doi:10.1093/imanum/drn000
Second-Order Elliptic PDE with Discontinuous Boundary Data
PAULHOUSTON†ANDTHOMASWIHLER‡
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK and
Mathematisches Institut, Universit¨at Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland [Received on 17 December 2009]
We shall consider the weak formulation of a linear elliptic model problem with discontinuous Dirichlet boundary conditions. Since such problems are typically not well-defined in the standard H1−H1setting, we will introduce a suitable saddle point formulation in terms of weighted Sobolev spaces. Furthermore, we will discuss the numerical solution of such problems. Specifically, we employ an hp–discontinuous Galerkin method and derive an L2-norm a posteriori error estimate. Numerical experiments demonstrate the effectiveness of the proposed error indicator in both the h– and hp–version setting. Indeed, in the latter case exponential convergence of the error is attained as the mesh is adaptively refined.
Keywords: Second-order elliptic PDE, discontinuous Dirichlet boundary conditions, inf-sup condition, hp-discontinuous Galerkin FEM, L2-norm a posteriori error analysis, exponential convergence.
1. Introduction
On a bounded polygonal domainΩ ⊂R2 with straight edges and N>1 corners C1,C2, . . . ,CN, we
consider the linear diffusion-reaction problem
−∆u+cu=f inΩ (1.1)
u=g onΓ, (1.2)
whereΓ =∂Ω denotes the boundary of Ω, c∈L∞(Ω) is a nonnegative function, f ∈L2(Ω), and g∈L2(∂Ω)is a possibly discontinuous function onΓ whose precise regularity will be specified later. Throughout the paper we shall use the following notation. For a domain D⊂Rn(n=1 or n=2) we denote by L2(D)the space of all square-integrable functions on D, with normk · k0,D. Furthermore, for an integer k∈N0, we let Hk(D)be the usual Sobolev space of order k on D, with normk · kk,D and
semi-norm| · |k,D. The space ˚H1(Ω)is defined as the subspace of H1(Ω)consisting of functions with
zero trace on∂Ω.
Several variational formulations for elliptic problems with discontinuous Dirichlet boundary condi-tions exist. We mention the very weak formulation which is to find a solution u∈L2(Ω)such that
−
Z
Ωu∆v dxxx+ Z
Ωcuv dxxx= Z
Ω f v dxxx− Z
Γg∇v·n ds
for any v∈H2(Ω)∩H˚1(Ω), where n denotes the unit outward normal vector to the boundaryΓ. It is based on twofold integration by parts of (1.1) and incorporates the Dirichlet boundary data in a natural
†[email protected] ‡[email protected]
c
way. On the other hand, however, the numerical solution by means of a conforming finite element discretisation would require continuously differentiable test functions. In order to avoid this problem, the following saddle point formulation can be used (see Neˇcas (1962)): provided that g∈H1/2−ε(∂Ω), for someε∈[0,1/2), find u∈H1−ε(Ω)with u|Γ=g such that
Z
Ω∇u·∇v dxxx+ Z
Ωcuv dxxx= Z
Ω f v dxxx (1.3)
for all v∈H1+ε(Ω)∩H˚1(Ω). We note that the bilinear form on the left hand side is formally symmetric and corresponds to the standard form for the Poisson equation. For results dealing with related finite element approximations, we refer to Babuˇska (1971).
In the present paper, a new variational formulation for (1.1)–(1.2) is presented and analysed. Here, the emphasis shall be on Dirichlet boundary conditions which may exhibit (isolated) discontinuities and are essentially continuous otherwise. The formulation in this article is closely related to the saddle point formulation (1.3), however, it features Sobolev spaces which describe the local singularities in the analytical solution resulting from the discontinuities in the boundary data in a more specific way. More precisely, weighted Sobolev spaces which have been used in the context of regularity statements for second-order elliptic boundary value problems, see, e.g., Babuˇska & Guo (1988); Babuˇska & Guo (1989); Guo & Schwab (2006), will be used. We will establish well-posedness of the weak formulation in terms of an appropriate inf-sup condition.
