3.3. Let u be the solution of (1.1) and uhp ∈ Sp(T) its DG
approx-imation obtained by (2.6) with γ ≥γ0. Let the local error estimatorsηK be defined
by (3.2) and the local data approximation errorΘK by (3.3). Then, for anyδ∈(0,12),
we have the local upper bound
ηK.pδK+1ku−uhpkE,wK+p
2δ+1 2
K ΘwK.
The proof of Theorem 3.3 follows in an analogous manner to the proofs of efficiency derived in [16, 21, 23, 32] for two-dimensional problems. Thereby, for the sake of brevity, we omit the proof of Theorem 3.3 and refer the reader to [31] for details.
Remark 3.4. As in the two-dimensional case considered in [16], our error
es-timator can be extended to the Poisson problem with the inhomogeneous boundary conditionu=g onΓforg∈H1/2(Γ). In this case, the local error indicatorsη
K have
to be modified by redefining the jump estimatorsηJK as
η2JK =1 2
X
F∈FI(K)
γ2p3
F
hF
k[[uhp]]k2L2(F)+
X
F∈FB(K)
γ2p3
F
hF
kuhp−ghpk2L2(F),
where ghp is a polynomial approximation of the boundary datum g. In this setting,
Theorem 3.1 and Theorem 3.3 still hold with the inclusion of an additional data-oscillation term on the boundary; see [16] for details.
4. Proof of Theorem 3.1. In this section, we present the proof of Theorem 3.1. To this end, we proceed in the following steps.
4.1. Edges and nodes. We begin by introducing the following sets associated with nodes. We denote byN(K) the set of eight vertices of an elementK ∈ T, and byN(F) the set of the four vertices of a faceF in F(T). We then introduce the set of all mesh nodes by
N(T) = [
K∈T
N(K).
We writeN(T) =NI(T)∪ NB(T), whereNI(T) andNB(T) are the sets of interior
and boundary mesh nodes, respectively.
Next, we introduce the following sets of edges. We denote E(K) the set of the twelve elemental edges of an element K∈ T, and by E(F) the set of the four edges of a mesh faceF ∈ F(T). We callEan edge of the meshT ifE =∂F∩∂F0is a line segment given by the intersection of two facesF, F0 in F(T) in such a way that its midpoint is not a mesh node ofN(T). We denote byE(T) the set of all mesh edges ofT. The length of an edgeE is denoted byhE.
4.2. Auxiliary meshes. As in [32], we shall make use of an auxiliary 1-irregular meshTe of affine hexahedra. We construct the auxiliary meshTe from the meshT as follows. LetK ∈ T. If all twelve elemental edges are edges of the meshT, that is, if E(K) ⊆ E(T) (in this case, we have also F(K)⊆ F(T)), we leave K untouched. Otherwise, at least one of the elemental edges of K contains a hanging node. In this case, we replaceKby the eight hexahedral elements obtained from bisecting the elemental edges of K; see [32] for an illustration of the analogous construction in two dimensions. Clearly, the mesh Te is a refinement of T; it is also shape-regular and 1-irregular. More importantly, the hanging nodes of T are not hanging nodes ofTe anymore. In the following, we shall writeR(K) for the elements inTe that are insideK. That is, ifK is unrefined, we haveR(K) ={K}. OtherwiseR(K) consists of eight newly created elements.
We denote by FR(T) the set of mesh faces in F(T) that have been refined in
the construction of Te. Furthermore, we denote by FH(T) the set of faces inFR(T)
that have at least one hanging node ofT on their edges, and byFN(T) the ones that
have no hanging nodes ofT on their edges. The sets of nodes, edges and faces of the auxiliary meshTe are denoted byN(Te),E(Te) andF(Te), respectively; these sets are defined in an analogous manner to the corresponding sets introduced for the meshT.
We then define the following subsets ofN(Te),E(Te) andF(Te):
NA(Te) ={νe∈ N(Te) : ∃K∈ T such thateν is inside K},
EA(Te) ={Ee∈ E(Te) : ∃K∈ T such thatEe is insideK},
FA(Te) ={Fe∈ F(Te) : ∃K∈ T such thatFeis inside K}.
We then introduce the following auxiliary DG finite element space on the meshTe:
S
ep(Te) ={v∈L
2(Ω) : v|
e
K◦TKe ∈ QpKf
(Kb), Ke ∈T }e ,
where the auxiliary polynomial degree vectorepis defined byp
e
K =pK forKe ∈ R(K) andT
e
Kis the affine mapping fromKb ontoKe. We clearly have the following inclusion:
Sp(T)⊆Sep(Te). (4.1)
In analogy with (3.1), the energy norm associated withTe is defined by
kuk2 E,Te
= X
e
K∈Te
k∇uk2
L2(Ke)
+ X
e
F∈F(Te)
γp2
e
F
h
e
F
k[[u]]k2
L2(Fe)
, (4.2)
where the auxiliary face polynomial degreesp
e
F for the jump terms overTe are defined as in (2.4), but using the auxiliary degreesp
e
K.
4.3. Averaging operator. Our analysis is based on an hp-version averaging operator that allows us to approximate discontinuous functions by continuous ones. Analogous operators are used in thehp-version approaches presented in [9, 16, 21, 32]. For the h-version of the DG method, we also refer the reader to [13, 22] and the references therein. To state our result, letSc
e
p(Te) be the conforming subspace ofS
e
p(Te) given by
Sc
ep(Te) =Sep(Te)∩H
1
0(Ω). (4.3)
Theorem 4.1. There exists an averaging operator Ihp : Sp(T) → Sc
e
p(Te) that satisfies
X
e
K∈Te
k∇(v−Ihpv)k2L2(Ke).
X
F∈F(T)
p2Fh−1F k[[v]]k2
L2(F), (4.4)
X
e
K∈Te
kv−Ihpvk2L2(Ke).
X
F∈F(T)
p−2F hFk[[v]]k2L2(F). (4.5)
The explicit construction of Ihp and the detailed proof of properties (4.4)–(4.5) are
presented in Section 5.
Remark 4.2. The result in Theorem 4.1 generalizes severalhp-version approxi-mation results of the same type to three dimensions. The analyses in [16, 21] showed theH1-norm estimate (4.4) on two-dimensional regular and irregular meshes, respec-tively. In [9, Lemma 3.2], both estimates in (4.4) and (4.5) were proven for regular two-dimensional meshes and a fixed polynomial degree. In [32], these results have been extended to two-dimensional 1-irregular meshes and variable polynomial degrees.
Remark 4.3. We emphasize that for partitions with no irregular nodes, the
auxiliary mesh Te coincides withT. In this case, Theorem 4.1 holds true directly on the original meshT.
