HP -DGFEM FOR SECOND ORDER ELLIPTIC PROBLEMS IN
POLYHEDRA II: EXPONENTIAL CONVERGENCE ∗
D. SCHÖTZAU† , C. SCHWAB‡ , AND T. P. WIHLER§
SIAM J. Numer. Anal., Vol. 51, No. 4, pp. 2005–2035, 2013
Abstract. The goal of this paper is to establish exponential convergence of hp-version interior
penalty (IP) discontinuous Galerkin (dG) finite element methods for the numerical approximation of
linear second-order elliptic boundary-value problems with homogeneous Dirichlet boundary conditions and piecewise analytic data in three-dimensional polyhedral domains. More precisely, we shall
analyze the convergence of the hp-IP dG methods considered in [23] based on axiparallel σ-geometric
anisotropic meshes and s-linear anisotropic polynomial degree distributions.
1. Introduction. Let Ω ⊂ R3 be an open bounded polyhedron with Lipschitz
boundary Γ = ∂Ω that consists of a finite union of plane faces. We consider the
Dirichlet problem for the diffusion equation
Lu ≡ −∇ · (A∇u) = f
u=0
in Ω,
on Γ = ∂Ω,
(1.1)
(1.2)
where A ∈ R3×3
sym is a symmetric positive definite coefficient matrix which is independent of the space coordinate x. Then, for every f ∈ H −1 (Ω), the boundary-value
problem (1.1)–(1.2) admits a unique solution u ∈ H01 (Ω). If f is analytic in Ω, then u
is analytic in Ω away from the singular parts of the boundary (i.e., away from edges
and corners of Ω).
The hp-version of the finite element method (FEM) for the numerical solution
of elliptic problems was proposed in the
√ mid 80ies by Babuška and his coworkers.
Exponential convergence rates exp(−b N ) with respect to the number of degrees of
freedom N for the hp-version of the FEM in one dimension were shown by Babuška
and Gui in [10] for the model singular solution u(x) = xα − x ∈ H01 (Ω) in Ω = (0, 1),
inspired by exponential convergence results in free-knot, variable order spline interpolation, e.g. [8, 20] and the references there. This result required σ-geometric meshes
with a fixed subdivision ratio σ ∈ (0, 1) (in particular, for σ = 1/2 geometric element
sequences Ωi are obtained by successive element
√ bisection towards x = 0) while the
constant b in the convergence estimate exp(−b N ) depends on the singularity exponent α as
√ well as on σ. Among all σ ∈ (0, 1), the optimal value was shown to be
σopt = ( 2 − 1)2 ≈ 0.17, see [10, Theorem 3.2], provided that the geometric mesh
refinement is combined with nonuniform polynomial degrees pi ≥ 1 in Ωi which are
s-linear, i.e., pi ≃ si, with the optimal slope √
s being sopt = 2(α − 1/2). p
In this case,
the finite element error converges as exp(−b N ) where b = 1.76 . . . × (α − 1/2).
∗ This work was initiated during the workshop “Adaptive numerical methods and simulation of
PDEs”, held from January 21-25, 2008, at the Wolfgang Pauli Institute in Vienna, Austria. This
work was supported in part under the FP7 programme of the EU under grant No. ERC AdG 247277,
by the Swiss National Science foundation under grant No. 200021 126594 and by the Natural Sciences
and Engineering Research Council of Canada (NSERC).
† Mathematics Department, University of British Columbia, Vancouver, BC V6T 1Z2, Canada,
(schoetzau@math.ubc.ca). This author was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).
‡ SAM, ETH Zürich, 8092 Zürich, Switzerland (schwab@math.ethz.ch).
§ Mathematisches Institut, Universität Bern, 3012 Bern, Switzerland (wihler@math.unibe.ch).
1
For the bisected geometric mesh where σ = 1/2 and for linear polynomialpdegree distributions with slope sopt = 0.39 . . . × (α − 1/2), one has b = 1.5632 . . . × p(α − 1/2),
whereas for σ = 1/2 and uniform polynomial degree, b = 1.1054 . . . × (α − 1/2);
see [10, Table 1].
In two
√ dimensions, exponential convergence (i.e., an upper bound of the form
C exp(−b 3 N ) on the error) for the hp-version FEM in polygons was obtained by
Babuška and Guo in the mid 80ies in a series of landmark papers ([3, 13, 14] and
the references therein). Key ingredients in the proof were geometric mesh refinement
towards the singular support S (the set of vertices of Ω) of the solution and nonuniform
elemental polynomial degrees which increase linearly with the elements’ distance to S.
Starting in the 90ies, steps were undertaken to extend the analytic regularity and
the a-priori error analysis of hp-FEM in [3, 13, 14] to polyhedra in R3 in [4, 12, 15,
16]. While these works were devoted to conforming hp-FEM with isotropic elemental
polynomial spaces for second-order elliptic problems, extensions to hp-version mixed
methods and conforming methods for higher-order problems in polygons were obtained
in [11, 21].
Discontinuous Galerkin (dG) FEM emerged in the 70ies as stable discretizations
of first-order transport-dominated problems, and as nonconforming discretizations of
second-order elliptic problems. We refer to [5] for a historical account of the development of these methods. In the 90ies, dG methods were studied within the hp-version
setting for first-order transport and for advection-reaction-diffusion problems in twoand three-dimensional domains (see, e.g., [17]). There, exponential convergence rates
were established for piecewise analytic solutions excluding, in particular, corner singularities as occurring in polygonal domains. In that context, exponential convergence
was proved in [25] for diffusion problems and in [24] for the Stokes equations. We also
refer to [9, 19, 26] for convergence of dG methods for solutions with low regularity.
This paper is a continuation of our work in [23] on hp-dGFEM for elliptic problems
in polyhedral domains with analytic regularity. In [23], we have shown the unique
solvability and consistency of hp-version interior penalty (IP) discontinuous Galerkin
discretizations for problem (1.1)–(1.2) on general combinations of anisotropically and
σ-geometrically refined meshes with s-linearly increasing and anisotropic elemental
polynomial degrees. The analysis in [23] allows for solutions which exhibit typical corner and edge singularities in polyhedra and which belong to suitably weighted Sobolev
spaces. In addition, we have derived abstract bounds for the hp-dG discretization errors measured in the dG energy norm.
In this paper
√ and starting from [23], we shall prove that exponential convergence
rates exp(−b 5 N ) in terms of the number of degrees of freedom N can be obtained
for hp-IP dGFEM discretizations of (1.1)–(1.2) on axiparallel and σ-geometric meshes
with s-linear anisotropic elemental polynomial degrees. The proof of this bound is
crucially based on the recent results in [6] which establish the analytic regularity of
solutions to problem (1.1)– (1.2) with piecewise analytic right-hand sides, extending
the pioneering work [2, 3] in two dimensions to the three-dimensional case. We point
out that our exponential convergence estimate covers, in particular, three-dimensional
and anisotropic generalizations of all mesh-degree combinations found to be optimal in
the univariate case in [10, Table 1]. Based on the analysis in [10] and on numerical
experiments we expect the flexibility afforded by the present convergence analysis,
i.e., combinations with geometric subdivision ratios σ 6= 1/2 and nonuniform, possibly anisotropic polynomial degrees,
to increase the value of the constant b in the
√
exponential error bound exp(−b 5 N ).
2
The outline of the article is as follows: In Section 2, we recapitulate the analytic regularity results of [6]. In Section 3, we define hp-dG finite element spaces on
σ-geometric axiparallel meshes with possibly s-linear anisotropic polynomial degree
distributions. Section 4 recalls the stability properties of hp-version IP dG discretizations obtained in [23], as well as the corresponding error estimates. Furthermore,
Section 5 is devoted to hp-interpolation estimates in the interior domains as well as
in the elements abutting the singular support of the solutions. Section 6 states and
proves the exponential convergence of hp-dGFEM in polyhedra.
Standard notation will be employed throughout the paper. The number of elements in a set A of finite cardinality is denoted by |A|. In Section 5 the function
Ψq,r =
Γ(q + 1 − r)
,
Γ(q + 1 + r)
0 ≤ r ≤ q,
(1.3)
shall be used frequently, where Γ is the Gamma function satisfying Γ(n + 1) = n! for
any n ∈ N. Occasionally, we shall use the notations ”.” or ”≃” to mean an inequality or an equivalence containing generic positive multiplicative constants which are
independent of any local mesh sizes, polynomial degrees, and regularity parameters,
but may in particular depend on σ and s. In addition, we shall use in several places
notations and results from [23] which will only be briefly mentioned.
2. Regularity. Under our assumptions on the matrix A and if the source term f
in (1.1) is analytic in Ω, the solution u of (1.1)–(1.2) is analytic away from any corners
and edges of Ω. To establish exponential convergence of hp-dGFEM, it is necessary to
specify the precise regularity of u in countably normed weighted Sobolev spaces. To
that end, we essentially follow [6], based on the notation already introduced in [23]. We
also mention the papers [12, 15, 16] where alternative definitions of countably normed
weighted Sobolev spaces in terms of local spherical coordinates have originally been
defined and studied. In addition, we refer to the recent work [18] for shift theorems
in weighted Sobolev spaces on polyhedral domains.
2.1. Subdomains and Weights. We denote by C the set of corners c, and by E