In order to discretise the underlying PDE problem, we exploit the hp–version of the symmetric interior penalty discontinuous Galerkin (dG) finite element method, cf. Arnold et al. (2001), and the references cited therein. DG methods are ideally suited for realising hp–adaptivity for second-order boundary-value problems, an advantage that has been noted early on in the recent development of these methods; see, for example, Baumann & Oden (1999); Cockburn et al. (2000); Houston et al. (2002, 2007, 2008); Perugia & Sch ¨otzau (2002); Rivi`ere et al. (1999); Stamm & Wihler (2010); Wihler et al. (2003) 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 ir-regularly refined meshes—the two key ingredients for hp–adaptive algorithms. A further advantage of interior penalty dG formulations is that they incorporate Dirichlet boundary conditions in a natural way irrespective of their smoothness (in fact, L1-regularity is sufficient for well-posedness). With this in mind, we shall derive a computable a posteriori bound for the error measured in terms of the L2–norm on Ω. On the basis of the resulting computable error indicators, adaptive h– and hp–mesh adapta-tion strategies will be investigated for a model second–order elliptic PDE with discontinuous boundary conditions. In particular, we shall show numerically that exploiting hp–mesh refinement leads to expo-nential convergence of the L2–norm of the error as the finite element space is enriched.
2. Variational Formulation
2.1 Weighted Sobolev Spaces LetA ={Ai}M
i=1⊂∂Ω, Ai6=Aj for i6= j, be a finite set of points on the boundary of the polygonal
domainΩ which are numbered in counter-clockwise direction along∂Ω; the points inA will signify
the locations of the discontinuities in the Dirichlet boundary condition g in (1.2). Furthermore, we denote byΓi⊂Γ, i=1,2, . . . ,M, the (open) subset ofΓ which connects the two points Ai and Ai+1; here, we set AM+1=A1. Moreover, letωi∈(0,2π]signify the interior angle of the polygonΩat Ai. To
each Ai∈A, i=1,2, . . . ,M, we associate a weightαi∈[0,1). These numbers are stored in a weight
vector
ααα= (α1,α2, . . . ,αM)∈[0,1)M. (2.1)
Moreover, for any number k∈R, we use the notation kααα = (kα1,kα2, . . . ,kαM)andααα+k= (α1+
k,α2+k, . . . ,αM+k). Furthermore, for a fixed number
η>0, (2.2)
we introduce the following weight function onΩ:
Φααα(xxx) =
M
∏
i=1
ri(xxx)αi, ri(xxx) =min{η−1|xxx−Ai|,1}.
Here, we assume thatηis small enough, so that the open sectors
Si={xxx∈Ω: |xxx−Ai|<η}, i=1,2, . . . ,M, (2.3)
do not intersect, i.e., Si∩Sj=/0 if i6=j. There holds, for xxx∈Ω, that
ri(xxx) =
(
η−1|xxx−Ai| if xxx∈S
i,
1 if xxx∈Ω\Si,
and ri∈C0(Ω), i=1,2, . . . ,M. Furthermore, setting
S =
M
[
i=1
Si, Ω0=Ω\S, we have
Φααα=
(
riαi if xxx∈Sifor some i=1,2, . . .M,
1 if xxx∈Ω0.
(2.4)
Note thatΦαααis continuous onΩ. Furthermore, forααα1,ααα2∈RM, we have
Φααα1+ααα2=Φααα1Φααα2, Φ −1
ααα =Φ−ααα.
Then, for any integers m>l>0, we define the weighted Sobolev spaces Hαααm,l(Ω)as the completion of the space C∞(Ω)with respect to the weighted Sobolev norms
kuk2
Hαααm,l(Ω)=kuk
2
l−1,Ω+
m
∑
k=l
|u|2
Hαααk,l(Ω), l>1,
kuk2
Hαααm,0(Ω)=
m
∑
k=0
Here,
|u|2
Hαααk,l(Ω)=
∑
|λλλ|=k
Φααα+k−l|Dλλλu|
2
0,Ω
is the Hαααk,l-seminorm inΩ, where
Dλλλu= ∂ |λλλ|u
∂xλ1 1 ∂xλ22
,
withλλλ = (λ1,λ2)∈N02and|λλλ|=λ1+λ2.
In addition, for m>l>1, let us define the space Hm− 1 2,l−
1 2
ααα (∂Ω)as the trace space of Hm,l(Ω), equipped with the norm
kuk
Hm−
1 2,l−12
ααα (∂Ω)
= inf
v∈Hαααm,l(Ω)
v|∂Ω=u
kvkHm,l
ααα (Ω).
Finally, we denote by ˚Hαααm,l(Ω)the subspace of Hαααm,l(Ω)consisting of functions with zero trace on∂Ω.
2.2 Inequalities in Hααα1,1(Ω)
In order to describe the well-posedness of (1.1)–(1.2), the weighted Sobolev space Hααα1,1(Ω)will play an important role. In the sequel, we shall collect a few inequalities which will be used for the analysis in this paper.
LEMMA 2.1 Let I= (a,b)⊂R, a<b, be an open interval. Then, there holds the Poincar´e-Friedrichs inequality
Z b
a φ(x)
2dx6(b−a)2
π2 Z b
a (φ
′(x))2dx
for allφ∈H1(a,b)withφ(a) =φ(b) =0.