4.4. Proof of Theorem 3.1. To prove Theorem 3.1, we follow [16, 28] and decompose the DG solutionuhp into a conforming part and a remainder:
uhp =uchp+u r hp,
whereuc
hp=Ihpuhp ∈Sc
e
p(Te)⊂H01(Ω) is defined using the averaging operatorIhp in
Theorem 4.1. The remainderur
hp is given byu r
hp=uhp−uchp ∈Sep(Te). Analogously
to [32, Lemma 4.4], one can show that
ku−uhpkE,T .ku−uhpkE,Te.
Therefore, by the triangle inequality,
ku−uhpkE,T .ku−uchpkE,Te+ku
r hpkE,Te.
Finally, since u−uc
hp ∈ H
1
0(Ω), we have ku−uchpkE,T = ku−uchpkE,Te. As the
starting point of our proof, we thus obtain the following inequality:
ku−uhpkE,T .ku−uchpkE,T +kurhpkE,Te. (4.6)
We first show thatkurhpkE,
e
T in (4.6) can be bounded by the error estimatorη.
Lemma 4.4. Under the foregoing assumptions, the following upper bound holds
kurhpkE,Te . η.
Proof. Recall from (4.2) that
kurhpk2 E,Te
= X
e
K∈Te
k∇urhpk2
L2(Ke)+
X
e
F∈F(Te)
γp2
e
F
h
e
F
k[[urhp]]k2
L2(Fe).
Since uhp ∈ Sp(T) and [[urhp]]|F = [[uhp]]|F for all F ∈ F(Te), an argument similar to [32, Lemma 4.3] allows us to bound the jump terms by
X
e
F∈F(Te)
γp2
e
F
h
e
F
k[[urhp]]k2L2(Fe).γ
−1 X
F∈F(T) γ2p2F
hF
k[[uhp]]k2L2(F).γ−1
X
K∈T η2JK,
where we have also used the fact that pF ≥ 1. To bound the volume terms, we
apply Theorem 4.1 and the last bound in the previous argument. This results in the estimate
X
e
K∈Te
k∇urhpk2L2(Ke).γ
−2 X
F∈F(T) γ2p2F
hF
k[[uhp]]k2L2(F).γ−2
X
K∈T ηJ2K.
This completes the proof. To boundku−uc
hpkE,T in (4.6), we shall make use of the following two auxiliary forms:
Dhp(u, v) =
X
K∈T Z
K
∇u· ∇v dx+ X
F∈F(T) γp2F
hF
Z
F
[[u]]·[[v]]ds,
Khp(u, v) =−
X
F∈F(T) Z
F
{{∇u}} ·[[v]]ds− X
F∈F(T) Z
F
{{∇v}} ·[[u]]ds.
The form Dhp(u, v) is well-defined for u, v ∈ Sp(T) +H01(Ω), whereas Khp(u, v) is
only well-defined for discrete functionsu, v∈Sp(T). Furthermore, we have
A(u, v) =Dhp(u, v) ∀u, v∈H01(Ω), (4.7)
as well as
Ahp(u, v) =Dhp(u, v) +Khp(u, v) ∀u, v∈Sp(T). (4.8)
We also recall the standardhp-version approximation result from [21, Lemma 3.7]: For anyv∈H1
0(Ω), there exists a functionvhp∈Sp(T) such that
p2Kh−2K kv−vhpk2L2(K).k∇vk2L2(K),
k∇(v−vhp)k2L2(K).k∇vk2L2(K),
pKh−1K kv−vhpk2L2(∂K).k∇vk2L2(K),
(4.9)
for any elementK∈ T.
Next, we prove the following auxiliary estimate.
Lemma 4.5. For any v∈H01(Ω), we have
Z
Ω
f(v−vhp)dx−Dhp(uhp, v−vhp) +Khp(uhp, vhp).(η+ Θ)kvkE,T.
Here,vhp∈Sp(T)is thehp-version approximation ofv defined in (4.9).
Proof. For notational convenience, let us set
T = Z
Ω
f(v−vhp)dx−Dhp(uhp, v−vhp) +Khp(uhp, vhp).
By writing out the formsDhp and Khp, integrating by parts the volume terms and
manipulating the resulting expressions, we readily obtain
T = X
K∈T Z
K
(f+ ∆uhp)(v−vhp)dx−
X
F∈F(T) γp2F
hF
Z
F
[[uhp]]·[[v−vhp]]ds
− X
F∈FI(T)
Z
F
[[∇uhp]]{{v−vhp}}ds−
X
F∈F(T) Z
F
{{∇vhp}} ·[[uhp]]ds
≡ T1+T2+T3+T4.
To bound termT1, we first add and subtract the approximationfhp tof:
T1= X
K∈T Z
K
(fhp+ ∆uhp)(v−vhp)dx+
X
K∈T Z
K
(f −fhp)(v−vhp)dx.
Using the Cauchy-Schwarz inequality and the approximation properties (4.9) shows that
T1. X
K∈T
η2RK+ Θ2K
1 2 X
K∈T
p2Kh−2K kv−vhpk2L2(K)
12
. X
K∈T
η2RK+ Θ
2
K
1 2
kvkE,T.
For termT2, we again exploit the Cauchy-Schwarz inequality to conclude that
T2≤
X
F∈F(T)
γ2p3Fh−1F k[[uhp]]k2L2(F)
12 X
F∈F(T)
pFh−1F k[[v−vhp]]k2L2(F) 12
.
Thus, by the shape-regularity of the meshes, the bounded variation property (2.3) of the polynomial degrees and the approximation properties (4.9), we get the bound
T2. X
K∈T η2JK
1 2
kvkE,T.
Similarly, termT3 can be bounded by
T3≤
X
F∈FI(T)
p−1F hFk[[∇uhp]]k2L2(F)
12 X
F∈FI(T)
pFh−1F k{{v−vhp}}k2L2(F)
12
. X
K∈T ηF2K
12
kvkE,T.
Finally, for term T4, we use the Cauchy-Schwarz inequality, the shape-regularity of the meshes, and the bounded variation property (2.3) of the polynomial degrees, to obtain
T4.γ−1
X
F∈F(T)
γ2p2Fh−1F k[[uhp]]k2L2(F)
12 X
K∈T
p−2K hKk∇vhpk2L2(∂K)
12
.
¿From the standardhp-version inverse trace inequality, see [29], we conclude that
T4.γ−1 X
K∈T η2JK
1 2 X
K∈T
k∇vhpk2L2(K)
12
.
¿From the approximation properties in (4.9) it follows that
X
K∈T
k∇vhpk2L2(K).
X
K∈T
k∇(v−vhp)k2L2(K)+
X
K∈T
k∇vk2L2(K).kvk
2 E,T.
Hence,
T4.γ−1 X
K∈T ηJK2
1 2
kvkE,T.
The above bounds for termsT1,T2,T3, andT4 now imply the assertion. We are now ready to boundku−uc
hpkE,T in (4.6).
Lemma 4.6. Under the foregoing assumptions, the following upper bound holds
ku−uchpkE,T . η+ Θ.