the set of (open) edges e of Ω. The singular support is given by
[ [ S=
c ∪
e ⊂ Γ.
(2.1)
c∈C
e∈E
The set S coincides with the singular support of the solution u of (1.1)–(1.2). For
c ∈ C, e ∈ E and x ∈ Ω, we define the following distance functions:
rc (x) = dist(x, c),
re (x) = dist(x, e),
ρce (x) = re (x)/rc (x).
(2.2)
We assume that vertices are separated:
\
∃ ε(Ω) > 0 :
c∈C
Bε (c) = ∅,
(2.3)
where Bε (c) denotes the open ball in R3 with center c and radius ε. For each corner c ∈ C, we define by Ec = { e ∈ E : c ∩ e 6= ∅ } the set of all edges of Ω which meet
at c. For any e ∈ E, the set of corners of e is given by Ce ≡ ∂e = { c ∈ C : c ∩ e 6= ∅ }.
Then, for c ∈ C, e ∈ E and ec ∈ Ec , we define
ωc = { x ∈ Ω : rc (x) < ε ∧ ρce (x) > ε ∀ e ∈ Ec },
ωe = { x ∈ Ω : re (x) < ε ∧ rc (x) > ε ∀ c ∈ Ce },
ωcec = { x ∈ Ω : rc (x) < ε ∧ ρcec (x) < ε }.
3
When clear from the context, we write ωce instead of ωcec . Possibly by reducing ε
in (2.3), we may partition the domain Ω into four disjoint parts,
.
.
.
Ω = Ω0 ∪ ΩC ∪ ΩE ∪ ΩCE ,
(2.4)
where
ΩC =
[
ωc ,
ΩE =
c∈C
[
ωe ,
ΩCE =
e∈E
[ [
ωce .
(2.5)
c∈C e∈Ec
We shall refer to the subdomains ΩC , ΩE and ΩCE as corner, edge and corner-edge
neighborhoods of Ω, respectively, and the remaining interior part of the domain Ω is
defined by
2.2. Weighted Sobolev Spaces. To each c ∈ C and e ∈ E we associate a
corner and an edge exponent βc , βe ∈ R, respectively. We collect these quantities
in the multi-exponent β := {βc : c ∈ C} ∪ {βe : e ∈ E} ∈ R|C|+|E|. Inequalities
of the form β < 1 and expressions like β ± s, where s ∈ R, are to be understood
componentwise. For example, β + s = {βc + s : c ∈ C} ∪ {βe + s : e ∈ E}.
Moreover, we recall from [23], for corners c ∈ C and edges e ∈ E, the local
coordinate systems in ωe and ωce which are chosen such that e corresponds to the
direction (0, 0, 1). Then, we denote quantities that are transversal to e by (·)⊥ , and
quantities parallel to e by (·)k . In particular, if α ∈ N30 is a multi-index corresponding
to the three local coordinate directions in a subdomain ωe or ωce , then we have α =
(α⊥ , αk ), where α⊥ = (α1 , α2 ) and αk = α3 . Likewise notation shall be employed
below in anisotropic quantities related to a face. Following [6, Definition 6.2 and
Equation (6.9)], we introduce the anisotropically weighted semi-norm
|u|2M m (Ω) = |u|2H m (Ω0 ) +
β
+
X X
c∈C
α∈N3
0
|α|=m
X X βe +|α⊥ |
2
re
Dα u 2
L (ωe )
e∈E
α∈N3
0
|α|=m
β +|α| α 2
rc c
D u 2
L
⊥
X
rcβc +|α| ρβcee +|α | Dα u2 2
+
(ωc )
L (ωce )
e∈Ec
2
for m ∈ N0 , and define the norm k ◦ kMβm (Ω) by kukM m (Ω) =
β
Pm
k=0
!
(2.6)
,
2
|u|M k (Ω) . Here,
β
|u|2H m (Ω0 ) is the usual Sobolev semi-norm of order m on Ω0 , and the operator Dα
denotes the derivative in the local coordinate directions corresponding to the multiindex α. Finally, Mβm (Ω) is the weighted Sobolev space obtained as the closure
of C0∞ (Ω) with respect to the norm k◦kM m (Ω) . For subdomains K ⊂ Ω we shall
β
denote by | ◦ |Mβm (K) the semi-norm (2.6) with all domains of integration replaced by
their intersections with K ⊂ Ω and likewise also for k ◦ kMβm (K) .
2.3. Analytic Regularity. We follow [6, Definition 6.3] and introduce the analytic class
\
Aγ (Ω) = v ∈
Mγm (Ω) : ∃ Cv > 0 s.t. |v|M m (Ω) ≤ Cvm+1 m! ∀ m ∈ N0 . (2.7)
γ
m≥0
Our hp-analysis is now based on the following analytic regularity result from [6,
Corollary 7.1]:
4
Proposition 2.1. There exist bounds βE , βC > 0 (depending on Ω and on the
matrix A in (1.1)) such that, for β satisfying
0 < βe < βE ,
0 < βc <
1
+ βC ,
2
e ∈ E, c ∈ C,
(2.8)
the following shift theorem holds for problem (1.1)– (1.2): if f ∈ A1−β (Ω), then
u ∈ A−1−β (Ω).
Remark 2.2. We note that Proposition 2.1 is a reformulation of [6, Corollary 7.1] which results from substituting the weights β by −1 − β. We note that, while
the upper bounds βE and βC correspond to the ones in [6], we exclude the cases βe = 0
or βc = 0 of lower bounds in our analysis. Incidentally, we have chosen this alternative formulation since our weights β in (2.8) are now strictly positive and thereby,
the ensuing error analysis is slightly facilitated.
3. hp-Subspaces in Ω. In [23], we introduced a class of hp-dG spaces which in(ℓ)
volve three basic ingredients: families Mσ = {Mσ }ℓ≥1 of σ-geometric meshes with ℓ
layers of refinement in Ω, polynomial degree distributions which are nonuniform and
s-linearly increasing between elements and possibly anisotropic within each element.
We gave a specific construction of such hp-space families in general Lipschitz polyhedra Ω ⊂ R3 with a boundary consisting of a finite number of plane faces. Here, we
recapitulate this construction in the particular case of axiparallel domains and meshes,
and refer to [23, Section 3] for details and proofs.
3.1. Geometric hp-Meshes in Ω. We start from any coarse regular quasiuniform partition M0 = {Qj }Jj=1 of Ω into J convex axiparallel hexahedra. Each of
these hexahedral elements Qj ∈ M0 is the image under an affine mapping Gj of
e = (−1, 1)3 , i.e. Qj = Gj (Q)
e for j = 1, . . . , J. In fact, since
the reference patch Q
the hexahedra {Qj }j are assumed axiparallel, the mappings Gj are compositions of
(isotropic) dilations and translations. Due to our assumption that the faces of Ω are
plane, it is geometrically exact.
e have been
In [23], canonical geometric mesh patches on the reference patch Q
constructed; see Figure 3.1. Geometric meshes towards corners and edges in Ω can
then be obtained by again applying the patch mappings Gj to transform these canone to the patches Qj ∈ M0 . It is
ical geometric mesh patches on the reference patch Q
important to note that the geometric refinements in the canonical patches have to be
suitably selected and oriented in order to achieve a proper geometric refinement towards corners and edges of Ω. In [23, Section 3.3], a specific construction of geometric
meshes has been introduced in terms of four different hp-extensions (Ex1)–(Ex4) as
displayed in Figure 3.1. This is, however, still covered by our exponential convergence
analysis below. Moreover, the patches Qj with Qj ∩ S = ∅ away from the singular
e
support S are left unrefined, i.e., no refinement is considered on Q.
Consider now the hexahedral patch Qj ∈ M0 . We denote the elements in the
fj = {K},
e where we allow
canonical geometric mesh patch associated with Qj by M
fj = {Q}
e in the case of unrefined patches. The elements in M
fj are then transported
M
to the physical domain Ω via the (finitely many) affine patch maps Gj . Moreover,
e ∈M
fj , we can write K
e = H e (K),
b where H e : K
b →K
e is a possibly
for each K
j,K
j,K
b = (−1, 1)3 (to
anisotropic dilation combined with a translation of the reference cube K
e
be distinguished from the
n reference patch Q). Thus, the elements
o in the patch Qj ⊂ Ω
b
e
f
will be given by Mj = K : K = (Gj ◦ Hj,Ke )(K), K ∈ Mj , j = 1, . . . , J.
5
e with
Fig. 3.1. Examples of three basic geometric mesh subdivisions in the reference patch Q
subdivision ratio σ = 21 : isotropic towards the corner c (left), anisotropic towards the edge e (center),
and anisotropic towards the edge-corner pair ce (right). The sets c, e, ce are shown in boldface.
SJ
j=1 Mj . By construction, each
b under an element
hexahedral element K ∈ M is the image of the reference cube K
b which is a possibly anisotropic dilation with a translation
mapping ΦK K = ΦK (K),
b to K. We collect all element mappings ΦK in the mapping vector Φ(M) :=
from K
{ ΦK : K ∈ M }.
With each hexahedral element K ∈ M, we associate a polynomial degree vector pK = (pK,1 , pK,2 , pK,3 ) ∈ N30 . Its components correspond to the coordinate dib = Φ−1 (K). The polynomial degree is called isotropic if pK,1 = pK,2 =
rections in K
K
pK,3 = pK . In the hp-error estimates, we shall be mainly concerned with the situation
k
⊥
where pK,1 = pK,2 =: p⊥
K ; in this case we simply write pK = (pK , pK ). Given a mesh
M of hexahedral elements in Ω, we combine the elemental polynomial degrees pK into
the polynomial degree vector p(M) := { pK : K ∈ M }. We remark that in addition
to the mesh refinements, the extensions (Ex1)–(Ex4) introduced in [23] also provide
appropriate polynomial degree distributions that increase s-linearly away from the
singular set S.
In the sequel, we shall be working with sequences of σ-geometrically refined meshes
(0)
(1)
(2)
(0)
denoted by Mσ , Mσ , Mσ , . . . , where Mσ := M0 . Here, σ ∈ (0, 1) is a fixed
parameter defining the ratio of subdivision in the canonical geometric refinements
in Figure 3.1. We shall refer to the index ℓ as refinement level and to the sequence
(ℓ)
Mσ = {Mσ }ℓ≥1 as a σ-geometric mesh family; see [23, Definition 3.4].
A geometric mesh in Ω is now given by M =
3.2. Mesh Layers. As in [23, Section 3], we shall use the concept of mesh layers:
(ℓ)
these are partitions of Mσ = {Mσ }ℓ≥1 into certain subsets of elements with identical
scaling properties in terms of their relative distance to the sets C and E. The following
result holds.
Proposition 3.1. Any σ-geometric mesh family Mσ obtained by iterating the
basic hp-extensions (Ex1)–(Ex4) in [23] can be partitioned into a countable sequence
of disjoint mesh layers {Ljσ }ℓ−1
j=0 , and a corresponding nested sequence of terminal
(ℓ)
layers Tℓσ , such that each Mσ ∈ Mσ , ℓ ≥ 1, can be written as
.
.
.
.
0
1
ℓ−1
M(ℓ)
∪ Tℓσ .
σ = Lσ ∪ Lσ ∪ . . . ∪ Lσ
(3.1)
Elements in the submesh
.
.
.
Oℓσ := L0σ ∪ L1σ ∪ . . . ∪ Lℓ−1
⊂ M(ℓ)
σ
σ ∈ Mσ ,