Proof. The bound follows from (Hardy et al., 1952, Theorem 257) and a scaling argument.
Applying the previous lemma, we shall prove the following result.
LEMMA2.2 Consider a sector S={(r,θ): 0<r<R,θ0<θ<θ1} ⊂R2, where(r,θ)denote polar co-ordinates inR2, and R>0, 06θ
0<θ162πare constants. Furthermore, let u∈L2(S)withkrα∇uk0,S<
∞for someα∈[0,1), and u|∂S<=0, where∂S<={(r,θ): 0<r<R,θ∈ {θ0,θ1}}. Then, there holds Z
S
r2α−2u(xxx)2dxxx6(θ1−θ0)
2
π2 Z
S
r2α|∇u|2dxxx.
Proof. Using integration in polar coordinates, we get Z
S
r2α−2u(xxx)2dxxx= Z R
0 r2α−1
Z θ1
θ0
u2dθdr. (2.5)
Then, since for any r∈(0,R)there holds u(r,θ0) =u(r,θ1) =0, we can apply Lemma 2.1. This implies Z θ1
θ0
u2dθ6(θ1−θ0)
2
π2 Z θ1
θ0
Furthermore, noticing that|∂θu|6r|∇xxxu|, we obtain
Z θ1
θ0
u2dθ6(θ1−θ0)
2
π2 r 2Z θ1
θ0
|∇xxxu|2dθ, 0<r<R.
Inserting this estimate into (2.5), leads to Z
S
r2α−2u(xxx)2dxxx6(θ1−θ0)
2
π2 Z R
0 r2α+1
Z θ1
θ0
|∇xxxu|2dθdr.
Changing back to Cartesian coordinates xxx, completes the proof.
LEMMA2.3 Given a weight vectorααα∈[0,1)M. Then, there holds kΦ−αααuk0,Ω 6Ckuk1,Ω
for any u∈H1(Ω), where the constant C>0 only depends onαααandΩ.
Proof. Let Si, i=1,2, . . . ,M, be the (sufficiently small) sectors from (2.3). Then, we recall the property
(2.4) to write
kΦ−αααuk20,Ω=kuk
2
0,Ω0+kΦ−αααuk 2
0,S =kuk
2 0,Ω0+
M
∑
i=1
r−i αiu2
0,Si
. (2.6)
If, for some 16i6M, we have thatαi>0, then
ri−αiu
2
0,Si
6C
kuk20,Si+r1−i αi∇u
2
0,Si
6Ckuk21,Si;
this follows from expressing the norms in terms of polar coordinates and from applying (Hardy et al., 1952, Theorem 330). Inserting this into (2.6), gives the desired inequality.
LEMMA2.4 Consider a function u∈H˚1,1
ααα (Ω), whereαi∈[0,1), i=1,2, . . . ,M. Then, there holds k|∇(Φααα)|uk0,Ω 6 1
π1max6i6Mαiωi|u|Hααα1,1(Ω).
Proof. Let Si, i=1,2, . . . ,M, be the (sufficiently small) sectors from (2.3). Then, due to (2.4), we have
|∇(Φααα)|=
(
|∇(rαi)|=αiriαi−1 if xxx∈Sifor some i=1,2, . . . ,M,
0 if xxx∈Ω0.
(2.7)
Hence,
Z
Ω|∇(Φααα)|
2u2dxxx=
∑
Mi=1
α2
i
Z
Si
r2iαi−2u2dxxx. (2.8) Then, applying Lemma 2.2, we have
Z
Si
r2iαi−2u2dxxx6ω
2
i π2
Z
Si
Thus, Z
Ω|∇(Φααα)|
2u2dxxx6
∑
Mi=1
α2
iωi2
π2 Z
Si
r2αi|∇u|2dxxx6
max 16i6Mα
2
iωi2
π2 Z
ΩΦ
2
ααα|∇u|2dxxx,
as required.
Furthermore, there holds the following Poincar´e-Friedrichs inequality. LEMMA 2.5 Consider a weight vectorααα∈[0,1)M andγ⊆∂Ω withR
γds>0. Then, there exists a
constant C>0 depending only onγ,Ω, andαααsuch that
kuk0,Ω6C|u|H1,1
ααα (Ω)
for all functions u∈Hααα1,1(Ω)with u|γ=0 (in the trace sense). In particular, we have that| · |
Hααα1,1(Ω)is a
norm on ˚Hααα1,1(Ω).
Proof. We first note that the embedding W1,1(Ω)֒→L2(Ω)is continuous for Lipschitz polygons inR2 (cf., e.g., (Adams & Fournier, 2003, Theorem 4.12)). Hence, there exists a constant C>0 depending onΩ such that
kuk0,Ω 6CkukW1,1(Ω).