Proof. Sinceu−uchp∈H01(Ω), we have that
ku−uchpkE,T =
A(u−uc hp, v)
kvkE,T
, (4.10)
where v = u−uhp. To bound the right-hand side of (4.10), we note that, by (1.2)
and property (4.7),
A(u−uchp, v) = Z
Ω
f v dx−A(uchp, v) = Z
Ω
f v dx−Dhp(uchp, v).
One can now readily see that
Dhp(uchp, v) =Dhp(uhp, v) +R,
with
R=− X
e
K∈Te
Z
e
K
∇urhp· ∇v dx.
Here, we have also used that the jumps of v vanish. Furthermore, from the DG method in (2.6) and property (4.8), we have
Z
Ω
f vhpdx=Dhp(uhp, vhp) +Khp(uhp, vhp),
where vhp ∈ Sp(T) is thehp-version approximation of v in (4.9). Combining these
results, we thus arrive at
A(u−uchp, v) = Z
Ω
f(v−vhp)dx−Dhp(uhp, v−vhp) +Khp(uhp, vhp)−R,
The estimate in Lemma 4.5 now yields
|A(u−uchp, v)|.(η+ Θ)kvkE,T +|R|. (4.11)
It remains to bound |R|; from the Cauchy-Schwarz inequality and Lemma 4.4, we readily obtain
|R|.kurhpkE,
e
TkvkE,T .ηkvkE,T. (4.12)
The desired result now follows from (4.10), (4.11) and (4.12).
The proof of Theorem 3.1 readily follows from (4.6), Lemma 4.4 and Lemma 4.6.
5. Proof of Theorem 4.1. In this section, we prove the result of Theorem 4.1.
5.1. Polynomial basis functions. As in the proof of [32, Theorem 4.5], we begin by introducing polynomial basis functions. To that end, let Ib= (−1,1) be the reference interval. We denote byZbp(bI) ={zb
p
0,· · ·,bz
p
p}the Gauss-Lobatto nodes
of order p ≥ 1 on Ib. Recall that bz
p
0 = −1 and bz
p
p = 1. We denote by Zb
p
int(bI) =
{zb1p,· · · ,zbpp−1}the interior Gauss-Lobatto nodes of orderponIb.
Now let E ∈ E(K) be an elemental edge of K ∈ T. The nodes in Zbp can be affinely mapped onto E and we denote by Zp(E) = {zE,p
0 ,· · · , z
E,p
p } the
Gauss-Lobatto nodes of orderponE. The pointsz0E,p andzE,p
p coincide with the two end
points ofE. The setZintp (E) ={zE,p1 ,· · ·, zpE,p−1}denotes the interior Gauss-Lobatto points of orderp. We writePp(E) for the space of all polynomials of degree less than
or equal toponE and define
Pint
p (E) ={q∈ Pp(E) : q(z E,p
0 ) =q(z
E,p p ) = 0},
Ppnod(E) ={q∈ Pp(E) : q(z) = 0, z∈ Z p
int(E)}.
By construction, we havePp(E) =Ppint(E)⊕ Ppnod(E).
On the reference square Ib2 = (−1,1)2, we define the tensor-product Gauss-Lobatto nodes of order p by Zbp(bI2) = {zb
p i,j = (zb
p i,bz
p
j)}0≤i,j≤p. These nodes can
be affinely mapped onto an elemental face F ∈ F(K) of K ∈ T and we define
Zp(F) ={zF,p
i,j }0≤i,j≤pto be the Gauss-Lobatto nodes of orderponF. Furthermore,
we writeZintp (F) ={zi,jF,p}1≤i,j≤p−1 for the interior Gauss-Lobatto points onF. We also define
Qint
pF(F) ={q∈ QpF(F) : q= 0 on∂F}.
Similarly, we define the interior Gauss-Lobatto nodes of orderpon the reference elementKb byZb
p
int(Kb) ={zb
p
i,j,k= (bz
p i,bz
p j,bz
p
k)}1≤i,j,k≤p−1. For an elementK∈ T and
a polynomial degreep≥1, we denote its interior Gauss-Lobatto points byZintp (K) =
{zi,j,kK,p}1≤i,j,k≤p−1. Here, the points z K,p
i,j,k are the affine mappings of bz
p
i,j,k onto the
elementK.
Suppose now that we are given edge and face polynomial degrees 1 ≤ pE ≤ p
and 1≤pF ≤p, associated with the elemental edgesE ∈ E(K) and elemental faces
F ∈ F(K). We assume thatpE ≤pF forE ∈ E(F). We shall define basis functions
for polynomialsv∈ Qp(K) with the restriction that
v|E∈ PpE(E), E∈ E(K), v|F ∈ QpF(F), F ∈ F(K). (5.1)
b ν4
b ν7
b ν3
b ν6
b ν8
b ν5
b
ν1 bν2
(a) Numbering of nodes
b E5
b E1
b E2 b
E4
b E6 b
E8 b
E9 Eb7
b E10 b E11
b E3 b
E12
(b) Numbering of edges
b
F6
bx1
bx2
b
x3
b
F1
b
F5
b
F3
b
F2
b
F4
(c) Numbering of faces
Fig. 5.1.Reference elementKb with the numbering of faces, edges and vertices.
As usual, we shall divide the basis functions into interior, face, edge and vertex basis functions. We first consider the reference element K = Kb = (−1,1)3. We denote its faces byFb1, . . .Fb6, its edges by Eb1, . . . ,Eb12 and its vertices bybν1, . . . ,νb8, numbered as in Figure 5.1. Let{ϕbpi}0≤i≤pbe the Lagrange basis functions associated
with the Gauss-Lobatto nodesZbp(bI) onIb. The interior basis functions are then
b
Φinti,j,k,p(xb1,bx2,bx3) =ϕb
p i(xb1)ϕb
p j(xb2)ϕb
p
k(bx3), 1≤i, j, k≤p−1.
Next, we define the face basis functions exemplary for the faceFb1 in Figure 5.1 with face polynomial degreep
b
F1. They are given by
b ΦFb1,pFb1
i,j (xb1,xb2,bx3) =ϕb
p b F1
i (xb1)ϕb
p
0(bx2)ϕb
p b F1
Note thatΦb
b
F1,pFb1
i,j vanishes onFb2throughFb6. The other face basis functions are then defined analogously. To define the edge basis functions, we consider exemplary the edgeEb1 in Figure 5.1 with edge degreepEb1. The edge basis functions forEb1 are
b ΦEb1,pEb1
i (bx1,bx2,bx3) =ϕb
pEb1 i (xb1)ϕb
pFb5
0 (xb2)ϕb
pFb1
0 (xb3), i= 1, . . . , pEb1−1.