ℓ ≥ 1,
(3.2)
are bounded away from C ∪ E, while all elements in the terminal layer Tℓσ have a
nontrivial intersection with C ∪ E.
6
.
(ℓ)
Oℓσ. into discrete corner, edge
Evidently, Mσ = Oℓσ ∪ Tℓσ for ℓ ≥ 1. We. partition
.
ℓ
ℓ
ℓ
ℓ
and corner-edge neighborhoods as Oσ = OC ∪ OE ∪ OCE ∪ Oℓint , where for ℓ ≥ 1,
Oℓint := K ∈ Oℓσ : K ∩ Ω0 6= ∅ ,
OℓC := K ∈ Oℓσ : K ∩ ΩC 6= ∅ \ Oℓint ,
(3.3)
OℓE := K ∈ Oℓσ : K ∩ ΩE 6= ∅ \ (Oℓint ∪ OℓC ),
OℓCE := K ∈ Oℓσ : K ∩ ΩCE 6= ∅ \ Oℓint ∪ OℓC ∪ OℓE .
Note that there exists ℓ0 ≥ 1 (depending on ε from (2.3) and on σ) such that Oℓint =
0
Oℓint
for ℓ ≥ ℓ0 . Without loss of generality, we shall assume that the initial mesh
is sufficiently fine so that we can choose ℓ0 = 2. Consequently, in what follows we
0
. In addition, we may assume without loss of
shall simply write Oint instead of Oℓint
0
ℓ
generality that Lσ ⊂ Oint for ℓ ≥ ℓ0 = 2.
.
For an element K, we set hK = diam(K). For anisotropic elements K ∈ OℓE ∪
k
OℓCE , we denote by hK and h⊥
K the elemental diameters of K parallel and transversal
to the singular edge e ∈ E nearest to K; cf. [23]. For isotropic elements K ∈ OℓC ,
(ℓ)
k
we have hK ≃ h⊥
K ≃ hK . In a sequence Mσ = {Mσ }ℓ≥1 of σ-geometric meshes, we
(ℓ)
define for any K ∈ Mσ , c ∈ C and e ∈ E the quantities:
deK := dist(K, e) = inf re (x),
dcK := dist(K, c) = inf rc (x).
x∈K
x∈K
(3.4)
(ℓ)
The geometric meshes Mσ constructed above and in [23] have the following scaling
properties.
Proposition 3.2. There exists a constant 0 < κ1 < 1 (depending only on σ and
M0 ) such that for all ℓ ≥ 1 we have
c
κ1 dcK ≤ rc |K ≤ κ−1
1 dK ,
∀K ∈ OℓC :
∀K ∈
OℓE
κ1 deK
:
≤ re |K ≤
κ1 hK ≤ dcK ≤ κ−1
1 hK ,
κ1−1 deK ,
(3.5)
κ1−1 h⊥
K,
(3.6)
c
κ1 dcK ≤ rc |K ≤ κ−1
1 dK ,
(3.7)
κ1 h⊥
K
≤
deK
≤
and
∀K ∈
OℓCE
:
e
κ1 deK ≤ re |K ≤ κ−1
1 dK ,
k
−1 ⊥
e
κ1 h⊥
K ≤ dK ≤ κ1 hK ,
k
κ1 hK ≤ dcK ≤ κ−1
1 hK .
Remark 3.3. It follows from Proposition 3.2 that there exists κ1 (σ, M0 ) > 0
such that for all ℓ ≥ 1 there holds
∀K ∈ OℓC ∩ Lℓ−1
:
σ
∀K ∈
OℓE
∩
Lℓ−1
σ
:
ℓ
κ21 σ ℓ ≤ hK ≤ κ−2
1 σ ,
κ21 σ ℓ
≤
h⊥
K
≤
ℓ
κ−2
1 σ ,
(3.8)
(3.9)
and, for some index i = i(K), 1 ≤ i ≤ ℓ,
∀K ∈ OℓCE \Oℓ−1
CE :
−1 ℓ
κ1 σ ℓ ≤ h⊥
K ≤ κ1 σ ,
k
i
κ1 σ i ≤ hK ≤ κ−1
1 σ .
(3.10)
.
Similarly, we partition the terminal layer Tℓσ into Tℓσ = VCℓ ∪ VEℓ , where
VCℓ := K ∈ Tℓσ : ∃c ∈ C with c ∈ K ,
VEℓ := K ∈ Tℓσ : ∃e ∈ E s. t. (K ∩ S)◦ ∩ e is an entire edge of K \ VCℓ .
7
(3.11)
k
ℓ
We notice that elements in VCℓ are isotropic with hK ≃ h⊥
K ≃ hK , while elements in VE
may be anisotropic. Analogous to Proposition 3.2, we have the following result.
Proposition 3.4. There exists a constant κ2 (σ, M0 ) > 0 such that for all ℓ ≥ 1
we have
∀ K ∈ VCℓ :
ℓ
κ2 σ ℓ ≤ hK ≤ κ−1
2 σ ,
(3.12)
and (cf. (3.10)) for all K ∈ VEℓ there holds
−1 ℓ
κ2 σ ℓ ≤ h⊥
K ≤ κ2 σ ,
k
ℓ−j
κ2 σ ℓ−j ≤ hK ≤ κ−1
,
2 σ
(3.13)
for an exponent j = j(K) with 0 ≤ j ≤ ℓ.
3.3. Finite Element Spaces. Let M be a geometric mesh of a σ-geometric
mesh family Mσ in Ω. Furthermore, let Φ(M) and p(M) be the associated element
mapping and elemental polynomial degree vectors, as introduced above. We then
introduce the discontinuous hp finite element space
(3.14)
V (M, Φ, p) = u ∈ L2 (Ω) : u|K ∈ QpK (K), K ∈ M .
Here, we define the local polynomial approximation space QpK (K) as follows: first,
b and for a polynomial degree vector p = (p1 , p2 , p3 ) ∈ N30 ,
on the reference element K
we introduce the anisotropic polynomial space:
b = span { x̂α : αi ≤ pi , 1 ≤ i ≤ 3 } .
b ⊗ Pp3 (I)
b ⊗ Pp2 (I)
b = Pp1 (I)
Qp (K)
(3.15)
b the space of all polynomials of degree at most p
Here, for p ∈ N0 , we denote by Pp (I)
on the reference interval Ib = (−1, 1). Then, if K is a hexahedral element of M with
b → K and polynomial degree vector pK =
associated elemental mapping ΦK : K
o
n
pK
b . In
(pK,1 , pK,2 , pK,3 ), we define Q (K) := u ∈ L2 (K) : (u|K ◦ ΦK ) ∈ QpK (K)
the case where the polynomial degree vector pK associated with K is isotropic, i.e.,
pK,1 = pK,2 = pK,3 = pK , we simply write QpK (K) = QpK (K). For technical reasons
that will become clear in the analysis, we will assume throughout the paper that all
polynomial degrees on elements K ∈ Oℓσ are greater than or equal to 3.
We now introduce two families of hp-finite element spaces for the discontinuous Galerkin methods; both yield exponentially convergent approximations and are
(ℓ)
based on the σ-geometric mesh families Mσ = {Mσ }ℓ≥1 . The first family of hp-dG
subspaces is defined by
(ℓ)
(ℓ)
Vσℓ := V (M(ℓ)
σ , Φ(Mσ ), p1 (Mσ )),
ℓ ≥ 1,
(3.16)
(ℓ)
where the elemental polynomial degree vectors pK in p1 (Mσ ) are isotropic and
uniform, given on each element K ∈ M (ℓ )σ as pK = max{3, ℓ}. The second family of
hp-dG subspaces is chosen as
ℓ
(ℓ)
(ℓ)
Vσ,s
:= V (M(ℓ)
σ , Φ(Mσ ), p2 (Mσ )),
ℓ ≥ 1,
(3.17)
(ℓ)
for an increment parameter s > 0. Here the polynomial degree vectors p2 (Mσ ) are
linearly increasing with slope s away from S, i.e., specifically, the polynomial degrees
within each element increase linearly with the number of mesh layers between that
element and the component of the singular set S nearest to it, with the factor of
proportionality (“slope” in the terminology of [10]) being s > 0; see [23, Section 3].
8
Remark 3.5. By construction, increasing the index j in the mesh layers Ljσ
corresponds to moving from inside the domain towards the singular set S, with L0σ
being the most inner layer, and the terminal layer Tℓσ being the most outer layer
abutting at S; see (3.1). While this numbering takes into account the scaling properties
of Ljσ , it is in contrast to the notion of s-linearly increasing polynomial degrees where
the polynomial degree increases s-linearly away from the singular set into the interior
of the domain; see also [23].
ℓ
3.4. Properties of Bounded Variation. The spaces Vσℓ and Vσ,s
defined
in (3.16) and (3.17), respectively, satisfy bounded variation properties with respect to
the local mesh sizes and polynomial degrees. These properties will be implicitly used
in our analysis. To describe them, let Mσ be the underlying σ-geometric mesh family.
For any M ∈ Mσ , we denote the set of all interior faces in M by FI (M), and the set
of all boundary faces by FB (M). In addition, let F (M) = FI (M) ∪ FB (M) denote
the set of all (smallest) faces of M. If the mesh M ∈ Mσ is clear from the context, we
shall omit the dependence of these sets on M. Furthermore, for an element K ∈ M,
we denote the set of its faces by FK = { f ∈ F : f ⊂ ∂K }. For K ∈ M and f ∈ FK ,
we denote by h⊥
K,f the height of K over the face f , i.e., the diameter of element K
k,(1)
k,(2)
in the direction transversal to f . Similarly, we denote by pK,f , pK,f the two components of the elemental degree pK parallel to f , and by p⊥
K,f the polynomial degree
of pK transversal to f (defined as the corresponding components of Φ−1
K (K)).
The geometric mesh family now satisfies the following property with respect to
the local mesh sizes: there is a constant µ1 ∈ (0, 1) only depending on σ and M0 such
◦
−1
⊥
that µ1 ≤ h⊥
K♯ ,f /hK♭ ,f ≤ µ1 , for all interior faces f = (∂K♯ ∩ ∂K♭ ) ∈ FI (M) and
(ℓ)
M ∈ Mσ . Furthermore, the family of degree vectors p2 (Mσ )ℓ≥1 introduced in (3.17)
satisfies a similar property with respect to the polynomial degree: there is a constant
−1
⊥
µ2 ∈ (0, 1) (depending on s) such that µ2 ≤ p⊥
K♯ ,f /pK♭ ,f ≤ µ2 , for all interior
◦
(ℓ)
(ℓ)
faces f = (∂K♯ ∩ ∂K♭ ) ∈ FI (Mσ ) and ℓ ≥ 1. Note that for the family p1 (Mσ )ℓ≥1
in (3.16) this property is trivially satisfied.
4. Discontinuous Galerkin Discretization. In this section we present the
hp-dG discretizations of (1.1)–(1.2) for which we shall prove exponential convergence.
In addition, we shall briefly recall the stability and quasi-optimality results from [23,
Section 4]. Throughout, M ∈ Mσ denotes a generic, σ-geometric mesh.
4.1. Face Operators. In order to define a dG formulation on a given mesh M
for the model problem (1.1)–(1.2), we shall first recall some element face operators.
◦
For this purpose, consider an interior face f = ∂K ♯ ∩ ∂K ♭ ∈ FI (M) shared by two
elements K ♯ , K ♭ ∈ M. Furthermore, let v, w be a scalar- respectively a vector-valued
function that is sufficiently smooth inside the elements K ♯ , K ♭ . Then we define the
following jumps and averages of v and w along f :
hhvii = 1/2 (v|K ♯ + v|K ♭ )
[[v]] = v|K ♯ nK ♯ + v|K ♭ nK ♭
[[w]] = w|K ♯ · nK ♯ + w|K ♭ · nK ♭
hhwii = 1/2 (w|K ♯ + w|K ♭ ) .
Here, for an element K ∈ M, we denote by nK the outward unit normal vector
◦
on ∂K. For a boundary face f = (∂K ∩ ∂Ω) ∈ FB (M) for K ∈ M, and sufficiently
smooth functions v, w on K, we let [[v]] = v|K nΩ , [[w]] = w|K · nΩ , and hhvii = v|K ,
hhwii = w|K , where nΩ is the outward unit normal vector on ∂Ω.
9
4.2. hp-IP dG Discretizations. The problem (1.1)–(1.2) will be discretized
using an interior penalty (IP) discontinuous Galerkin method. Let V (M, Φ, p) be an
hp-dG finite element space on a σ-geometric mesh M ∈ Mσ with a degree vector p.
For a fixed parameter θ ∈ R, we define the hp-discontinuous Galerkin solution uDG
by
Z
f v dx
∀ v ∈ V (M, Φ, p),
(4.1)
uDG ∈ V (M, Φ, p) :
aDG (uDG , v) =
Ω
where aDG (u, v) is given by
Z
Z
hhA∇h wii · [[v]] ds
aDG (w, v) = (A∇h w) · ∇h v dx −
F (M)
Ω
Z
Z
+θ
hhA∇h vii · [[w]] ds + γ
α[[v]] · [[w]] ds.
F (M)
F (M)
Here, ∇h is the elementwise gradient, and γ > 0 is a stabilization parameter that will
be chosen sufficiently large. Furthermore, α ∈ L∞ (F ) is the discontinuity stabilization
function defined facewise as