Moreover, applying the Poincar´e-Friedrichs inequality in W1,1(Ω), it follows that
kuk0,Ω 6CkukW1,1(Ω)6C′k∇ukL1(Ω),
for a constant C′>0 depending onγandΩ. Therefore, using H¨older’s inequality, we obtain
kuk0,Ω 6C′ Z
Ω|∇u|dxxx6C
′Z
ΩΦ
−2
ααα dxxx
1 2Z
ΩΦ
2
ααα|∇u|2dxxx
1 2
.
Then, employing (2.4) yields Z
ΩΦ
−2
ααα dxxx= M
∑
i=1
Z
Si
ri−2αidxxx+ Z
Ω0 1 dxxx,
and using integration in polar coordinates, it follows that the above integrals are all bounded forαi<1,
i=1,2, . . . ,M. This completes the proof.
To close this section, we shall prove the following Green’s type formula:
LEMMA 2.6 Letααα ∈[0,1)M be a weight vector, and consider two functions u∈H1,1
ααα (Ω) andφ ∈
H2(Ω). In addition, suppose that the trace of u|∂Ω∈L2(∂Ω). Then, Z
Ω∆φu dxxx= Z
∂Ω(∇φ·nnn)u ds− Z
Ω∇φ·∇u dxxx (2.9)
holds true, where nnn denotes the outward unit vector to∂Ω.
Proof. Due to the density of C∞(Ω)in Hααα1,1(Ω)we can choose a sequence{un}n>0⊂C∞(Ω)such that limn→∞ku−unkH1,1
ααα (Ω)=0. Then, using Green’s formula for smooth functions, we have Z
Ω∆φundxxx= Z
∂Ω(∇φ·nnn)unds− Z
for any functionφ∈C∞(Ω). Furthermore, there holds
Z
Ω∆φ(un−u)dxxx
6kφk2,Ωku−unk0,Ω
n→∞
−→0,
and, using Lemma 2.3,
Z
Ω∇φ·∇(un−u)dxxx
6kΦ−ααα∇φk0,ΩkΦααα∇(u−un)k0,Ω 6Ckφk2,Ωku−unkHααα1,1(Ω)
n→∞
−→0.
Furthermore, applying the trace theorem in W1,1(Ω), yields
Z
∂Ω(∇φ·nnn)(un−u)ds
6supΩ |∇φ| ku−unkL1(∂Ω)
6C sup
Ω
|∇φ|ku−unkL1(Ω)+k∇(u−un)kL1(Ω)
6C sup
Ω |∇φ|
ku−unk0,Ω+kΦ−αααk0,ΩkΦααα∇(u−un)k0,Ω
6C sup
Ω |∇φ| k
u−unkH1,1
ααα (Ω)
n→∞
−→0.
This implies the identity (2.9) for u∈Hααα1,1(Ω)andφ∈C∞(Ω).
Forφ∈H2(Ω), the density of C∞(Ω)in H2(Ω)guarantees the existence of a sequence{φn}n>0⊂ C∞(Ω)with limn→∞kφn−φk2,Ω =0. Then,
Z
Ω∆φnu dxxx= Z
∂Ω(∇φn·nnn)u ds− Z
Ω∇φn·∇u dxxx
for all u∈Hααα1,1(Ω). Similarly, as before, we have
Z
Ω∆(φn−φ)u dxxx
6kφn−φk2,Ωkuk0,Ω
n→∞
−→0,
and, with Lemma 2.3,
Z
Ω∇(φn−φ)·∇u dxxx
6kΦ−ααα∇(φn−φ)k0,ΩkΦααα∇uk0,Ω 6kφn−φk2,ΩkukHααα1,1(Ω)
n→∞
−→0.
Moreover, using the trace theorem again, we obtain
Z
∂Ω(∇(φn−φ)·nnn)u ds
6k∇(φn−φ)kL2(∂Ω)kukL2(∂Ω)
6Ckφn−φk2,ΩkukL2(∂Ω)
n→∞
−→0.
2.3 Weak Formulation
The aim of this section is to introduce a weak formulation for the boundary value problem (1.1)–(1.2) and to discuss its well-posedness.
Let g∈H 1 2,12
ααα (∂Ω)in (1.2), whereαααis the weight vector from (2.1) withαi∈[0,1), i=1,2, . . . ,M.