Note thatΦb
b
E1,pEb1
i vanishes on all the other edges and on the facesFb2,Fb3,Fb4andFb6. Moreover, it vanishes on the interior nodes{bzFb1,pFb1
i,j }
p b F1−1
i,j=1 and {zb
b
F5,pFb5
i,j }
p b F5−1
i,j=1 of the facesFb1 andFb5, respectively. The other edge basis functions are then defined analo-gously. Finally, we consider the vertexbν1, which is shared by the edgesEb1,Eb4andEb5; see Figure 5.1. The associated vertex basis function is then defined by
b Φbν1
b
K(bx1,bx2,bx3) =ϕb
p b E1
0 (xb1)ϕb
p b E4
0 (xb2)ϕb
p b E5
0 (bx3).
The vertex basis functions associated with the other vertices of Kb can be defined analogously. This completes the definition of the shape functions on the reference elementKb.
For an arbitrary elementK, the basis functions Φ on Kcan be defined from the analogous ones on Kb by the pull-back map TK: Φ(x1, x2, x3) =Φb ◦TK−1(x1, x2, x3), giving rise to shape functions Φν
K, Φ E,pE
i , Φ
F,pF i,j and Φ
int,p
i,j,k onK. Therefore, a
poly-nomialv∈ Qp(K) satisfying (5.1) can be expanded in the following form:
v(x) = X
ν∈N(K)
v(ν) ΦνK(x) +
X
E∈E(K)
pE−1
X
i=1
v(zE,pE
i ) Φ
E,pE i (x)
+ X
F∈F(K)
pF−1 X
i,j=1
cFi,jΦF,pFi,j (x) + X 1≤i,j,k≤p−1
ci,j,kΦinti,j,k,p(x),
with coefficientscF
i,j andci,j,k.
In the sequel, we will make use of the following two estimates for polynomials, which are proven in Lemma 3.1 of [9]; see also [32].
Lemma 5.1. For an element K, we have the following estimates:
(i) Ifv ∈ QpK(K) vanishes at the interior tensor-product Gauss-Lobatto nodes
of K, then there holds
kvk2
L2(K).hKpK−2kvk2L2(∂K).
(ii) If the vertex ν ofK is shared by the elemental edgesEi,Ej andEk, then the
vertex basis functionΦν
K can be bounded by
kΦνKkL2(K).h
3/2
K p
−1
Eip
−1
Ejp
−1
Ek.
(iii) Let the elemental face F be spanned by the two elemental edges Ei and Ej.
Suppose that the vertexν is given by the intersection ofEi andEj. Then the
vertex basisΦν
K can be bounded by
kΦνKkL2(F).hKp−1Eip
−1
Ej.
5.2. Edge extension operators. In this section, we define extension operators over an edgeE. To that end, fix an elementK∈ T. We discuss three cases where we shall employ edge extensions. First, ifE∈ E(K) is an elemental edge ofK without a hanging node, we define the edge extension operatorLE
p by
LEp,K :Pint
p (E)−→ Qp(K), q(x)7−→ p−1 X
i=1
q(ziE,p)ΦE,pi (x). (5.2)
Second, if the edgeE∈ E(K) contains a hanging node located in the middle ofE, thenE=E1∪E2for two mesh edgesE1andE2inE(T). In this case, we partitionK into two parallelepipeds, K =K1∪K2, by connecting the hanging node onE with the midpoint of the edge parallel toE, as illustrated in Figure 5.2. Forq1∈ Ppint(E1) andq2∈ Ppint(E2), we then define the extension operatorLEp,K(q1, q2) by
LEp,K(q1, q2) =LEp,K1 1(q1) +L
E2
p,K2(q2), (5.3)
withLE1
p,K1(·) andL
E2
p,K2(·) given in (5.2).
The third case arises if the edgeE belongs to the space
EF(K) ={E ∈ E(T) : E is insideF} (5.4)
for an elemental faceF ∈ F(K). That is,E ∈ EF(K) is one of the four mesh edges
whose intersection is a hanging node located in the middle of F. This situation is depicted in Figure 5.3. In this case, we partition K = ∪4
i=1Ki into four elements,
as illustrated in Figure 5.3. If E is shared by K1 and K2 and if q ∈ Ppint(E), the
extensionLE
p,K(q) is then defined by
LEp,K(q) =LEp,K1(q) +LEp,K2(q), (5.5)
withLEp,K1 andLEp,K2 given in (5.2) and extended by zero to the other two elements.
u E2 E1
E K2
K1
Fig. 5.2. Case 2: The elemental
edgeE ∈ E(K) has a hanging node located
in its midpoint.
u
E4 E2
E1
E3
K1 K2
K3 K4
Fig. 5.3.Case 3: The mesh edgesEi
be-long toEF(K)for the elemental faceF. The
elementKis then divided into four elements.
By construction, the extension operatorsLE
p,K(q) in (5.2), (5.5) andL E
p,K(q1, q2) in (5.3) are continuous onKand satisfy
LEp,K(q)|E=q, LEp,K(q1, q2)|E1=q1, L
E
Moreover,LE
p,K(q) and LEp,K(q1, q2) both vanish at the interior Gauss-Lobatto nodes inZintp (K), on the other edges ofE(K) and the elemental faces inF(K) not contain-ingE. From [9, Lemma 3.1], we have the following inequalities.
Lemma 5.2. The linear edge extension operators LEp introduced above satisfy
kLEp,K(q)kL2(K).p−2hKkqkL2(E), E∈ E(K),
kLEp,K(q)kL2(K).p−2hKkqkL2(E), E∈ EF(K), F ∈ F(K),
kLEp,K(q1, q2)kL2(K).p−2hK
2 X
i=1
kqikL2(Ei), E∈E1∪E2, E1, E2∈ E(T).
5.3. Face extension operators. Next, we define extension operators over faces. To that end, fix an element K ∈ T and letF ∈ F(K) be an elemental face of K. Again, we shall discuss three cases of face extensions. First, if there is no hanging node ofT located onF (i.e.,F ∈ F(T)∩ F(Te) orF ∈ FN(T)), we defineLFp,K by
LFp,K :Qint
p (F)−→ Qp(K), q(x)7−→ p−1 X
i,j=1
q(zi,jF,p)ΦF,pi,j(x). (5.6)
Second, ifF has a hanging node in its midpoint (i.e.,F /∈ F(T)), we writeF as F =∪4
i=1Fi, for four facesFi ∈ F(T). We then partitionKinto four parallelepipeds,
K=∪4
i=1Ki, as illustrated in Figure 5.4. For polynomials qi ∈ Qpint(Fi),i= 1, . . . ,4,
we define the operatorLF
p,K(q1, q2, q3, q4) by
LFp,K(q1, q2, q3, q4) = 4 X
i=1 LFi
p,Ki(qi), (5.7)
withLEi
p,Ki, i= 1, . . . ,4, given in (5.6).