2
⊥
⊥


max
p
,
p
K♯ ,f
K♭ ,f



if f = (∂K♯ ∩ ∂K♭ )◦ ∈ FI (M),

⊥
⊥
min
h
,
h
αf =
(4.2)
K♯ ,f
K♭ ,f

⊥
2

(p
)

K,f


if f = (∂K ∩ ∂Ω)◦ ∈ FB (M).
 ⊥
hK,f
The parameter θ allows us to describe a whole range of interior penalty methods:
for θ = −1 we obtain the standard symmetric interior penalty (SIP) method while
for θ = 1 the non-symmetric (NIP) version is obtained; cf. [1] and the references
therein. Let us further remark that, by our assumption that A is symmetric positive
definite and constant, we omit the explicit dependence of the penalty jump terms on
the diffusivity.
4.3. Anisotropic Trace Inequality. In order to analyze the numerical fluxes
in the dG formulation, we next recall the anisotropic trace inequality proved in [23]:
let M ∈ Mσ for 0 < σ < 1, K ∈ M axiparallel, f ∈ FK and s ≥ 1. Then, for
any v ∈ W 1,s (K), there holds
s
−1 s
s
s
kvkLs (K) + h⊥
(4.3)
kvkLs (f ) ≤ Cs h⊥
k∂K,f,⊥ vkLs (K) .
K,f
K,f
The constant Cs > 0 depends only on σ and on the mapping vector Φ through the
patch maps Gj , but is independent of the element size and element aspect ratio.
Here, ∂K,f,⊥ signifies the partial derivative in direction transversal to f ∈ FK .
4.4. Stability and Galerkin Orthogonality. To formulate the well-posedness
of the hp-dGFEM, we use the standard dG norm defined by
Z
Z
2
2
|||v|||2DG =
|∇h v| dx + γ
α |[[v]]| ds,
(4.4)
Ω
F
for any v ∈ V (M, Φ, p) + H 1 (Ω). In [23], we have shown the the following result.
10
Theorem 4.1. For any σ-geometric mesh M with 0 < σ < 1 and any degree
vector p, the dG bilinear form aDG (·, ·) is continuous and coercive on V (M, Φ, p):
there exist constants 0 < C1 ≤ C2 < ∞ independent of the refinement level ℓ, the
element aspect ratios, the local mesh sizes, and the local polynomial degree vectors
such that
|aDG (v, w)| ≤ C1 |||v|||DG |||w|||DG
∀ v, w ∈ V (M, Φ, p),
and, for γ > 0 sufficiently large independent of the refinement level ℓ, the element
aspect ratios, the local mesh sizes, and the local polynomial degree vectors,
aDG (v, v) ≥ C2 |||v|||2DG
∀ v ∈ V (M, Φ, p).
In particular, there exists a unique solution uDG of (4.1).
Moreover, the following Galerkin orthogonality property holds: suppose that the
2
solution u of (1.1)–(1.2) belongs to M−1−β
(Ω), where β is a weight vector satisfying (2.8). Then, the dG approximation uDG ∈ V (M, Φ, p) satisfies
aDG (u − uDG , v) = 0
∀ v ∈ V (M, Φ, p).
(4.5)
We point out that for the non-symmetric interior penalty method corresponding to
θ = 1, any value of γ > 0 is sufficient for the method to be coercive.
4.5. Error Estimates. Although the error of the dG method (4.1) is not quasioptimal with respect to the energy norm, the Galerkin orthogonality (4.5) of the
error implies that the error can be bounded by a certain interpolation error of the
exact solution in the dG subspace. We proceed in a standard way and split the
error eDG = u − uDG into two parts η and ξ, eDG = η + ξ, with
η = u − Πu ∈ H01 (Ω) + V (M, Φ, p),
ξ = Πu − uDG ∈ V (M, Φ, p).
(4.6)
Here, Π is a discontinous hp-version (quasi)interpolant into V (M, Φ, p) which is stable
2
(Ω). A specific choice for Π
in H 2 (K) for each K ∈ Oℓσ and bounded for u ∈ M−1−β
will be made in Section 5. In [23], we have proved the following error estimate.
(ℓ)
Theorem 4.2. On any σ-geometric and axiparallel mesh Mσ and any degree
vector p, there holds the error bound
(4.7)
|||u − uDG |||2DG ≤ Cp4max ΥOℓσ [η] + ΥTℓσ [η] ,
where
ΥOℓσ [η] =
X K∈Oℓσ
+
X
max h⊥
K,f
f ∈FK
X
h⊥
K,f
K∈Oℓσ f ∈FK
and
ΥTℓσ [η] =
X K∈Tℓσ
+
max h⊥
K,f
f ∈FK
X X
K∈Tℓσ f ∈FK
−2
|f |
−1
2
2
2
kηkL2 (K) + k∇ηkL2 (K)
2
(4.8)
k∂K,f,⊥ ∇ηkL2 (K) ,
−2
2
2
kηkL2 (K) + k∇ηkL2 (K)
2
h⊥
K,f k∇ηkL1 (f ) .
(4.9)
Here, C > 0 is a constant independent of the refinement level ℓ, the element aspect
ratios, the local mesh sizes, and the local polynomial degree vectors. Furthermore, |f |
is the surface measure of a face f , and pmax = maxK∈M(ℓ) max pK .
σ
11
5. hp-Approximations. It is now sufficient to construct an exponentially convergent interpolant Π for the bounds in (4.8) and (4.9) in Theorem 4.2. Since Πu ∈
V (M, Φ, p) may be discontinuous across element interfaces, we can construct Πu
separately for each element K of the discrete neighborhoods in (3.3) and (3.11). We
then estimate η|K = u|K − Πu|K for every element K under the regularity property
u ∈ A−1−β (Ω) stated in Proposition 2.1, that is,
m+1
m
|u|M−1−β
m!
(Ω) ≤ Cu
∀ m ∈ N0 ,
(5.1)
for a constant Cu > 0 and weights β satisfying (2.8).
5.1. Anisotropic Polynomial Approximation. We begin by developing a
polynomial approximation analysis based on tensor-product interpolation operators.
5.1.1. Univariate hp-Projectors and Error Bounds. Let I = (−1, 1) be the
unit interval. For any k ≥ 1, we write H k (I) for the usual Sobolev space endowed with
the norm kukH k (I) . For q ≥ 0, we denote by π
bq,0 : L2 (I) → Pq (I) the L2 (I)-projection.
k−1
The following C
-conforming and univariate projector has been constructed in [7,
Section 8].
Lemma 5.1. For any p, k ∈ N with p ≥ 2k −1, there is a projector π
bp,k : H k (I) →
p
(k)
(k)
(j)
(j)
P (I) that satisfies (b
πp,k u) = π
bp−k,0 (u ), and (b
πp,k ) u(±1) := u (±1), for any
j = 0, . . . , k − 1.
Note that although the above conditions formally overspecify the interpolating
polynomial, the projector is, in fact, well defined. Moreover, we have the following
stability and approximation properties, which have been proved in [7, Proposition 8.4]
and [7, Theorem 8.3], respectively.
Proposition 5.2. There holds:
(i) For every k ∈ N, there exists a constant Ck > 0 such that
∀u ∈ H k (I) , ∀p ≥ 2k − 1 :
kb
πp,k ukH k (I) ≤ Ck kukH k (I) .
(5.2)
(ii) For integers p, k ∈ N with p ≥ 2k − 1, κ = p − k + 1 and for u ∈ H k+s (I)
with any k ≤ s ≤ κ there holds the error bound
k(u − π
bp,k u)(j) k2L2 (I) ≤
(κ − s)! (k+s) 2
ku
kL2 (I) ,
(κ + s)!
j = 0, 1, . . . , k.
(5.3)
5.1.2. Tensor-interpolation and Error Bounds. Let now I d = I × · · · × I be
the unit cube in Rd , d ≥ 1. Points x in I d have coordinates xi , i.e., x = (x1 , . . . , xd ).
For k = (k1 , . . . , kd ) ∈ Nd , we define the space
k
Hmix
(I d ) = H k1 (I) ⊗ · · · ⊗ H kd (I),
(5.4)
where ⊗ denotes the tensor-product of separable Hilbert spaces. If k1 = . . . = kd = k,
k
we write Hmix
(I d ) = H k (I) ⊗ · · · ⊗ H k (I). These tensor-product spaces are isomork
k
phic to standard Bochner spaces. For example, Hmix
(I d ) ≃ H k (I; Hmix
(I d−1 )) ≃
k
d−1
k
d
p d
p1
Hmix (I
; H (I)). For p = (p1 , . . . , pd ) ∈ N , we set Q (I ) := P (I) ⊗ · · · ⊗ Ppd (I).
d
In I of dimension d > 1 and for pi ≥ 2ki −1, we now define the interpolation operator
bd =
Π
p,k
d
O
i=1
12
(i)
π
bpi ,ki ,
(5.5)
(i)
where π
bpi ,ki is the univariate operator from Lemma 5.1, acting in the variable xi . If
b d in place of Π
bd .
pi = p and ki = k for all i, we also write Π
p,k
p,k
d−1
In what follows, we denote by ki ∈ N
the multiindex k with component ki
deleted from it.
Proposition 5.3. For d ≥ 1 there holds:
b dp,k ukH k (I d ) ≤ Ck,d kukH k (I d )
kΠ
mix
mix
and
b dp,k u
u − Π
k (I d )
Hmix
≤ Ck,d
d
X
i=1
(5.6)
(i)
ku − π
bpi ,ki ukH ki (I;H ki
mix (I
d−1 ))
.
(5.7)
Proof. This follows readily from property (5.2) of the one-dimensional interpolation operators and by induction over the dimension d exploiting the tensor-product
structure. We refer to [22] for details.
5.1.3. Application to the reference cube. We now consider the reference
b = (−1, 1)3 = K
b⊥ × K
b k , with K
b ⊥ = (−1, 1)2 , K
b k = (−1, 1). The superscripts
cube K
⊥
k
⊥
· and · denote, respectively, the coordinates x = (x1 , x2 ) and xk = x3 . For a
b = Qp⊥ (K
b ⊥ ) ⊗ Ppk (K
b k) =
polynomial degree vector p = (p⊥ , pk ), we write Qp (K)
⊥ k
⊥
⊥
k
(k ,k ) b
(K) =
Pp (I) ⊗ Pp (I) ⊗ Pp (I), where again I = (−1, 1). We then set Hmix
⊥
k
⊥
⊥
⊥
k
⊥
k
k
k
k
k
b
b
Hmix (K ) ⊗ H (K ) = H (I) ⊗ H (I) ⊗ H (I), for regularity parameters
(k ⊥ , k k ) ∈ N2 .
Following the definitions and the analysis above, choosing k = k k = k ⊥ = 2,
and p⊥ , pk ≥ 3, there is a tensor-product projector
2
b 3 ⊥ k : Hmix
b → Q(p⊥ ,pk ) (K).
b
Π
(K)
(p ,p ),2
(5.8)
(1)
(2)
(2)
b3 ⊥ k
Recall that Π
= π
bp⊥ ,2 ⊗ π
bp⊥ ,2 ⊗ π
bpk ,2 . This operator is well-defined and
(p ,p ),2
2
b
bounded in Hmix
(K).
By combining (5.7) and (5.3), we now obtain the following approximation estimate, see [22] for details.
Proposition 5.4. For any integers 3 ≤ s⊥ ≤ p⊥ , 3 ≤ sk ≤ pk , there holds
X
⊥
sk +1
b 3 ⊥ k uk2 2
kDα
uk2L2 (K)
ku − Π
⊥ Dk
b
b . Ψpk −1,sk −1
(p ,p ),2
H
(K)
mix
⊥
α⊥
1 ≤2,α2 ≤2
+ Ψp⊥ −1,s⊥ −1
X
k
α⊥
1 ≤2,α ≤2
+ Ψp⊥ −1,s⊥ −1
X
k
α⊥
2 ≤2,α ≤2
(α⊥ ,s⊥ +1)
2
Dα
b
k ukL2 (K)
(s⊥ +1,α⊥
2 )
2
Dα
b .
k ukL2 (K)
kD⊥ 1
kD⊥
k
k
Remark 5.5. We note that in the isotropic case, we have p = p⊥ = pk and
s = s⊥ = sk , so that the error bound in Proposition 5.4 becomes
b 3 uk2 2
ku − Π
p,2
H
b
mix (K)
. Ψp−1,s−1 kuk2H s+5 (K)
b
for any 3 ≤ s ≤ p. s⊥ = s⊥ + 1, sk = sk + 1
13
(5.9)
5.1.4. Anisotropic interpolation on axiparallel hexahedra in Oℓσ . We
construct the hp-interpolant Πu on an axiparallel element K ∈ Oℓσ as follows. Let
k
the elemental polynomial degree vector be given by pK = (p⊥
K , pK ). On the reference
b we then define Πb
b u := Π
b3
b3
element K,
b, with Π
the projector defined
k u
k
⊥
⊥
(pK ,pK )
(pK ,pK )
in (5.8). Then we set u
b := u ◦ ΦK , and define the elemental interpolation operator
(Πu)|K by
b u.
(Πu) |K ◦ ΦK := Πb
(5.10)
Upon possibly translating the element K, we may assume without loss of generalk
⊥
ity that K = (0, h⊥
K ) × (0, hK ) × (0, hK ) be an axiparallel element that is possi
bly anisotropic in the third coordinate direction. We denote by x = x⊥ , xk and
b respectively. Similarly, for a multi-index
b= x
b⊥ , x
x
bk the coordinates on K and K,
k
⊥
k
α⊥
α = α , α , we write D⊥ respectively Dα
k for partial derivatives perpendicular
respectively parallel to the third coordinate direction. On the reference element, we
k
⊥
b α⊥ and D
b αk . Then, db
use D
x⊥ = (hK/2)−2 dx⊥ and db
xk = (hK/2)−1 dxk . Similarly,
⊥
k
|α⊥ | α⊥
b α⊥ = (h⊥
b αk = (hkK/2)αk Dαk . Thus, we obtain
K/2)
D
D
and D
⊥
⊥
k
k
h⊥ 2|α⊥ |−2 hk 2αk −1
⊥
2
αk
2
K
K
b αk u
b α⊥ D
kDα
b
k
=
kD
⊥ Dk ukL2 (K) ,
b
⊥
k
L2 (K)
2
2
(5.11)
for any α⊥ ∈ N20 and αk ∈ N0 . In particular, we also have
⊥
2
⊥ 2|α⊥ |−2 k 2αk −1
αk
2
b α⊥ D
b αk u
kD
(hK )
kDα
⊥ Dk ukL2 (K) ,
⊥
b ≤ C(hK )
k bkL2 (K)
(5.12)
with a constant C > 0 that is independent of the differentiation orders.
Lemma 5.6. For an axiparallel element K as above and η = u − Πu, there holds
k
kb
ηk2H 2
. Epk ,sk (K) + Ep⊥⊥ ,s⊥ (K),
b
mix (K)
(5.13)
for any 3 ≤ s⊥ ≤ p⊥ and 3 ≤ sk ≤ pk , with
k
Epk ,sk (K) . Ψpk −1,sk −1
X
⊥
2|α
(h⊥
K)
|−2
k
k
(hK )2s
+1
⊥
α⊥
1 ≤2,α2 ≤2
X
Ep⊥⊥ ,s⊥ (K) . Ψp⊥ −1,s⊥ −1
⊥
2|α
(h⊥
K)
|−2
k
⊥
s
kDα
⊥ Dk
k
k
(hK )2α
−1
s⊥ +1≤|α⊥ |≤s⊥ +3,αk ≤2
b we see that kb
Proof. From Proposition 5.4 on K,
ηk2H 2
+1
uk2L2 (K) ,
⊥
k
α
2
kDα
⊥ Dk ukL2 (K) .
can be bounded by
b
mix (K)
three terms. Using (5.12) the first of these terms scales as follows:
Ψpk −1,sk −1
X
⊥
α⊥
1 ≤2,α2 ≤2
. Ψpk −1,sk −1
b α⊥ D
b sk +1 u
bk2L2 (K)
kD
⊥
b
k
X
⊥
2|α
(h⊥
K)
⊥
α⊥
1 ≤2,α2 ≤2
14
|−2
k
k
(hK )2s
+1
⊥
k
s
kDα
⊥ Dk
+1
uk2L2 (K) .
Similarly, we have for the second term:
X
Ψp⊥ −1,s⊥ −1
k
α⊥
1 ≤2,α ≤2
. Ψp⊥ −1,s⊥ −1
X
(α⊥ ,s⊥ +1) b αk
Dk u
bk2L2 (K)
b
b 1
kD
⊥
⊥
2(s
(h⊥
K)
+α⊥
1 )
k
k
(hK )2α
−1
k
α⊥
1 ≤2,α ≤2
X
. Ψp⊥ −1,s⊥ −1
⊥
2|α
(h⊥
K)
|−2
(α⊥ ,s⊥ +1)
kD⊥ 1
k
k
(hK )2α
s⊥ +1≤|α⊥ |≤s⊥ +3,αk ≤2
−1
k
2
Dα
k ukL2 (K)
⊥
k
2
α
kDα
⊥ Dk ukL2 (K) .
The third term scales as the second term.
5.2. Error Bounds on Oℓσ . We are now ready to estimate ΥOℓσ [η] in (4.8) for
η = u − Πu and Π in (5.10). Due to the various element scalings, we give a separate
error analysis on each of the submeshes in (3.3). To that end, we write
ΥOℓσ [η] ≤ ΥOint [η] + ΥOℓC [η] + ΥOℓE [η] + ΥOℓCE [η],
(5.14)
where the consistency errors on the respective submeshes are defined as in (4.8), but
with sums that are taken only over the elements in the submeshes.
We remark that we first analyze the interpolant Πu ∈ Vσℓ with variable and
ℓ
anisotropic polynomial degree vector that defines the space Vσ,s
in (3.17). The
ℓ
space Vσ in (3.16) with uniform, isotropic polynomial degree distributions will be
addressed after that in Corollary 5.19 below.
In what follows, it will be convenient to introduce the notation
T K [η] := T1K [η] + T2K [η] + T3K [η],
(5.15)
−2
kηk2L2 (K) , T2K [η] := k∇ηk2L2 (K) , and T3K [η] :=
where T1K [η] := maxf ∈FK (h⊥
K,f )
P
⊥
2
2
f ∈FK (hK,f ) k∂K,f,⊥ ∇ηkL2 (K) . Hence, in light of Theorem 4.2, for any of the neighP
borhoods D in (5.14) there holds ΥD [η] = K∈D T K [η].
Lemma 5.7. Let K ∈ Oℓσ be an axi-parallel element. Then we have
k
2 k −1
T K [η] . (h⊥
+ hK )kb
η k2H 2
K ) (hK )
b
mix (K)
.
Proof. We separately estimate the expressions TiK [η], i = 1, 2, 3, defined above
using the scaling in (5.11). We start with the most involved term T3K [η]. To that
end, letf ∈ FK and e ∈ E the nearest singular edge to K. We further distinguish
k
the two cases f ⊥ e and f k e. In the former case, f ⊥ e, we have h⊥
K,f ≃ hK , and
∂K,f,⊥ is the derivative parallel to the edge. Thus, by applying the scaling in (5.11)
and absorbing the factors 1/2 appearing there in the constants, we obtain
2
2
(h⊥
K,f ) k∂K,f,⊥ ∇ηkL2 (K) ≃
k
(hK )2
k
≃ (hK )