Then, we call u∈Hααα1,1(Ω)with u|∂Ω =g a weak solution of (1.1)–(1.2) if Z
Ω∇u·∇v dxxx+ Z
Ωcuv dxxx= Z
Ω f v dxxx ∀v∈H˚
1,1
−ααα(Ω). (2.10) Writing the solution in the form u=u0+G, where u0∈H˚ααα1,1(Ω)and G∈Hααα1,1(Ω)is a lifting of the boundary data g, i.e., G|Γ =g, there holds
Z
Ω∇u0·∇v dxxx+ Z
Ωcu0v dxxx= Z
Ω f v dxxx− Z
Ω∇G·∇v dxxx− Z
ΩcGv dxxx ∀v∈H˚
1,1 −ααα(Ω).
We note that this is a saddle point formulation on ˚Hααα1,1(Ω)×H˚1,1
−ααα(Ω). Its well-posedness will be discussed in the following.
We first show that the bilinear form a(u,v) =
Z
Ω∇u·∇v dxxx+ Z
Ωcuv dxxx
and the linear functional
ℓ(v) = Z
Ω f v dxxx− Z
Ω∇G·∇v dxxx− Z
ΩcGv dxxx= Z
Ω f v dxxx−a(G,v)
are continuous. Here, we suppose that the lifting G is chosen such that
kGkH1,1
ααα (Ω)6CkgkH12,12
ααα (Γ)
(2.11)
for some fixed constant C>1 independent of g.
PROPOSITION2.1 There is a constant C>0 (depending onΩ andααα) such that
|a(u,v)|6C|u|
Hααα1,1(Ω)|v|H−1,1ααα(Ω)
for all u∈H˚1,1
ααα (Ω), v∈H˚−1,ααα1(Ω). Furthermore, for f ∈L2(Ω)and g∈H 1 2,12
ααα (Γ)we have
|ℓ(v)|6C
kfk0,Ω+kgk H
1 2,12
ααα (Γ)
|v|H1,1
−ααα(Ω)
for any v∈H˚1,1 −ααα(Ω). Proof. There holds
|a(u,v)|6kΦααα∇uk0,ΩkΦ−ααα∇vk0,Ω+kckL∞(Ω)kuk0,Ωkvk0,Ω 6C|u|H1,1
ααα (Ω)|v|H−1,1ααα(Ω)+kuk0,Ωkvk0,Ω
Furthermore, using the Poincar´e-Friedrichs inequality and Lemma 2.5, we get
kuk0,Ωkvk0,Ω6C|u|H1,1
ααα (Ω)|v|1,Ω6C|u|Hααα1,1(Ω)|v|H−1,1ααα(Ω).
Hence,
|a(u,v)|6C|u|H1,1
ααα (Ω)|v|H−1,1ααα(Ω).
Moreover, employing the previous estimate and proceeding as before to estimate the L2-norm, we obtain
|ℓ(v)|6kfk0,Ωkvk0,Ω+|a(G,v)|6kfk0,Ω|v|H1,1
−ααα(Ω)+C|G|Hααα1,1(Ω)|v|H−1,1ααα(Ω).
Then, applying (2.11), yields the stability bound forℓ.
Furthermore, the following inf-sup stability holds.
PROPOSITION2.2 Letααα∈[0,1)Mbe a weight vector. Suppose that the weightsα
i, i=1,2, . . .M, are
sufficiently small so that
µ:= 1
π1max6i6Mαiωi<
1 2. Then, there holds
inf 06≡u∈H˚ααα1,1(Ω)
sup 06≡v∈H˚1,1
−ααα(Ω)
a(u,v)
|u|H1,1
ααα (Ω)|v|H−1,1ααα(Ω)
>δ, (2.12)
where
δ=p1−2µ
2(4µ2+1). Furthermore, we have that
sup
u∈H˚ααα1,1(Ω)
a(u,v)>0 ∀v∈H˚1,1
−ααα(Ω),v6≡0. (2.13)
Proof. For u∈H˚1,1
ααα (Ω), we defineev=Φ2
αααu. Then, there holds
|ev|2H1,1
−ααα(Ω)=
Z
ΩΦ
2
−ααα|∇ev|2dxxx62 Z
ΩΦ
−2
ααα ∇(Φααα2)2u2+Φααα4|∇u|2dxxx
62
4 Z
Ω|∇Φααα|
2u2dxxx+|u|2
Hααα1,1(Ω)
.