Third, if F contains a hanging node located on one of its elemental edges (i.e., F ∈ FH(T)), we divideFinto four facesF1, . . . , F4∈ F(Te) and again partitionKinto four parallelepipeds,K=∪4
i=1Ki, as shown in Figure 5.5. We denote byνcthe center
ofF. Ifq∈ Qp(F) withq= 0 on∂F, we define the extension operator LFp,K(q) by
LFp,K(q) = 4 X
i=1
LFip,Ki(q|Fi), (5.8)
where, for 1≤i≤4,
LFi
p,Ki(q|Fi) = p−1 X
k,l=1
q(zk,lFi,p)ΦFi,pk,l + X
E∈E(Fi)
p−1 X
k=1
q(zkE,p)ΦE,pk +q(νc)ΦνKci.
By definition, the face extensions LF
p,K(q) in (5.7), (5.8) and LFp,K(q1, q2, q3, q4) in (5.7) are continuous onKand satisfy
LFp,K(q)|F =q, LFp,K(q1, q2, q3, q4)|Ei =qi, i= 1, . . . ,4.
Moreover,LF
p,K(q) and L F
p,K(q1, q2, q3, q4) both vanish in the interior Gauss-Lobatto nodes inZintp (K) and on the elemental faces ofK not equal toF. From [9, Lemma 3.1], we have the following inequalities.
u
F1 F2
F4 F3
K1
K2
K4 K3
Fig. 5.4. Case 2: Partition ofK
asso-ciated with the partition of faceF.
u
F1 F2
F4 F3
K1
K2
K4 K3
Fig. 5.5. Case 3: Partition ofK
asso-ciated with the partition of faceF.
Lemma 5.3. The linear face extension operatorsLFp,K introduced above satisfy
kLFp,K(q)kL2(K).p−1h
1/2
K kqkL2(F), F ∈ F(T)∩ F(Te)orF ∈ FN(T),
kLFp,K(q)kL2(K).p−1h
1/2
K kqkL2(F), F ∈ FH(T),
kLFp,K(q1, . . . , q4)kL2(K).p−1h1K/2
4 X
i=1
kqikL2(F
i), F =∪
4
i=1Fi, F1, . . . , F4∈ F(T).
5.4. Decomposition of functions in Sp(T). We shall now decompose
func-tions inSp(T), in a similar manner to the construction in [32, Section 5.3]. To this
end, we first define the minimal edge and face degrees. For an edgeE∈ E(T)∪ E(Te) and a faceF ∈ F(T)∪ F(Te), we set
pE= min{p
e
K : Ke ∈ T ∪Te, E∈ E(Ke)}, pF = min{pKe : Ke ∈ T ∪Te, F ∈ F(Ke)}.
(5.9)
Letv∈Sp(T). We denote byvK the restriction of v to an elementK∈ T ∪Te. We decomposev into a nodal, edge, face and interior part, respectively:
v=vnod+vedge+vface+vint, (5.10)
withvnod,vedge,vface andvint inS
e
p(Te) introduced below.
5.4.1. Nodal part. First, we construct the nodal part vnod ∈S
ep(Te) in (5.10).
For each element K ∈ T and Ke ∈ R(K), we will construct the restriction vnod
e
K of
vnod toKe such thatvnod
e
K ∈ QpfK
(Ke) (note thatpK=pKe) and
vnod
e
K |E∈ PpE(E), E ∈ E(Ke), v
nod
e
K |F ∈ PpF(F), F ∈ F(Ke),
withpE andpF given in (5.9). To definevnod
e
K , we distinguish the following two cases.
Case 1: IfR(K) ={K} (i.e., ifK is unrefined), the interpolant vnod
e
K =v
nod
K is
simply defined by
vKnod(x) = X
ν∈N(K)
vK(ν) ΦνK(x). (5.11)
Case 2: IfR(K) consists of eight newly created elements, we definevnod
e
K on each
element Ke ∈ R(K) separately. To do so, fix Ke ∈ R(K). Without loss of generality, we may consider the situation shown in Figure 5.6, where we denote byνei,Eej andFek
the vertices, edges and faces ofKe, respectively, numbered as in Figure 5.1. Similarly, we denote byνi,Ej andFk the vertices, edges and faces ofK, respectively.
In this configuration, notice that we have eν8 ∈ NA(Te), Fe3,Fe4,Fe6 ∈ FA(Te), as well asEe8,Ee11,Ee12∈ EA(Te). Hence, the polynomial degrees are given by
p
e
Fi=pEje =pKe =pK, i∈ {3,4,6}, j∈ {8,11,12}.
Let us now define the value ofvnod
e
K at the nodes located on∂Ke. At the interior nodes
shared byFei andEej fori∈ {3,4,6}andj∈ {8,11,12}, we set
vnod
e
K (z) =vK(z), z∈ {Z p
e Fi
int (Fei)}i∈{3,4,6}∪ {Z
p e Ej
int (Eej)}j∈{8,11,12}. (5.12)
Similarly, we setvnod
e
K (ν) =vK(ν) for the verticesν =νe2andν =eν8.
ν1 =ν2
ν5 ν6
E1
E5 E6
E9
e
ν1 eν2 e E1
e F1 e ν5 eν6
e
E5 Ee6
e E9
Fig. 5.6. The elementKis refined into8elementsKe∈ R(K).
It remains to define vnod
e
K on the nodes located on the facesFe1, Fe2 and Fe5
(ex-cluding the vertex eν2). We only consider Fe1 (the construction for Fe2 and Fe5 is completely analogous); see Figure 5.6. IfFe1∈ F(T), then we haveeν1,νe5,eν6∈ N(T). The four edges Eei ∈ E(Fe1) for i ∈ {1,5,6,9} belong to E(T). For i ∈ {1,5,6,9} andj∈ {1,5,6}, we define
vnod
e
K (z) = 0, z∈ Z
p e F1
int (Fe1)∪ {Z
p e Ei
int (Eei)}Eie∈E(Fe1), (5.13)
vnod
e
K (νej) =vK(νej). (5.14)
Otherwise, if Fe1 ∈ F/ (T), then the large elemental face F1 belongs to FR(T).
Moreover, we have that eitherF1∈ FN(T) orF1∈ FH(T). We distinguish these two
subcases. First, ifF1∈ FN(T), then there is no hanging node ofT located onF1 or any edge ofF1, and we havep
e
F1 =pF1. In this case, we interpolate the values of the
nodal interpolant over the face F1 at the Gauss-Lobatto nodes on Fe1. That is, we
define
vnod
e
K (z) =
X
ν∈N(F1)
vK(ν) ΦνK(z), (5.15)
for allz∈ {ZpEe
int(Ee)}
e
E∈E(Fe1)∪ Z
pFe1
int (Fe1)∪ {νei}i∈{1,5,6}.
Second, ifF1∈ FH(T), thenνe5∈ N/ (T), buteν1andνe6may or may not belong to
N(T). We define the value ofvnod
e
K at the nodes located onFe1for this case as follows.