X
⊥
|α⊥ |=1
X
|α⊥ |=1

2
2
2

kDα
⊥ Dk ηkL2 (K) + kDk ηkL2 (K)
⊥ 2 k −1 b 2
b k ηbk2 2
b α⊥ D
kDk ηbk2L2 (K)
kD
b .
b + (hK ) (hK )
⊥
L (K)
15
⊥
In the latter case, f k e, we h⊥
K,f ≃ hK , and ∂K,f,⊥ is the derivative perpendicular to
the edge. Therefore,


X
X
⊥
⊥
⊥ 2
2
2
2
2

(h⊥
kDα
kDα
K,f ) k∂K,f,⊥ ∇ηkL2 (K) . (hK )
⊥ ηkL2 (K) +
⊥ Dk ηkL2 (K)
k
≃ hK
X
|α⊥ |=2
⊥
|α⊥ |=2
k
⊥ 2
−1
b α ηbk2 2
kD
⊥
b + (hK ) (hK )
L (K)
X
|α⊥ |=1
⊥
|α⊥ |=1
bα D
b k ηbk2 2 .
kD
⊥
b
L (K)
k
2 k −1
The two bounds above imply T3K [η] . (h⊥
)
(h
)
+
h
η |2H 2 (K)
.
K
K |b
K
b
k
k
2
−1
Proceeding similarly, we obtain T1K [η] . (h⊥
+ hK kb
η k2L2 (K)
K ) (hK )
b , as well
k
k
2
−1
b η k2
as T2K [η] . (h⊥
+ hK k∇b
K ) (hK )
b ; see [22]. This yields the assertion.
L2 (K)
5.2.1. Submesh Oint . The submesh Oint consists only of a finite number of
shape-regular elements away
from the singular support S of the solution u. That is,
S
on Ωint := int
K∈Oℓ K the distance functions re , rc and ρce in (2.2) appearing
int
in the definition of the norm (2.6) of |u|2M m (Ω) are bounded away from zero, and (5.1)
β
implies that |ukH m (Ωint ) ≤ C m+1 m!, m ∈ N0 , for a constant C > 1 (possibly different
from Cu in (5.1)). This estimate implies that there exists another constant C > 1
such that
m+1
ku ◦ ΦK kH m (K)
m!
b ≤ C
∀m ∈ N0 ,
(5.16)
for any K ∈ Oint . We now note that the hp-extensions (Ex1)–(Ex4) in [23] produce a
uniform and isotropic polynomial degree distribution in Oint , which is denoted by p
and satisfies p ≃ ℓ.
Lemma 5.8. There is a constant C > 1 such that
ΥOint [η] ≤ C 2s Ψp−1,s−1 Γ(s + 6)2
for any s ∈ [3, p].
Proof. Since there are only a finite number of shape-regular elements in Oint ,
Lemma 5.7, the approximation property in (5.9), and the regularity (5.16) yield
X
X
kb
η k2H 2 (K)
T K [η] .
ΥOint [η] =
b
K∈Oint
.
X
K∈Oint
mix
K∈Oint
2
2s
Ψp−1,s−1 kb
uk2H s+5 (K)
b ≤ C Ψp−1,s−1 ((s + 5)!) .
(5.17)
This is the desired bound for integer exponents 3 ≤ s ≤ p.
Next, we interpolate the bound (5.17) to fractional regularity order s. To do so,
let s = k + θ ∈ [3, p] with k = ⌊s⌋ and θ = s − k ∈ (0, 1). We note that there exist
constants 0 < C1 < C2 < ∞ such that
0 < C1 ≤
n!(n + 1)θ
≤ C2 < ∞
Γ(n + 1 + θ)
∀ n ∈ N, 0 ≤ θ ≤ 1.
(5.18)
Using the real method of interpolation with indices 0 < θ < 1 and 2 on (5.17) for the
integers k = ⌊s⌋ and k + 1 = ⌊s⌋ + 1, we readily obtain the bound
θ
1−θ
Γ(k + 1 + q)2 Ψp−1,k ,
ΥOint [η] ≤ C 2(k+θ) Γ(k + q)2 Ψp−1,k−1
16
where we have replaced 6 by q ≥ 0 for future reference. Elementary manipulations
using (5.18) reveal that as p → ∞, we have
ΥOint [η] . C 2(k+θ) Γ(k + θ + q)2 Ψp−1,k+θ−1 ≃ C 2s Γ(s + q)2 Ψp−1,s−1 ,
where the additional constants introduced depend only on C1 , C2 in (5.18).
To prove exponential convergence, we let p → ∞, and for each given p, minimize
the bound in Lemma 5.8.
Lemma 5.9. For any c > 0 and any q ∈ N, there exist constants b > 0 and C > 0
(depending only on c and q) such that
(5.19)
∀p ≥ 3 :
min c2s Γ(s + q)2 Ψp−1,s−1 ≤ C 2 exp(−2bp).
s∈[3,p]
Proof. We begin by claiming that there are constants b > 0 and C > 0 such that
∀p ≥ 3 :
min c2s Γ(s + q)2 Ψp,s ≤ C 2 exp(−2bp) .
(5.20)
s∈[3,p]
To
(5.20), we first consider the case q = 1. By Stirling’s inequalities we have
√ prove
2πss+1/2 e−s ≤ Γ(s + 1) ≤ ess+1/2 e−s for any s ≥ 1. Hence, we estimate
1
(p − s)p−s p − s 2
e3
c2s Γ(s + 1)2 Ψp,s ≤ √ c2s s2s+1
(p + s)p+s p + s
2π
2s p−s
p−s
cs
.
≤ Cs
p+s
p+s
(5.21)
Upon increasing the value of the constant c, we may absorb the linear factor in s
in (5.21) into c2s . In addition, we may assume that p ≥ ⌈3(c + 1)⌉ (otherwise, we
would simply increase the constant C in (5.21)). Then we choose s = p/(c+1) ∈ [3, p].
Inserting this into the previous bound gives
2s
2
c Γ(s + 1) Ψp,s ≤ C
c
c+1
1 + ps
2p
! c+1
1−
1+
s
p
s
p
!p(1− ps )
≤C
c
c+2
1
p(1+ c+1
)
.
1
c
) log c+2
> 0.
The bound (5.20) for q = 1 thus follows with b(c) := − 21 (1 + c+1
The case q > 1 is obtained analogously upon noting that by Stirling’s formula,
Γ(s + q) ≤ C(q)sq−1 Γ(s + 1), and the polynomial factor in s may again be absorbed
into c2s by suitably increasing the value of c, which proves (5.20).
Now, to prove the assertion (5.19), we recall that Γ(x + 1) = xΓ(x) for all x ≥ 1.
Hence, Ψp−1,s−1 = (p − 1 + s)(p + s)Ψp,s ≤ 4p2 Ψp,s , for any p ≥ 3 and s ∈ [3, p].
Combining this estimate with (5.20), and absorbing the quadratic factor in p into the
exponential term (by modifying the constants b and C) yield the desired result.
Lemma 5.8 and Lemma 5.9 immediately yield the following estimate.
Proposition 5.10. Under the regularity assumption (5.1), the hp-dG interpolation operator Πu defined in (5.10) satisfies the error bound
ΥOint [η] . exp(−2bℓ)
for some constant b > 0 independent of ℓ ≥ 2.
17
5.2.2. Submesh OℓC . Next, we consider the corner neighborhood OℓC in (3.3)
and establish the analog of Proposition 5.10 for ΥOℓC [η]. We may assume without
loss of generality that C = {c} (the general case of an arbitrary, but finite number of
corners follows readily by superposition). Elements K ∈ OℓC are shape-regular with
k
hK ≃ h⊥
K ≃ hK
and
rc |K ≃ dcK ≃ hK ,
(5.22)
according to (3.5) (see also Figure 3.1, left). We shall also use that
∀ 2 ≤ j ≤ ℓ, ∀ K ∈ Lℓ−j+1
∩ OℓC :
σ
dcK ≃ σ ℓ−j .
(5.23)
Here we recall that the index j = ℓ + 1 corresponds to mesh layer L0σ , whose
elements are assumed to belong to Oint , and hence it can be omitted. Let us further
ℓ
recall that the hp-extensions (Ex1)–(Ex4) introduced in [23] for the subspace Vσ,s
in (3.17) produce isotropic polynomial degrees that are uniform in each mesh layer.
In the following, we simply denote these degrees by pj , i.e.,
∀ 2 ≤ j ≤ ℓ, ∀ K ∈ OℓC ∩ Lℓ−j+1
:
σ
pK = pj ≥ 3 .
(5.24)
By construction and for ℓ → ∞, the elemental polynomial degrees {pj }j≥2 form a
sequence of s-linear growth:
∃ χ ∈ (0, 1) :
∀j ≥ 2 :
χ ≤ pj /js ≤ χ−1 .
(5.25)
We can now bound the consistency term ΥOℓC [η].
Lemma 5.11. Under the regularity assumption (5.1), we have
ΥOℓC [η] .
ℓ
X
σ 2(ℓ−j) min β Ψpj −1,sj −1 Cu2sj Γ(sj + 6)2
j=2
for any sj ∈ [3, pj ] and the constant Cu in (5.1).
k
Proof. Recalling that the elements in OℓC are shape-regular with hK ≃ h⊥
K ≃ hK ,
ℓ−j+1
∩ OℓC
we conclude from Lemma 5.7, Lemma 5.6 and (5.22) that, for any K ∈ Lσ
and 2 ≤ j ≤ ℓ,
X
(dcK )2|α|−2 kDα uk2L2 (K) . (5.26)
T K [η] . dcK kb
ηk2H 2 (K)
b . Ψpj −1,sj −1
mix
sj +1≤|α|≤sj +5
Since possibly K ∩ ωce 6= ∅ we write
X
kDα uk2L2 (K∩ωe ) + kDα uk2L2 (K∩wce ) . (5.27)
kDα uk2L2 (K) = kDα uk2L2 (K∩ωc ) +
e∈Ec
By inserting the weight rc , we first obtain
kDα uk2L2 (K∩ωc ) . (dcK )2+2βc −2|α| krc−1−βc +|α| Dα uk2L2 (K∩ωc )
. (dcK )2+2 min β−2|α| krc−1−βc +|α| Dα uk2L2 (K∩ωc ) .
(5.28)
Let now e be an edge in Ec . By noticing that re |K ≃ deK ≃ dcK ≃ hK , we then have
⊥
⊥
kDα uk2L2 (K∩ωe ) . (dcK )2+2βe −2|α | kre−1−βe +|α | Dα uk2L2 (K∩ωe )
⊥
. (dcK )2+2 min β−2|α| kre−1−βe +|α | Dα uk2L2 (K∩ωe ) .
18
(5.29)
Similarly, since ρce ≃ 1 on K,
⊥
e +|α |
Dα uk2L2 (K∩ωce )
kDα uk2L2 (K∩ωce ) . (dcK )2+2βc −2|α| krc−1−βc +|α| ρ−1−β
ce
⊥
−1−βe +|α | α
. (dcK )2+2 min βc −2|α| krc−1−βc +|α| ρce
D uk2L2 (K∩ωce ) .
(5.30)
Combining (5.26)–(5.30) and using (5.1) yields
T K [η] . Ψpj −1,sj −1 (dcK )2 min β Ψpj −1,sj −1 kuk2
s +5
j
M−1−β
(K)
(5.31)
≤ Cu2sj Γ(sj + 6)2 (dcK )2 min β Ψpj −1,sj −1
for any K ∈ Lσℓ−j+1 ∩ OℓC with 2 ≤ j ≤ ℓ.
Next, we sum the estimate (5.31) over all mesh layers. Referring to (5.23), we
arrive at
ΥOℓC [η] .
.
ℓ
X
j=2
K∈Lℓ−j+1
∩OℓC
σ
ℓ
X
X
X
T K [η]
(5.32)
Ψpj −1,sj −1 σ
2(ℓ−j) min β
Cu2sj Γ(sj
2
+ 6) ,
ℓ+j−1
j=2 K∈Lσ
∩Oℓ
C
for any integers 3 ≤ sj ≤ pj . The conclusion now follows by noting that the cardinality
of the set Lℓ+j−1
∩ OℓC is uniformly bounded in j and by using an interpolation
σ
argument as in Lemma 5.8.
Finally, to show that the bound in Lemma 5.11 is exponentially convergent, we
shall make use of the following refinement of Lemma 5.9.
Lemma 5.12. For every 0 < σ < 1, constants β > 0, c > 0, q ≥ 0 and sequences
{pj }∞
j=2 , pj ≥ 3, of s-linear growth as in (5.25), there exist constants b > 0 and C > 0
(depending only on q, σ, s, χ and β) such that