Hence, applying Lemma 2.4, results in
|ev|2H1,1
−ααα(Ω)62(4µ
2+1)|u|2
Hααα1,1(Ω). (2.14)
In particular, it follows thatev∈H−1,ααα1(Ω). Moreover, we observe that
a(u,v) =e
Z
Ω∇u·∇ev dxxx+ Z
Ωcuv dxxxe = Z
Ω∇u·∇ Φ
2
αααudxxx+ Z
ΩcΦ
2
Thus, since c>0, we get a(u,ev)>
Z
Ω ∇u·∇ Φ
2
αααu+Φααα2|∇u|2dxxx
=2 Z
ΩΦααα∇u·∇(Φααα)u dxxx+ Z
ΩΦ
2
ααα|∇u|2dxxx
>−1 µ
Z
Ω|∇(Φααα)|
2u2dxxx+ (1−µ)Z
ΩΦ
2
ααα|∇u|2dxxx.
Recalling Lemma 2.4, leads to a(u,ev)>−µ|u|2
Hααα1,1(Ω)+ (1−µ)|u|
2
Hααα1,1(Ω)>(1−2µ)|u|
2
Hααα1,1(Ω). (2.15)
Now, combining (2.14) and (2.15), it follows that
sup
v∈H˚1,1
−ααα(Ω)
a(u,v)
|u|H1,1
ααα (Ω)|v|H−1,1ααα(Ω)
> |u|
Hααα1,1(Ω) |ev|H1,1
−ααα(Ω)
a(u,ev)
|u|2 Hααα1,1(Ω)
>δ
for any u∈H˚1,1
ααα (Ω), u6≡0. Taking the infimum over all u∈H˚1,1
ααα (Ω)results in (2.12). In addition, let v∈H˚1,1
−ααα(Ω), v6≡0. Then, sup
u∈H˚ααα1,1(Ω)
a(u,v)>a(v,v)>
Z
Ω|∇v|
2 dxxx.
Due to v|Γ=0 and v6≡0, there holdsk∇vk0,Ω >0, and hence (2.13) holds.
The above results, Propositions 2.1 and 2.2, imply the well-posedness of the variational formula-tion (2.10); cf., e.g., (Schwab, 1998, Theorem 1.15).
THEOREM2.3 Letααα∈[0,1)Mbe a weight vector, withα
i, i=1,2, . . . ,M sufficiently small such that
max 16i6Mαiωi<
π
2 is satisfied. Furthermore, suppose that g∈H
1 2,12
ααα (∂Ω)and f ∈L2(Ω)in (1.1)–(1.2). Then, there exists exactly one solution of the weak formulation (2.10) in Hααα1,1(Ω).
3. Numerical Approximation
We shall now discuss the numerical approximation of the problem (1.1)–(1.2). To this end, we will consider hp-version interior penalty discontinuous Galerkin finite element methods. Particularly, we will derive an L2-norm a posteriori error estimate which can be applied for adaptive purposes.
3.1 Meshes, Spaces, and Element Edge Operators
We consider shape-regular meshesThthat partitionΩ ⊂R2into open disjoint triangles and/or paral-lelograms{K}K∈Th, i.e.,Ω =SK∈T
hK. Each element K∈Thcan then be affinely mapped onto the
denote the diameter of an element K∈Th. We assume that these quantities are of bounded variation,
i.e., there is a constantρ1>1 such that
ρ−1
1 6hK♯/hK♭6ρ1, (3.1)
whenever K♯ and K♭ share a common edge. We store the elemental diameters in a vector hhh given by
h h
h={hK : K∈Th}. Similarly, to each element K∈Thwe assign a polynomial degree pK>1 and
define the degree vector ppp={pK : K∈Th}. We suppose that ppp is also of bounded variation, i.e., there
is a constantρ2>1 such that
ρ−1
2 6pK♯/pK♭6ρ2, (3.2)
whenever K♯and K♭share a common edge.
Moreover, we shall define some suitable element edge operators that are required for the dG method. To this end, we denote byEI the set of all interior edges of the partitionThofΩ, and byEBthe set of
all boundary edges ofTh. In addition, letE =EI∪EB. The boundary∂K of an element K and the sets
∂K\∂Ωand∂K∩∂Ωwill be identified in a natural way with the corresponding subsets ofE.
Let K♯and K♭be two adjacent elements of Th, and xxx an arbitrary point on the interior edge e∈EI
given by e=∂K♯∩∂K♭. Furthermore, let v and qqq be scalar- and vector-valued functions, respectively,
that are sufficiently smooth inside each element K♯/♭. By(v♯/♭,qqq♯/♭), we denote the traces of(v,qqq)on e
taken from within the interior of K♯/♭, respectively. Then, the averages of v and qqq at xxx∈e are given by
hhvii=1
2(v♯+v♭), hhqqqii= 1
2(qqq♯+qqq♭), respectively. Similarly, the jumps of v and qqq at xxx∈e are given by
[[v]] =v♯nnnK♯+v♭nnnK♭, [[qqq]] =qqq♯·nnnK♯+qqq♭·nnnK♭,
respectively, where we denote by nnnK♯/♭ the unit outward normal vector on∂K♯/♭, respectively. On a
boundary edge e∈EB, we sethhvii=v,hhqqqii=qqq, and[[v]] =vnnn,[[qqq]] =qqq·nnn, with nnn denoting the unit
outward normal vector on the boundary∂Ω.