First, noticing thatp
e
E5=pEe9 =pFe1 =pF1 andνe2∈ N(T), we set
vnod
e
K (z) = 0, z∈ Z pF
1
int (Fe1)∪ Z
pF
1
int (Ee5)∪ Z
pF
1
int (Ee9)∪ {νe5}, (5.16)
Next, we define the values ofvnod
e
K on the nodes of the edgesEe1andEe6, as well as on
the nodesνe1and eν6. We only considereν1 andEe1 (the construction foreν6 and Ee6is completely analogous). Ifeν1∈ N(T) (i.e.,eν1is a hanging node inT), then we define
vnod
e
K (z) = 0, z∈ Z pEe1
int (Ee1), v nod
e
K (eν1) =vK(νe1). (5.17)
If νe1 ∈ N/ (T), then we have E1 ∈ E(T) and ν1 ∈ N(T). In this case, pEe1 =pE1,
and we interpolate the values of the nodal interpolant over the long edgeE1 at the Gauss-Lobatto nodes onEe1. That is, we set
vnod
e
K (z) =vK(ν1) Φ ν1
K(z) +vK(ν2) ΦKν2(z), z∈ Z p
e E1
int (Ee1)∪ {eν1}. (5.18)
With the nodal values ofvnod
e
K constructed in (5.12)-(5.18), we have
vnod
e
K (x) =
X
ν∈N(Ke)
vnod
e
K (ν) Φ ν
e
K(x) +
X
E∈E(Ke)
pE−1 X
i=1
vnod
e
K (z E,pE
i )Φ
E,pE i (x)
+ X
F∈F(Ke)
pF−1 X
i,j=1
vnod
e
K (z F,pF i,j )Φ
F,pF i,j (x)
.
This finishes the construction of the interpolantvnod. Notice thatvnod ∈S
e
p(Te); it is continuous over facesF ∈ FA(Te) and over edges inside facesF ∈ F(T). Moreover, it satisfies
vK(ν)−vKnod(ν) = 0, ν ∈ N(T) located on∂K,
and
vnod
e
K |E∈ P
nod
pE (E), E∈ E(T), Ke ∈weE,
withweE defined by
e
wE={Ke ∈ T ∪Te : E∈ E(Ke)}, ∀E∈ E(T).
5.4.2. Edge part. Second, we construct the edge functionvedge∈S
e
p(Te) in the decomposition (5.10). To do so, fix an element K ∈ T. For an edge E on ∂K, we definevKE by
vEK=
LE
pK,K((vK−v
nod
K )|E), E∈ E(K)∩ E(T),
LEp
K,K((vK−v
nod
K )|E), E∈ EF(K), F ∈ F(K),
LE
pK,K((vK−vKnod)|E1,(vK−v
nod
K )|E2), E=E1∪E2, E1, E2∈ E(T),
with LEpK,K(·) defined for Case 1 in (5.2) or for Case 3 in (5.5), and LEpK,K(·,·) for Case 2 in (5.3), respectively. We then definevedge on each element as:
vKedge(x) = X
E∈E(K)
vKE(x) + X
F∈F(K) X
E∈EF(K)
vEK(x).
5.4.3. Face part. Third, we construct the face functionvface∈S
e
p(Te) in (5.10). Fix an element K ∈ T and let F be an elemental face in F(K). IfF ∈ F(T), we definevKF by
vFK= (
LFp
K,K((vK−v
nod
K −v
edge
K )|F), F /∈ FH(T),
LF
pK,K((vK−v
nod
K −v
edge
K )|F), F ∈ FH(T),
with LF
pK,K(·) defined for Case 1 in (5.6) and for Case 3 in (5.8). Otherwise, there
exists four facesFi∈ F(T),i= 1, . . . ,4, such thatF =∪4i=1{Fi}. We definevKF by
vFK=LFpK,K((vK−vKnod−v
edge
K )|F1, . . . ,(vK−v
nod
K −v
edge
K )|F4),
withLF
pK,K(·,·,·,·) defined for Case 2 in (5.7). We then definev
face elementwise as
vKface(x) = X
F∈F(K) vFK(x).
5.5. Interior part. Finally, the interior functionvint∈S
ep(Te) in (5.10) is simply
obtained by setting on each element
vKint=vK−vKnod−v
edge
K −v
face
K , K∈ T.
Notice thatvintK belongs toH01(K). Hence, we havevint ∈Sc
e
p(Te).
5.6. Proof of Theorem 4.1. In this section, we outline the proof of Theo-rem 4.1. Some of the auxiliary results are postponed to Sections 5.7.1, 5.7.2 and 5.7.3.
Forv ∈Sp(T), we writev =vnod+vedge+vface+vint, according to (5.10). We
shall define the averaging operatorIhpv in four parts:
Ihpv=ϑnod+ϑedge+ϑface+ϑint, (5.19)
with ϑnod, ϑedge, ϑface, ϑint ∈ Sc
e
p(Te). Since vint ∈ Sc
ep
(Te), we simply take ϑint =
vint. Below we further constructϑnod, ϑedge, andϑface such that the following three approximation results hold.
Proposition 5.4.
(i) Nodal approximation: There is a conforming approximation ϑnod ∈ Sc
e
p(Te) that satisfies:
X
e
K∈Te
kvnod−ϑnodk2L2(Ke).
X
F∈F(T) p−2F hF
Z
F
[[vnod]]2 ds,
X
e
K∈Te
k∇(vnod−ϑnod)k2
L2(Ke).
X
F∈F(T) p2Fh−1F
Z
F
[[vnod]]2ds.
(5.20)
(ii) Edge approximation: There is a conforming approximation ϑedge ∈ Sc
e
p(Te) that satisfies:
X
e
K∈Te
kvedge−ϑedgek2
L2(Ke).
X
F∈F(T) p−2F hF
Z
F
([[v]]2+ [[vnod]]2)ds,
X
e
K∈Te
k∇(vedge−ϑedge)k2L2(Ke).
X
F∈F(T) p2Fh−1F
Z
F
([[v]]2+ [[vnod]]2)ds. (5.21)
(iii) Face approximation: There is a conforming approximationϑface∈Sc
e
p(Te)that satisfies:
X
e
K∈Te
kvface−ϑfacek2
L2(Ke).
X
F∈F(T) p−2F hF
Z
F
([[v]]2+ [[vnod]]2)ds,
X
e
K∈Te
k∇(vface−ϑface)k2
L2(Ke).
X
F∈F(T) p2Fh−1F
Z
F
([[v]]2+ [[vnod]]2)ds. (5.22)
By the triangle inequality and Proposition 5.4, we then obtain
X
e
K∈Te
kv−Ihpvk2L2(Ke).
X
F∈F(T)
p−2F hF(k[[v]]k2L2(F)+k[[v
nod]]
k2L2(F)),
X
e
K∈Te
k∇(v−Ihpv)k2L2(Ke).
X
F∈F(T)
p2Fh−1F (k[[v]]k2L2(F)+k[[v
nod]]
k2L2(F)).