ℓ

X
σ 2(ℓ−j)β min c2sj Γ(sj + q)2 Ψpj −1,sj −1 ≤ C 2 exp(−2bℓ)
(5.33)


sj ∈[3,pj ]
j=2
for every ℓ ≥ 2.
Proof. By Lemma 5.9,
min c2sj Γ(sj + q)2 Ψpj −1,sj −1 ≤ C exp(−2bpj ),
sj ∈[3,pj ]
with constants b > 0 and C > 0 only depending on the values of q and c in that
Lemma. Hence, since pj ≃ sj, the sum (5.33) can be bounded by
ℓ
X
j=2
σ 2(ℓ−j)β
min c2sj Γ(sj + q)2 Ψpj −1,sj −1 ≤ C
sj ∈[3,pj ]
ℓ
X
e−2β| log σ|(ℓ−j)−2bsj .
(5.34)
j=2
We split the sum into two partial sums as follows: first, a sum over 2 ≤ j ≤ θℓ
(corresponding to mesh layers with ‘small’ elements) and second, a sum over θℓ ≤ j ≤
19
ℓ with a parameter 0 < θ < 1 which is independent of ℓ. Then we estimate the sum
in (5.34),
ℓ
X
⌊θℓ⌋
e
−2β| log σ|(ℓ−j)−2bsj
j=2
≤
X
e−2β| log σ|(ℓ−j)−2bsj +
j=2
ℓ
X
e−2β| log σ|(ℓ−j)−2bsj
j=⌈θℓ⌉
. ⌈θℓ⌉e
−2β| log σ|(1−θ)ℓ
+ ⌈(1 − θ)ℓ⌉e−2bsθℓ ,
from where the assertion (5.33) follows.
Lemma 5.11 and Lemma 5.12 give the following result.
Proposition 5.13. Under the regularity assumption (5.1), the hp-dG interpolation operator Πu defined in (5.10) satisfies the error bound
ΥOℓC [η] . exp(−2bℓ)
for some constant b > 0 independent of ℓ ≥ 2.
5.2.3. Submesh OℓE . In this section, we consider the edge neighborhood ΩE
in (2.5) and shall bound ΥOℓE [η]. As before we may assume without loss of generality
k
that E = {e}. We recall that for elements K ∈ OℓE , the diameters hK parallel to e
are of order one, while the diameters h⊥
K perpendicular to edge e satisfy
re |K ≃ deK ≃ h⊥
K,
(5.35)
according to (3.6) (see also Figure 3.1, middle). We also observe that
∩ OℓE :
∀ 2 ≤ j ≤ ℓ, ∀ K ∈ Lℓ−j+1
σ
deK ≃ σ ℓ−j .
(5.36)
As in (5.23), we can omit j = ℓ + 1 corresponding to layer L0σ ⊂ Oint . The hpℓ
extensions (Ex1)–(Ex4) of [23] for the space Vσ,s
in (3.17) yield anisotropic elemental
k
ℓ−j+1
polynomial degrees p⊥
∩
K and pK that are identical over all elements K in Lσ
k
ℓ
⊥
OE ; we thus simply denote them by pj and pj , respectively. The variation of the
polynomial degrees p⊥
j across mesh layers is s-linear as in (5.25), while the polynomial
k
k
degrees pj ≥ 3 parallel to the edge e are constant and proportional to ℓ, i.e., pj =
∩ OℓE .
pk ≥ 3, with pk ≃ ℓ. Next, we bound T K [η] for K ∈ Lℓ−j+1
σ
ℓ
Lemma 5.14. Let K ∈ Lℓ−j+1
∩
O
for
2
≤
j
≤
ℓ.
Then,
under the regularity
σ
E
assumption (5.1), there holds
2sk p
k
2
2s⊥
j Γ(s⊥ + 6)2 + Ψ k
C
Γ(s
+
6)
⊥
C
T K [η] . σ 2(ℓ−j)βe Ψp⊥
k
p −1,s −1
j
j −1,sj −1
⊥
k
k
for any s⊥
j ∈ [3, pj ] and s ∈ [3, p ].
k
Proof. Using that hK is of order one, Lemma 5.7 and Lemma 5.6 yield
T K [η] . kb
η k2H 2
k
b
mix (K)
. Epk ,sk (K) + Ep⊥⊥ ,s⊥ (K),
j
j
k
with Epk ,sk (K) and Ep⊥⊥ ,s⊥ (K) defined in Lemma 5.6. Taking into account (5.35) and
j
j
k
k
that hK is of order one, we can bound Epk ,sk (K) as follows:
k
Epk ,sk (K) ≃ Ψpk −1,sk −1
X
⊥
(deK )2|α
⊥
α⊥
1 ≤2,α2 ≤2
20
|−2
⊥
k
s
kDα
⊥ Dk
+1
uk2L2 (K) .
Since K ∩ Ω0 = ∅ and K ∩ ωc = ∅ for all c ∈ C, we may write
X
kDα uk2L2 (K) = kDα uk2L2 (K∩ωe ) +
kDα uk2L2 (K∩ωce )
c∈Ce
We then proceed by inserting the weight function re and employing (5.35):
⊥
⊥
kDα uk2L2 (K∩ωe ) . (deK )2+2βe −2|α | kre−1−βe +|α | Dα uk2L2 (K∩ωe ) .
k
If K ∩ ωce 6= ∅ for a corner c ∈ Ce , then rc |K ≃ hK is bounded away from zero, and
ρce ≃ re . Hence, we readily obtain
⊥
⊥
−1−βe +|α | α
kDα uk2L2 (K∩ωce ) . (deK )2+2βe −2|α | krc−1−βc +|α| ρce
D uk2L2 (K∩ωce ) .
Combining these estimates with (5.36) and (5.1), we find that
k
Epk ,sk (K) . Ψpk −1,sk −1 (deK )2βe kuk2
k
s +5
(K)
M−1−β
e
(5.37)
k
. Ψpk −1,sk −1 σ 2(ℓ−j)βe C 2s Γ(sk + 6)2 .
Similarly, we bound Ep⊥⊥ ,s⊥ (K):
j
j
X
Ep⊥⊥ ,s⊥ (K) . Ψp⊥
⊥
j −1,sj −1
j
j
⊥
(deK )2|α
|−2
⊥
⊥
k
s⊥
j +1≤|α |≤sj +3,α ≤2
⊥
k
α
2
kDα
⊥ Dk ukL2 (K) .
Proceeding as before, we see that
Ep⊥⊥ ,s⊥ (K) . Ψp⊥
⊥
(deK )2βe kuk2
j −1,sj −1
j
s⊥ +5
j
(K)
M−1−β
e
j
⊥
(5.38)
2
≤ Ψp⊥
⊥
σ 2(ℓ−j)βe C 2sj Γ(s⊥
j + 6) .
j −1,sj −1
The bounds (5.2.3), (5.37) and (5.38) imply the desired estimate for integer regularity
exponents. An interpolation argument as in (5.18) proves the assertion.
Proposition 5.15. Under the regularity assumption (5.1), the hp-dG interpolation operator Πu defined in (5.10) satisfies the error bound
ΥOℓE [η] . exp(−2bℓ)
for some constant b > 0 independent of ℓ ≥ 2.
Proof. Summing the bound in Lemma 5.14 over all mesh layers and noting that
the cardinality of the sets Lℓ−j+1
∩ OℓE are uniformly bounded in j result in
σ
ΥOℓC [η] . S k + S ⊥ ,
(5.39)
where
S⊥ =
ℓ
X
2s⊥
j
Ψp⊥
⊥
σ 2(ℓ−j)βe Cu
j −1,sj −1
2
Γ(s⊥
j + 6) ,
j=2
k
S =
ℓ
X
k
Ψpk −1,sk −1 σ 2(ℓ−j)βe Cu2s Γ(sk + 6)2 .
j=2
21
OℓC
OℓCE
11111111111
00000000000
00000000000
11111111111
00000000000
11111111111
ℓ=1
00000000000
11111111111
00000000000
11111111111
ℓ
00000000000
11111111111
OℓCE ⊃ Oce
00000000000
11111111111
ℓ
00000000000
11111111111
⊃ Oce′
00000000000
11111111111
000000
111111
00000000000 ℓ = 2
11111111111
000000
111111
e′
000000
111111
000000
111111
ℓ = 3 i = 2, . . . , ℓ + 1
000000
111111
000000
111111
ℓ=4
000000
111111
0000
1111
0000
1111
0000
1111
00
11
00
11
j = 1, . . . , ℓ + 1
e
00
11
00
11
00
11
00
11
c
Fig. 5.1. Elements in OℓCE .
Due to the s-linearity of the degrees p⊥
j and the fact that βe > 0, Lemma 5.12 allows
⊥
us to find parameters s⊥
∈
[3,
p
]
such
that S ⊥ . e−2bℓ .
j
j
In the sum S k , the polynomial degree pk parallel to the edge e is constant and
proportional to ℓ. Applying Lemma 5.9, we can find sk ∈ [3, pk ] such that
S k . e−2bℓ
ℓ
X
σ 2(ℓ−j)βe . e−2bℓ ,
j=2
which completes the proof.
5.2.4. Submesh OℓCE . Finally, we consider the corner-edge neighborhood ΩCE
defined in (3.3) and prove the exponential convergence of ΥOℓCE [η]. Again, it is sufficient to consider a single corner c with a single edge e = ec originating from it, i.e.,
C = {c} and E = {e}. In view of (3.3), an element K ∈ OCE has empty intersection
with Ω0 , ΩC , and ΩE . Hence, if the edges and vertices are sufficiently separated by
the initial mesh, we may assume K ⊆ ωce .
It will be convenient in the error analysis to group elements K ∈ OℓCE into sets Lij
CE
of elements whose aspect ratios are equivalent uniformly with respect to ℓ. To this
end, we observe that there exists κ(σ) > 0 such that for all elements K ∈ OℓCE there
are indices 2 ≤ i ≤ j ≤ ℓ + 1 such that
k
ℓ−i
re |K ≃ deK ≃ h⊥
,
K ≃σ
rc |K ≃ dcK ≃ hK ≃ σ ℓ−j .
(5.40)
We say that K ∈ OℓCE belongs to Lij
CE if it satisfies (5.40) with indices (i, j); cf. (3.10)
and Remark 3.5. Then, for ℓ sufficiently large, we may write (with a possibly nondisjoint union)
OℓCE
=
ℓ+1
[
[ n(j)
Lij
CE ,
(5.41)
j=2 i=2
with indices n(j) satisfying 2 ≤ n(j) ≤ j. We refer to Figure 5.1 for the notation
and illustration and observe that the cardinality of all Lij
CE is uniformly bounded,
independently of i, j.
22
Moreover, from the hp-extensions (Ex1)–(Ex4) in [23] we obtain polynomial deℓ
gree distributions for the dG subspace Vσ,s
that satisfy
k
∀ K ∈ Lij
CE :
pK = (p⊥
i , pj ) ≃ (si, sj),
2 ≤ i ≤ j ≤ ℓ + 1.
(5.42)
k
k
⊥
⊥
Lemma 5.16. Let K ∈ Lij
CE with pK = (pi , pj ) and with pi , pj ≥ 3. Then, under
k
k
⊥
the regularity assumption (5.1), for any s⊥
i ∈ [3, pi ], sj ∈ [3, pj ], there holds
T K [η] ≤ σ 2(ℓ−j)(βc −βe ) σ 2(ℓ−i)βe (1 + σ 2(j−i) )N [u]ij ,
(5.43)
where
k
⊥
k
2sj
2
N [u]ij = Ψp⊥
⊥
C 2si Γ(s⊥
Ψpk −1,sk −1 Γ(sj + 6)2 .
i + 6) + C
i −1,si −1
j
(5.44)
j
Proof. By combining Lemma 5.7, (5.40), and Lemma 5.6, we obtain
η k2H 2 (K)
T K [η] . (deK )2 (dcK )−1 + dcK kb
b
mix
k
⊥
e 2 c −2
c
E k k (K) + Ep⊥ ,s⊥ (K) ,
. dK 1 + (dK ) (dK )
pj ,sj
with E
k
k
k
pj ,sj
E
k
k
j