Given a finite element meshTh and an associated polynomial degree vector ppp= (pK)K∈T
h, with
pK>1 for all K∈Th, consider the hp-discretisation space
VDG(Th,ppp) ={v∈L2(Ω): v|K∈SpK(K),K∈Th}, (3.3)
for the dG method. Here, for K∈Th,Sp
K(K)is either the space PpK(K)of all polynomials of total
degree at most pKon K or the spaceQpK(K)of all polynomials of degree at most pKin each coordinate
direction on K.
3.2 hp-dG Discretisation
We will now consider the following hp-dG formulation for the numerical approximation of (1.1)–(1.2): find uDG∈VDG(Th,ppp)such that
aDG(uDG,v) =ℓDG(v) ∀v∈VDG(Th,ppp). (3.4)
Here,
aDG(w,v) = Z
Ω∇hw·∇hv dxxx− Z
Ehh∇h
wii ·[[v]]ds−
Z
E[[w]]· hh∇h
viids+γ Z
Eσ[[w]]·[[v]]
and
ℓDG(v) = Z
Ω f v dxxx− Z
EB(∇h
v·nnn)g ds+γ Z
EBσ
gv ds (3.6)
are hp-version symmetric interior penalty dG forms. In these forms,∇hdenotes the elementwise gradi-ent operator,γ>0 is a stability constant, and the function
σ=p 2
h (3.7)
is defined by means of the two functionsh∈L∞(E)andp∈L∞(E)given by
h(xxx) =
(
min(hK♯,hK♭) for xxx∈∂K♯∩∂K♭∈EI, hK for xxx∈∂K∩∂Ω∈EB, p(xxx) =
(
max(pK♯,pK♭) for xxx∈∂K♯∩∂K♭∈EI, pK for xxx∈∂K∩∂Ω∈EB.
REMARK3.1 Provided thatγ>0 is chosen sufficiently large (independently of the local element sizes and polynomial degrees), it is well-known that the dG form aDGis coercive. More precisely, there is a constant C>0 independent ofThand ppp such that
aDG(v,v)>C
k∇hvk20,Ω+γ
Z
Eσ|[[v]]|
2ds
for any v∈VDG(Th,ppp). In particular, the dG method (3.4) admits a unique solution uDG∈VDG(Th,ppp);
see, e.g., Stamm & Wihler (2010) and the references therein.
3.3 A Posteriori Error Estimation in the L2-Norm
We shall now derive a residual-based hp–a posteriori error estimate in the L2-norm for the dG formula-tion (3.4). In this secformula-tion we suppose that the dual problem
−∆φ+cφ=eDG inΩ, (3.8)
φ=0 onΓ, (3.9)
has a solutionφ∈H2(Ω)∩H˚1(Ω)with continuous dependence on the data, i.e., there exists a con-stant C>0 such that
kφkH2(Ω)6CkeDGk0,Ω. (3.10)
This is the case, for example, ifΩ is a convex polygon since then∆ : H2(Ω)∩H˚1(Ω)→L2(Ω)is an isomorphism; cf. Babuˇska & Guo (1988); Dauge (1988); Grisvard (1985). Here, eDG=u−uDGdenotes the error, where u∈Hααα1,1(Ω)is the solution of (1.1)–(1.2) and uDG∈VDG(Th,ppp)is the dG solution
defined in (3.4).
Furthermore, we assume that the Dirichlet boundary data satisfies g=u|Γ ∈L2(Γ).
We start the development of the L2-norm a posteriori error estimate by writing
keDGk20,Ω =
Z
Ω(−∆φ+cφ)eDGdxxx= Z
Ω(−∆φ+cφ)u dxxx− Z
Applying Lemma 2.6 in the first integral and integrating by parts elementwise in the second integral, and noticing that[[∇φ]] =0 onEI results in
keDGk20,Ω=
Z
Ω(∇u·∇φ+cuφ)dxxx− Z
Ω(∇huDG·∇φ+cuDGφ)dxxx
+ Z
EI ∇φ·[[uDG]]ds−
Z
EB(∇φ·n
nn)(u−uDG)ds =
Z
Ω fφdxxx− Z
Ω(∇huDG·∇φ+cuDGφ)dxxx
+ Z
EIhh∇φii ·[[uDG]]ds−
Z
EB(∇φ·
n n
n)(g−uDG)ds.