Hence, Theorem 4.1 follows if we show that
k[[vnod]]k2L2(F).k[[v]]k
2
L2(F), F ∈ F(T). (5.23)
To prove (5.23), we define the set
NT(F) ={ν ∈ N(T) : ν is located on∂F}, F ∈ F(T).
By the construction ofvnod, the jump overF satisfies
[[vnod]](ν) = [[v]](ν), ν ∈ NT(F).
IfF∈ F(T)∩F(Te) orF ∈ FN(T), then we haveN(F) =NT(F). Lemma 5.1(iii) and the bounded local variation ofpin (2.3) yield
k[[vnod]]kL2(F).
X
ν∈N(F)
|[[vnod(ν)]]|kΦνKkL2(F).p−2F hF max
ν∈NT(F)
|[[vnod]](ν)|,
withK one of the elements of whichF is an elemental face.
Otherwise, we have F ∈ FH(T). In this case, F is divided into four faces
e
Fi ∈ F(Te), i = 1, . . . ,4, and the middle points of the elemental edges of F may or may not belong toN(T). This situation is the same as the one discussed for the two-dimensional case in [32, Section 5.5 (Case 2)]. Thus, proceeding as in the corre-sponding proof of Lemma 5.4 of [32], we obtain from (2.3) and the construction of vnod that
k[[vnod]]kL2(F)=
4 X
i=1
k[[vnod]]kL2(Fie).p
−2
F hF max
ν∈NT(F)
|[[vnod]](ν)|.
Thus, for any faceF ∈ F(T), we have
k[[vnod]]kL2(F).p−2F hF max
ν∈NT(F)
|[[vnod]](ν)|=p−2F hF max ν∈NT(F)
|[[v]](ν)|.
Without loss of generality, we suppose that |[[vnod]](ν)| reaches its maximum at the vertexν1, an end point of an edgeE∈ E(T) which lies on∂F. From [29, Theorem 3.92], [9, Lemma 3.1] and (2.3), we further have the inverse estimate
max
ν∈NT(F)
k[[v]](ν)k=k[[v]](ν1)k.pEh
−1/2
E k[[v]]kL2(E).pF2h−1F k[[v]]kL2(F).
This, together with the bounded local variation of p in (2.3), implies (5.23). To complete the proof of Theorem 4.1, it remains to prove Proposition 5.4, which will now be undertaken in the next section.
5.7. Proof of Proposition 5.4. In this section, we present the proofs of the three approximation results in Proposition 5.4.
5.7.1. Nodal approximation. Let vnod ∈ S
e
p(Te) be the nodal part of v ∈ Sp(T) in the decomposition (5.10). We shall now construct the conforming
approxi-mationϑnod inSc
ep
(Te). For simplicity, we shall omit the superscript “nod” and, in the
sequel, writev forvnod andϑforϑnod. We introduce the sets:
e
w(ν) ={Ke ∈Te : ν ∈ N(Ke)}, wF(ν) ={F ∈ F(T) : ν ∈F}.
FixK∈ T andKe ∈ R(K). We proceed by distinguishing the same two cases as in Subsection 5.4.
Case 1: IfR(K) ={K}, we have K=Ke. Then, any elemental faceFe∈ F(Ke) belongs to F(T) and we have v
e
K|Fe ∈ QpFe
(Fe). Moreover, any elemental edge Ee ∈
E(Ke) belongs toE(T) and v
e
K|Ee ∈ P
nod
p e
E (Ee). For any Gauss-Lobatto nodeν located
on∂Ke, we define the value ofϑ(ν) by
ϑ(ν) =
|we(ν)|−1 X
e
K∈we(ν)
v
e
K(ν), ν ∈ NI(T),
0, otherwise.
(5.24)
Here,|we(ν)|denotes the cardinality of the setwe(ν). Note that we have|we(ν)|= 8 for ν∈ NI(T). Then we defineϑonKe by:
ϑ(x) = X
ν∈N(Ke)
ϑ(ν) Φν
e
K(x). (5.25)
¿From (5.11) and (5.25), we have
kv
e
K−ϑkL2(Ke).
X
ν∈N(Ke)
|v
e
K(ν)−ϑ(ν)| kΦ ν
e
KkL2(Ke). (5.26)
Analogously to [9, Pages 1125-1126], we conclude that
|v
e
K(ν)−ϑ(ν)|.
X
F∈wF(ν)
p2Fh−1F k[[v]]kL2(F). (5.27)
Hence, by combining (5.26), (5.27), Lemma 5.1(ii) and the bounded variation property ofpin (2.3), we obtain
kv
e
K−ϑkL2(Ke).
X
F∈{wF(ν)} ν∈N(Kf)
p−1F h1F/2k[[v]]kL2(F). (5.28)
Case 2:IfR(K) consists of eight elements, we defineϑon each elementKe ∈ R(K) separately, analogously to the construction of the nodal interpolant in Subsection 5.4. Without loss of generality, we may again consider the case illustrated in Figure 5.6. Since the faces Fe3,Fe4,Fe6 belong to FA(Te), the function v is continuous over them. The values ofϑon the face nodesz∈ {ZpKf
int(Fei)}i∈{3,4,6}∪ {Z
p f K
int(Eej)}j∈{8,11,12}and the vertexeν8 are defined byϑ(eν8) =vKe(νe8) and
ϑ(z) =v
e
K(z), z∈ {Z p
f K
int(Fei)}i∈{3,4,6}∪ {Z
p f K
int(Eej)}j∈{8,11,12}. (5.29)
We further define the value ofϑon the vertexνe2by (5.24). It remains to define the values of v
e
K on the nodes located on the faces Fe1, Fe2 andFe5, excluding the vertexνe2. We only considerFe1(the construction forFe2andFe5 is completely analogous); see Figure 5.6. If Fe1 ∈ F(T), then for any Gauss-Lobatto
node onFe1,z∈ Z
p e F1
int (Fe1)∪ {Z
pE
int(E)}E∈E(Fe1)∪ {ν1} ∪ {ν5} ∪ {ν6}, the value ofϑ(z)
is taken as in (5.24).
Otherwise, ifFe1∈ F/ (T), thenF1∈ FR(T) andF1 belongs toFN(T) orFH(T).