(5.45)
j
(K) and Ep⊥⊥ ,s⊥ (K) defined in Lemma 5.6. Using (5.40), we estimate
j
j
(K) as follows:
k
pj ,sj
E
k
k k
pj ,sj
(K) ≃ Ψpk −1,sk −1
j
j
X
⊥
(deK )2|α
⊥
α⊥
1 ≤2,α2 ≤2
|−2
k
⊥
k
s +1
(dcK )2sj +1 kDα Dkj
uk2L2 (K) .
Then, inserting the weight ρce ,
⊥
k
s +1
kDα Dkj
⊥
uk2L2 (K) . (dcK )2+2βc −2(|α
k
|+sj +1)−2−2βe +2|α⊥ |
k
k
−1−βc +(|α⊥ |+sj +1) −1−βe +|α⊥ | α⊥ sj +1
ρce
D Dk uk2L2 (K) .
⊥
× (deK )2+2βe −2|α | krc
Hence,
E
k
k
k
pj ,sj
(K) . Ψpk −1,sk −1 (dcK )2(βc −βe )−1 (deK )2βe kuk2
j
j
k
s +5
.
(5.46)
j
M−1−β
(K)
Similarly,
X
⊥
Ep⊥⊥ ,s⊥ (K) ≃ Ψp⊥
i −1,si −1
j
j
⊥
(deK )2|α
⊥
k
⊥
s⊥
i +1≤|α |≤si +3,α ≤2
|−2
k
(dcK )2α
−1
⊥
k
2
α
kDα
⊥ Dk ukL2 (K) ,
where
⊥
k
k
2
α
c 2α
kDα
⊥ Dk ukL2 (K) . (dK )
⊥
−1+2+2βc −2(|α⊥ |+αk )−2−2βe +2|α⊥ |
⊥
× (deK )2+2βe −2|α | krc−1−βc +(|α
23
|+αk ) −1−βe +|α⊥ | α⊥ αk
ρce
D Dk uk2L2 (K) .
Thus,
Ep⊥⊥ ,s⊥ (K) . Ψpk −1,sk −1 (dcK )2(βc −βe )−1 (deK )2βe kuk2
j
j
i
s⊥ +5
i
M−1−β
(K)
i
.
(5.47)
Referring to (5.45), (5.46), (5.47), and to the regularity property (5.1) shows that
(5.48)
T K [η] . (dcK )2(βc −βe ) (deK )2βe 1 + (deK )2 (dcK )−2 N [u]ij ,
which is the assertion for integer regularity exponents. An interpolation argument as
in Lemma 5.8 and using the relations in (5.40) once more finish the proof.
We are now ready to prove exponential convergence of ΥOℓCE [η].
Proposition 5.17. Under the regularity assumption (5.1) and as ℓ → ∞, the
hp-dG interpolation operator Πu defined in (5.10) satisfies the error bound
ΥOℓCE [η] . exp(−2bℓ)
for some constant b > 0 independent of ℓ.
Proof. Summing up the estimate in Lemma 5.16 over all the mesh layers (using
the fact that the cardinalities of the sets Lij
CE are uniformly bounded) results in
ΥOℓCE [η] .
ℓ+1 n(j)
X
X
σ 2(ℓ−j)(βc −βe ) σ 2(ℓ−i)βe (1 + σ 2(j−i) )N [u]ij
j=2 i=2
.
ℓ+1 n(j)
X
X
σ 2(ℓ−j)(βc −βe ) σ 2(ℓ−i)βe N [u]ij . S ⊥ + S k ,
j=2 i=2
where
S⊥ =
ℓ+1
X
n(j)
σ 2(ℓ−j)(βc −βe )
⊥
2
⊥
C 2si Γ(s⊥
σ 2(ℓ−i)βe Ψp⊥
i + 6) ,
i −1,si −1
i=2
j=2
Sk =
X
n(j)
ℓ+1
X
X
k
k
σ 2(ℓ−i)βe Ψpk −1,sk −1 C 2sj Γ(sj + 6)2 .
σ 2(ℓ−j)(βc −βe )
j
i=2
j=2
j
Let us first bound S ⊥ . To do so, we write
⊥
S .
ℓ+1
X
σ
2(ℓ−j)(βc −βe +βe )
σ
2(ℓ−i)βe
σ
2(j−ℓ)βe
Ψp⊥
C
⊥
i −1,si −1
i=2
j=2
=
j
X
ℓ+1
X
σ 2(ℓ−j)βc
j=2
j
X
⊥
C
σ 2(j−i)βe Ψp⊥
i −1,si −1
2s⊥
i
2
Γ(s⊥
i + 6)
i=2
!
2s⊥
i
Γ(s⊥
i
2
+ 6)
!
.
Then, we use the fact that βe > 0, Lemma 5.12 with ℓ replaced by j, and the s⊥
⊥
linearity of p⊥
i to obtain parameters si ∈ [3, pi ] and a constant b1 > 0 such that
j
X
⊥
2
−2b1 j
⊥
C 2si Γ(s⊥
,
σ 2(j−i)βe Ψp⊥
i + 6) . e
i −1,si −1
i=2
24
j ≥ 2.
Therefore, we conclude that there is second constant b2 > 0 such that
S⊥ .
ℓ+1
X
σ 2(ℓ−j)(βc −βe ) e−2b1 j .
ℓ+1
X
σ 2(ℓ−j)(βc −βe +βe ) e−2b1 j =
e−2b2 (ℓ−j)−2b1 j .
j=2
j=2
j=2
ℓ+1
X
With b = min{b1 , b2 }, we thus obtain
S⊥ .
ℓ+1
X
e−2b(ℓ+j−j) . ℓe−2bℓ . e−2bℓ .
(5.49)
j=2
To prove exponential convergence of S k , we proceed similarly and note that
n(j)
X
i=2
σ 2(ℓ−i)βe ≤
j
X
σ 2(ℓ−i)βe = σ 2(ℓ−j)βe
i=2
1 − σ 2βe (j−1)
≤ C(σ, βe )σ 2(ℓ−j)βe ,
1 − σ 2βe
for j ≥ 2. Hence, by Lemma 5.12,
Sk .
ℓ+1
X
k
k
σ 2(ℓ−j)βc Ψpk −1,sk −1 C 2sj Γ(sj + 6)2 . e−2bℓ .
j=2
j
j
This completes the proof.
5.3. Approximation in Oℓσ . By combining the bound in (5.14) with the results
in Propositions 5.10, 5.13, 5.15 and 5.17, we now immediately obtain the following
approximation property in Oℓσ .
(ℓ)
Theorem 5.18. Consider a family Mσ = {Mσ }∞
ℓ=1 of axi-parallel σ-geometric
(ℓ)
meshes with an anisotropic s-linear polynomial degree vector p2 (Mσ ) as in (3.17)
with degrees greater than or equal to 3. Then for u ∈ A−1−β (Ω) and ℓ ≥ 2, there is a
projection Πℓ : A−1−β (Ω) → V (Oℓσ , Φ(Oℓσ ), p2 (Oℓσ )) that satisfies the error bound
ΥOℓσ [u − Πℓ u] ≤ C exp(−2bℓ),
ℓ ≥ 2.
(5.50)
Here, the constants b > 0 and C > 0 are independent of ℓ (but depend on the weight
vector β, the geometric grading factor σ, the slope s, the regularity constant Cu
in (5.1), and on the initial mesh M0 ).
ℓ
Moreover, as ℓ → ∞, Nℓ (s) := dim(Vσ,s
) ≃ ℓ5 + O(ℓ4 ) , and the approximation
bound (5.50) can be written as
√
5
ΥOℓσ [u − Πℓ u] ≤ C exp(−2b N ),
ℓ
N = dim(Vσ,s
) → ∞.
(5.51)
By repeating verbatim the proofs of Propositions 5.13, 5.15 and 5.17 with uniform
polynomial degrees, and by replacing in these proofs each reference to Lemma 5.12
by a reference to Lemma 5.9, we obtain the following corollary.
Corollary 5.19. Under the assumptions of Theorem 5.18, but for uniform
polynomial degrees pK = p ≃ ℓ (with p ≥ 3) in all elements K ∈ Oℓσ , ℓ ≥ 2, the error
bounds (5.50), (5.51) remain valid, however, in (5.51), the constant b is replaced by
a smaller value b.
25
5.4. Error Bounds on Tℓσ . We now address the error in elements in the ter(ℓ)
minal layers Tℓσ ⊂ Mσ . Again, we proceed separately for elements near edges and
vertices and recall, to this end, the partition (3.11): Tℓσ = VCℓ ∪ VEℓ . Based on this, we
write
ΥTℓσ [η] ≤ ΥVCℓ [η] + ΥVEℓ [η],
(5.52)
where ΥTℓσ [η] is the term (4.9) in Theorem 4.2, and the consistency errors ΥVCℓ [η],
ΥVEℓ [η] are taken on the respective submeshes of Tℓσ as defined in (3.11).
The construction of the hp-interpolant Πu from (4.6) in Tℓσ will exploit the homo1
geneous essential boundary conditions: for functions in M−1−β
(Ω), the corresponding
2
1
1
L - and H -terms in the norm (2.6) on M−1−β (Ω) carry weights with negative exponents. Consequently, it will be sufficient to approximate the solution of (1.1)–(1.2)
on the exponentially small elements in Tℓσ by the zero function: we set Πu|K ≡ 0 for
all K ∈ Tℓσ and, hence, η = u in (4.6), and (4.9).
In the sequel, the interpolation errors on the subsets VCℓ and VEℓ will be analyzed
separately. We first prove the following auxiliary result:
2
Lemma 5.20. Let K ∈ Tℓσ , u ∈ M−1−β
(K), and 0 ≤ j . ℓ be chosen such
k
ℓ
ℓ−j
that h⊥
(cf. Proposition 3.4). Then
K ≃ σ and hK ≃ σ
j
3
k∇ukL1 (K) ≤ Cσ ℓ( 2 +min β)− 2 kukM 1
−1−β (K)
,
and
j
1
σ −ℓ Dk ∇uL1 (K) + kD⊥ ∇ukL1 (K) ≤ Cσ ℓ( 2 +min β)− 2 kukM 2
−1−β (K)
,
where the constant C > 0 does not depend on u, σ, ℓ, p and β.
Proof. We may assume that there is at most one corner c ∈ C such that K ∩ωc 6= ∅
or K ∩ (ωce ∪ ωe ) 6= ∅ for some edge e ∈ Ec . Then we write
X
k∇ukL1 (K∩ωe ) + k∇ukL1 (K∩ωce ) .
k∇ukL1 (K) = k∇ukL1 (K∩ωc ) +
e∈Ec
k
ℓ
First, note that if K ∩ ωc 6= ∅, then K must be isotropic with h⊥
K ≃ hK ≃ hK ≃ σ
ℓ
3ℓ
(i.e., j = 0). Thus, Hölder’s inequality and the fact that rc . σ , |K| ≃ σ yield
3
k∇ukL1 (K∩ωc ) ≤ rcβc L2 (K∩ω ) rc−1−βc +1 ∇uL2 (K∩ω ) . σ ℓ( 2 +βc ) kukM 1
(K∩ωc ) .
c
c
Then, if K ∩ ωe 6= ∅ for
X
k∇ukL1 (K∩ωe ) ≤
|α|=1
−1−β
e ∈ Ec , we have similarly re . σ ℓ and |K| ≃ σ 2ℓ σ ℓ−j so that
−1−βe +|α⊥ | α⊥ αk 1+βe −|α⊥ | D⊥ Dk u re
re
2
2
L (K∩ωe )
L (K∩ωe )
. reβe L2 (K∩ωe ) kukM 1
3
−1−β (K∩ωe )
j
. σ ℓ( 2 +βe )− 2 kukM 1
−1−β (K∩ωe )
Furthermore, if K ∩ ωce 6= ∅ for e ∈ Ec , we have
X βc 1+βe −|α⊥ | k∇ukL1 (K∩ωce ) ≤
rc ρce
L2 (K∩ωce )
|α|=1
⊥
α⊥ αk e +|α |
u
× rc−1−βc +1 ρ−1−β
D
D
ce
⊥
k
L2 (K∩ωce )
26
.
.
With rc ≃ σ ℓ−j on K ∩ ωce and re . σ ℓ , we find that
βc 1+βe −|α⊥ | . rcβc ρβcee L2 (K∩ωce ) = rcβc −βe reβe L2 (K∩ωce )
rc ρce
2
L (K∩ωce )
. σ (ℓ−j)(βc −βe ) reβe L2 (K∩ωce )
1
. σ (ℓ−j)(βc −βe )+ℓβe σ ℓ σ 2 (ℓ−j)
j
3
3
j
= σ 2 ℓ− 2 σ (ℓ−j)βc +jβe . σ 2 ℓ− 2 σ ℓ min β .
Hence,
j
3
k∇ukL1 (K∩ωce ) . σ ℓ( 2 +min β)− 2 kukM 1
−1−β (K∩ωce )
,
and thus the first bound follows. The proof of the second inequality is similar.
5.4.1. Interpolation on VCℓ . We first consider elements K ∈ VCℓ which abut at
k
ℓ
exactly one corner c ∈ C. Such elements are isotropic with hK ≃ h⊥