We distinguish these two subcases. First, ifF1∈ FN(T), we defineϑ(ν),ν ∈ N(F1),
by (5.24). Then we interpolate the values of the nodal interpolant over the faceF1at the Gauss-Lobatto nodes onFe1. That is, we set
ϑ(z) = X
ν∈N(F1)
ϑ(ν) ΦνK(z), z∈ {Z pEe
int(Ee)}
e
E∈E(Fe1)∪ Z
pFe1
int (Fe1)∪ {νei}i∈{1,5,6}. (5.30)
Second, ifF1 ∈ FH(T), theneν5 ∈ N/ (T), butνe1 andeν6 may or may not belong toN(T). We first define
ϑ(z) = 0, z∈ ZpF1
int (Fe1)∪ Z
pF
1
int (Ee5)∪ Z
pF
1
int (Ee9)∪ {eν5}, (5.31)
Next, we define the values of ϑ on the nodes of the edges Ee1 and Ee6, as well as onνe1andeν6. We only considereν1andEe1 (the definition foreν6andEe6 is completely analogous). If eν1 ∈ N(T) (i.e.,νe1 is a hanging node of T), then we defineϑ(z) for
z∈ ZpEe1
int (Ee1)∪ {νe1} by (5.24). If eν1 ∈ N/ (T), thenE1∈ E(T) and ν1 ∈ N(T). We
define ϑ(ν1) again by (5.24). Recall that ϑ(ν2) = ϑ(νe2) has already been defined. Then, for the nodes onEe1, we set
ϑ(z) =ϑ(ν1)Φν1
K(z) +ϑ(ν2)Φ ν2
K(z), z∈ Z p
e E1
int (Ee1)∪ {eν1}. (5.32)
Now we constructϑonKe by setting
ϑ(x) = X
ν∈N(Ke)
ϑ(ν) Φν
e
K(x) +
X
E∈E(Ke)
pE−1 X
i=1
ϑ(zE,pE
i )Φ
E,pE i (x)
+ X
F∈F(Ke)
pF−1 X
i,j=1
ϑ(zF,pF i,j )Φ
F,pF i,j (x)
.
(5.33)
This completes the construction ofϑ. It can be readily seen thatϑ∈Sc
ep
(Te).
We shall now derive an estimate analogous to (5.28) for Case 2. To do so, we estimate the difference betweenv
e
K andϑonKe as follows:
kv
e
K−ϑkL2(Ke).
X
e
ν∈N(Ke)
kς
e
νkL2(Ke)+
X
e
E∈E(Ke)
kς
e
EkL2(Ke)+
X
e
F∈F(Ke)
kς
e
FkL2(Ke), (5.34)
with
ς
e
ν(x) = vKe(νe)−ϑ(νe)
Φeν e
K(x),
ς
e
E(x) = p e E−1 X i=1 v e
K(z
e
E,pEe i )−ϑ(z
e
E,pEe
i )
ΦE,pe Ee
i (x)
,
ς
e
F(x) = p e F−1 X i,j=1 v e
K(z
e
F ,p e F i,j )−ϑ(z
e F ,p e F i,j ) ΦF ,pe
e F i,j (x)
.
Proceeding as in the two-dimensional proof in [32, Lemma 5.4], we obtain the following estimates. First, we have thatkς
e
νkL2(Ke)= 0 foreν∈ NA(Te) and
kς
e
νkL2(Ke).
X
F∈wF(
e
ν)
p−1F h1F/2k[[v]]kL2(F), eν∈ N(T).
Second, foreν /∈ N(T), we have
kς
e
νkL2(Ke).
X
F∈{wF(ν)} ν∈∂E
p−1F h1F/2k[[v]]kL2(F), ∃E∈ E(K),
e
ν is insideE,
X
F∈{wF(ν)} ν∈N(F ?)
p−1F h1F/2k[[v]]kL2(F), ∃F?∈ F(K),
e
ν insideF?.
Similarly, forς
e
E in (5.34), we have that ςEe= 0 if Ee∈ EA(Te) or ifEe∈ EF
?(K) for a
faceF?∈ F
H(T)∩ F(K). Moreover, if Ee∈ EF?(K) for a faceF?∈ FN(T)∩ F(K),
we have
kς
e
EkL2(Ke).
X
F∈{wF(ν)} ν∈N(F ?)
p−1F h1F/2k[[v]]kL2(F).
For the situation when there exists an edgeE∈ E(T) such thatEe⊆E, we have
kς
e
EkL2(Ke).
X
F∈{wF(ν)} ν∈∂E
p−1F h1F/2k[[v]]kL2(F).
Now we only need to boundkς
e
FkL2(Ke)in (5.34) for any faceFe∈ F(Ke). IfFe∈ F(T) or Fe ∈ FA(Te), by the construction of v and ϑ, we havekς
e
FkL2(Ke) = 0. Otherwise,
there exist a face F ∈ F(K) such thatF ∈ FR(T) andFe is obtained by refining F. Without loss of generality, we may again consider the case illustrated in Figure 5.6, with the facesF andFediscussed beingF1andFe1, respectively. IfF1∈ FH(T), then
kς
e
F1kL2(Ke)= 0. Otherwise,F1∈ FN(T). SinceςFe1 vanishes at all the interior
tensor-product Gauss-Lobatto nodes in Ke and on the faces of Z
p f K
int(Ke) that are different fromFe1, we obtain from Lemma 5.1(i) and the construction ofv andϑthat
kς
e
F1kL2(Ke).p
−1
e
K h
1/2
e
K kςFe1kL2(Fe1)
.p−1
e
K h
1/2
e
K kvKe−ϑkL2(Fe1)+
X
i∈{1,5,6,9}
kς
e
EikL2(Fe1)+
X
j∈{1,2,5,6}
kς
e
νjkL2(Fe 1)
.p−1K h1K/2kv
e
K−ϑkL2(F 1)+p
−1
K h
1/2
K
X
i∈{1,5,6,9}
kς
e
EikL2(Fe1)+
X
j∈{1,2,5,6}
kς
e
νjkL2(Fe1)
≡T1+T2.
Using (5.27), Lemma 5.1(iii) and (2.3), we get
T1.p−1K h
1/2
K
X
ν∈N(F1)
kςνkL2(F1).p−1K h
1/2
K
X
ν∈N(F1)
(|vK(ν)−ϑ(ν)| kΦνKkL2(F1))
. X
F∈{wF(ν)} ν∈N(F1 )
p−1F h1F/2k[[v]]kL2(F).
In an analogous manner to the two-dimensional proof in [32, Lemma 5.4], termT2is bounded by
T2.
X
F∈{wF(ν)} ν∈N(F1 )
p−1F h1F/2k[[v]]kL2(F).
Hence,ς
e
F in (5.34) can be bounded by
kς
e
FkL2(Ke).
X
F∈{wF(ν)} ν∈N(F1 )
p−1F h1F/2k[[v]]kL2(F).
To combine the bounds for ς
e
ν, ςEe and ςFe, we define the set N
?(
e
K) as follows. We start fromN(Ke) and first remove all the vertices belonging toNA(Te). Then, any vertexeν∈ N(Ke) witheν /∈ N(T)∪ NA(Te) is replaced by the vertexν∈ N(K) which lies on the same elemental edge ofKas eν; see [32, Section 5.5]. We also set
F?(
e
K) ={F ∈wF(ν) : ν ∈ N?(
e K)}.
Thus, we have
kv
e
K−ϑkL2(Ke).
X
F∈F?(Ke)
p−1F h1F/2k[[v]]kL2(F). (5.35)