K ≃ hK ≃ σ . For
ℓ
convenience, let us suppose that the mesh is fine enough, so that VC ⊂ ΩC ∪ ΩCE .
2
Proposition 5.21. Let K ∈ VCℓ and u ∈ M−1−β
(K). Then there holds
ΥVCℓ [η] ≤ Cσ 2ℓ min β kuk2M 2
−1−β (Ω)
. exp(−2bℓ) kuk2M 2
−1−β (K)
.
(5.53)
Proof. Let K ∈ VCℓ abut at corner c ∈ C with K ∩ ωc 6= ∅. We shall use
that supK rc ≃ hK ≃ σ ℓ . Then,
2
−2
2+2βc −1−βc 2
h−2
u L2 (K∩ω
rc
K kηkL2 (K∩ωc ) . hK sup rc
K
c)
. σ 2ℓβc kuk2M 1
−1−β (K∩ωc )
,
and
2
2
2
k∇ηkL2 (K∩ωc ) . sup rc2βc rc−1−βc +1 ∇uL2 (K∩ωc ) . σ 2ℓβc kukM 1
−1−β (K∩ωc )
K
.
Furthermore, if K ∩ ωce 6= ∅ for e ∈ Ec , we use that ρce is bounded on K ∩ ωce to
conclude that
2
−2
2+2βe −1−βc −1−βe 2
h−2
sup rc2+2βc ρce
u L2 (K∩ωce )
ρce
rc
K kηkL2 (K∩ωce ) . hK
.σ
K∩ωce
2
kukM 1
(K∩ωce )
−1−β
2ℓβc
,
and
2
k∇ηkL2 (K∩ωce )
. sup rc2βc
K
X
⊥
|α|=1
⊥
2+2βe −2|α | −1−βc +|α| −1−βe +|α | α
krc
ρce
D uk2L2 (K∩ωce )
sup ρce
K∩ωce
. σ 2ℓβc kuk2M 1
−1−β (K∩ωce )
.
Moreover, for f ∈ FK , we have that
−1
|f |
2
2
−1
h⊥
K,f k∇ηkL1 (f ) . hK k∇ukL1 (f ) ,
27
and employing the trace inequality (4.3) with s = 1, we obtain
−1
2
2
−3
−1 2 2
|f | h⊥
∇ u L1 (K) .
K,f k∇ηkL1 (f ) . hK k∇ukL1 (K) + hK
Next, using Lemma 5.20 with j = 0 results in
|f |
−1
2
2
2ℓ min β
h⊥
kukM 2
K,f k∇ηkL1 (f ) . σ
−1−β (K)
.
Summing up the above bounds completes the proof.
5.4.2. Interpolation on VEℓ . Elements K along Dirichlet edges may be anisok
tropic. They are parallel to some edge e ∈ E, with maximal length hK ≃ σ ℓ−j ,
for some 0 ≤ j . ℓ, in the direction parallel to e; their diameter in the direction
ℓ
orthogonal to e is h⊥
K ≃ σ ; see Proposition 3.4.
2
Proposition 5.22. Let K ∈ VEℓ and u ∈ M−1−β
(K). Then there holds
ΥVEℓ [η] ≤ Cσ 2ℓ min β kuk2M 2
−1−β (Ω)
. exp(−2bℓ) kuk2M 2
−1−β (K)
,
where σ ∈ (0, 1) is the geometric refinement parameter and where ℓ denotes the refinement level. The constant C > 0 is independent of u, σ, ℓ, the polynomial degree
vector p, and min β > 0 (cf. (2.8)), and the constant b > 0 depends on σ, β.
Proof. We distinguish three cases.
Case 1. If K ∩ ωc 6= ∅, for some c ∈ C, then K is isotropic, and we may proceed
as in the previous section. This leads to an estimate very similar to (5.53) (with the
left-hand side restricted to VEℓ ∩ ΩC ).
Case 2. If K ∩ωe 6= ∅, for some e ∈ Ec , then the weighted Sobolev norm from (2.6)
close to an edge e ∈ E behaves locally like
X −1−βe +|α⊥ | α 2
D
u
.
≃
kuk2M 2
r
e
(K∩ωe )
L2 (K∩ωe )
−1−β
|α|≤2
−1
−1
. h⊥
, there holds:
Noting that supf ∈FK h⊥
K
K,f
max h⊥
K,f
f ∈FK
−2
. h⊥
K
2
2
2
kηkL2 (K∩ωe ) + k∇ηkL2 (K∩ωe ) + |f |−1 h⊥
K,f k∇ηkL1 (f ∩ωe )
−2
2
2
2
kukL2 (K∩ωe ) + k∇ukL2 (K∩ωe ) + |f |−1 h⊥
K,f k∇ukL1 (f ∩ωe ) ,
ℓ
for any f ∈ FK . Here, using that supK∩ωe re ≃ h⊥
K ≃ σ , we have
h⊥
K
and
−1
kukL2 (K∩ωe ) . h⊥
K
k∇ukL2 (K∩ωe ) .
X
|α|=1
βe −1−βe re
uL2 (K∩ωe ) . σ ℓβe kukM 1
−1−β (K∩ωe )
h⊥
K
,
1+βe −|α⊥ | −1−βe +|α⊥ | α D u
re
L2 (K∩ωe )
. σ ℓβe kukM 1
−1−β (K∩ωe )
.
Next, by means of (4.3) with s = 1, we see that
−1 2
2
2
2
−1
kD⊥ ∇ukL1 (K) .
|f |−1 h⊥
h⊥
k∇ukL1 (K) + h⊥
K,f k∇ukL1 (f ∩ωe ) . |f |
K,f
K,f
28
−1
⊥
ℓ
−1
h
≃
Therefore, if f is parallel to e, there holds h⊥
≃
σ
and
thus
|f
|
K,f
K,f
−1
σ ℓ σ ℓ−j
σ −ℓ = σ j−3ℓ . In this case,
2
j−3ℓ
k∇uk2L1 (K) + σ 2ℓ kD⊥ ∇uk2L1 (K) .
|f |−1 h⊥
K,f k∇ukL1 (f ∩ωe ) . σ
Invoking Lemma 5.20 leads to
2
2ℓ min β
kuk2M 2
|f |−1 h⊥
K,f k∇ukL1 (f ∩ωe ) . σ
−1−β (K)
.
ℓ−j
If f is orthogonal to e, there holds h⊥
and therefore |f |−1 h⊥
K,f ≃ σ
K,f
σ −2ℓ σ −(ℓ−j) = σ j−3ℓ . In this case,
2
2
2
j−3ℓ
k∇ukL1 (K) + σ 2(ℓ−j) Dk ∇uL1 (K) .
|f |−1 h⊥
K,f k∇ukL1 (f ∩ωe ) . σ
−1
≃
Thence, with the aid of Lemma 5.20, we obtain
2
2ℓ min β 2(ℓ−j)
|f |−1 h⊥
σ
kuk2M 2
K,f k∇ukL1 (f ∩ωe ) . σ
−1−β (K)
. σ 2ℓ min β kuk2M 2
−1−β (K)
.
Case 3. Finally, if K ∩ ωce 6= ∅, then the norm from (2.6) behaves locally as
follows:
X −1−βc +|α| −1−βe +|α⊥ | α 2
ρ
D
u
.
kuk2M 2
≃
r
2
c
ce
(K∩ωce )
L (K∩ωce )
−1−β
|α|≤2
Here, for the volume terms, we may proceed similarly as in Case 2 by carefully taking into account the weight ρce . More precisely, using that supK∩ωce rc ≃ σ ℓ−j
ℓ
and supK∩ωce re ≃ h⊥
K ≃ σ , we have
h⊥
K
−1
kukL2 (K∩ωce ) . h⊥
K
e −1−βc −1−βe u L2 (K∩ω
sup rc1+βc ρ1+β
ρce
rc
ce
−1
K∩ωce
ce )
. σ (ℓ−j)(βc −βe ) σ ℓβe kukM 1
−1−β (K∩ωce )
.σ
ℓ min β
kukM 1
−1−β (K∩ωce )
,
and
k∇ukL2 (K∩ωce ) . sup rcβc ρβcee
K∩ωce
X −1−βc +1 −1−βe +|α⊥ | α ρce
D u
rc
L2 (K∩ωce )
|α|=1
. σ (ℓ−j)(βc −βe )+ℓβe kukM 2
−1−β (K∩ωce )
.σ
ℓ min β
kukM 2
−1−β (K∩ωce )
.
Furthermore, the face expressions are again estimated by employing Lemma 5.20.
This leads to
−2
2
2
2
kηkL2 (K∩ωce ) + k∇ηkL2 (K∩ωce ) + |f |−1 h⊥
max h⊥
K,f k∇ηkL1 (f ∩ωce )
K,f
f ∈FK
. σ 2ℓ min β kuk2M 2
−1−β (K)
This completes the proof.
29
.
5.4.3. Exponential Convergence in Terminal Layers. Summing up the
above estimates for all terminal layer elements, we have proved the following result:
2
Proposition 5.23. Let u ∈ M−1−β
(Ω). Then there holds
2
ΥVCℓ [η] + ΥVEℓ [η] ≤ Cσ 2ℓ min β kukM 2
−1−β (Ω)
2
≃ exp(−2bℓ) kukM 2
−1−β (Ω)
,
where σ is the geometric refinement parameter and ℓ the refinement level. The constant C > 0 is independent of u, σ, ℓ, the polynomial degree vector p, and min β > 0
(cf. (2.8)), and the constant b > 0 depends on σ, β.
6. Exponential Convergence. The hp-interpolation error estimates from the
previous section together with Theorem 4.2 allow us to deduce exponential convergence rates of the hp-dG discretizations in (4.1), provided the solution u of (1.1)–(1.2)
admits analytic regularity.
Theorem 6.1. Assume that the right-hand side f of the boundary value problem (1.1)–(1.2) in the axiparallel polyhedron Ω ⊂ R3 belongs to the analytic space
A1−β (Ω) with a weight vector β satisfying (2.8), and that the diffusion matrix A satisfies the properties from Section 1; then the solution u is in A−1−β (Ω) according to
Proposition 2.1.
(ℓ)
Furthermore, let Mσ = {Mσ }ℓ≥0 be a family of axiparallel σ-geometric meshes
as in Section 3.1, and consider the hp-dG discretizations in (4.1) based on the seℓ
quences of approximating subspaces Vσℓ and Vσ,s
defined in (3.16) respectively (3.17),
(ℓ)
(ℓ)
with the mapping vectors Φ(Mσ ) and with the vector p1 (Mσ ) in (3.16) of constant,
isotropic and uniform polynomial degrees equal to ℓ for the space Vσℓ , respectively the
(ℓ)
ℓ
s-linear, anisotropic degree distribution p2 (Mσ ) for Vσ,s
. All polynomial degrees are
assumed greater than or equal to 3.
Then for each ℓ ≥ 0, the hp-dG approximation uDG is well-defined, and as ℓ → ∞,
the approximate solutions uDG satisfy the error estimate
√ 5
(6.1)
|||u − uDG |||DG ≤ C exp −b N
(ℓ)
(ℓ)
(ℓ)
where N = dim(V (Mσ , Φ(Mσ ), p(Mσ ))) denotes the number of degrees of freeℓ
dom of the discretization for any of the two spaces Vσℓ or Vσ,s
.
The constants b > 0 and C > 0 are independent of N , but depend on σ, M0 , θ, γ,
(ℓ)
(ℓ)
α0 , min β > 0, and on which of the polynomial degree vectors p1 (Mσ ) or p2 (Mσ )
are used.
Proof. This follows readily from Theorem 4.2, Theorem 5.18, and Proposition 5.23. Note that the constants b > 0 and C > 0 have to be suitably modified to
absorb the algebraic factor p4max in (4.7).
Remark 6.2. The hp-dG interpolant constructed herein yields exponential convergence as in (6.1) for any u ∈ A−1−β (Ω). In particular, for diffusion-reaction
problems as in [23] or for second-order elliptic systems [27], exponential convergence
of hp-IP dGFEM can be achieved for such solutions.
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