Moreover, for an arbitrary functionφh∈VDG(Th,ppp), exploiting (3.4) with v=φh, gives keDGk20,Ω =
Z
Ω f(φ−φh)dxxx− Z
Ω(∇huDG·∇(φ−φh) +cuDG(φ−φh))dxxx
+ Z
EIhh∇φii ·[[uDG]]
ds−
Z
EB(∇φ·
n n
n)(g−uDG)ds +
Z
EB(∇φh·n
nn)g ds−γ
Z
EBσgφhds−
Z
Ehh∇huDGii ·[[φh]]ds −
Z
Ehh∇hφhii ·[[uDG]]ds+γ
Z
Eσ[[uDG]]·[[φh]]ds.
Using Green’s formula in the second integral, leads to Z
Ω∇huDG·∇(φ−φh)dxxx=− Z
Ω∆huDG(φ−φh)dxxx+K
∑
∈ThZ
∂K
(∇uDG·nnnK)(φ−φh)ds
=−
Z
Ω∆huDG(φ−φh)dxxx+ Z
Ehh∇h
uDGii ·[[φ−φh]]ds
+ Z
EI[[∇h
uDG]]hhφ−φhiids,
where∆his the elementwise Laplace operator. Hence, using that[[φ]] =000 onE, yields keDGk20,Ω
= Z
Ω(f+∆huDG−cuDG)(φ−φh)dxxx− Z
EI[[∇h
uDG]]hhφ−φhiids +
Z
EIhh∇φii ·[[uDG]]ds−
Z
EB(∇φ·
n n
n)(g−uDG)ds+ Z
EB(∇hφh·
n nn)g ds
−γ
Z
EBσ
gφhds−
Z
Ehh∇hφhii ·[[uDG]]ds+γ
Z
Eσ[[uDG]]·[[φh]]ds
= Z
Ω(f+∆huDG−cuDG)(φ−φh)dxxx− Z
EI[[∇h
uDG]]hhφ−φhiids +
Z
EIhh∇h(φ−φh)ii ·[[uDG]]
ds−
Z
EB(∇h(φ−φh)·
n
nn)(g−uDG)ds
−γ
Z
EBσ
(g−uDG)(φh−φ)ds+γ
Z
EIσ
Now, applying the Cauchy-Schwarz inequality and noting that pK>1, K∈Th, gives
keDGk20,Ω 6
∑
K∈Th
h4Kp−4K kf+∆huDG−cuDGk20,Ω+
∑
K∈Th
h3Kp−3K k[[∇huDG]]k20,∂K\∂Ω + (γ2+1)
∑
K∈Th
hKpKk[[uDG]]k20,∂K\∂Ω
+ (γ2+1)
∑
K∈Th
hKpKkg−uDGk20,∂K∩∂Ω
!1 2
×
∑
K∈Th
h−4K p4Kkφ−φhk20,K+
∑
K∈Th
h−3K p3Kkφ−φhk20,∂K
+
∑
K∈T
h
h−1K pKk∇h(φ−φh)k20,∂K
!1 2
.
Then, choosingφh∈VDG(Th,ppp)to be an elementwise optimal hp-interpolant (see, e.g., Babuˇska & Suri (1987a,b)), i.e., for any K∈Th,
h−4K p4Kkφ−φhk20,K+h−3K p3Kkφ−φhk
2
0,∂K+h−1K pKk∇h(φ−φh)k20,∂K6Ckφk
2
H2(K),
and recalling the regularity estimate (3.10), gives
keDGk20,Ω 6CkeDGk0,Ω
∑
K∈Th
η2
K
!1 2
,
with
η2
K=
∑
K∈T
h
h4Kp−4K kf+∆huDG−cuDGk20,Ω+
∑
K∈T
h
h3Kp−3K k[[∇huDG]]k20,∂K\∂Ω
+
∑
K∈T
h
hKpKk[[uDG]]k20,∂K\∂Ω+
∑
K∈T
h
hKpKkg−uDGk20,∂K∩∂Ω.
(3.11)
Hence, dividing both sides of the above inequality bykeDGk0,Ω leads to the following result.
THEOREM3.1 Suppose that the dual problem (3.8)–(3.9) fulfils (3.10), and that the Dirichlet boundary data g∈L2(Γ). Furthermore, let uDG∈VDG(Th,ppp)denote the hp-dG solution from (3.4), and u∈
Hααα1,1(Ω)the analytical solution of (1.1)–(1.2) for some weight vectorααα∈[0,1)M. Then, the following
a posteriori error estimate holds
ku−uDGk20,Ω 6C
∑
K∈T
h η2
K,
where C>0 is a constant independent of the local element sizes hhh and polynomial degrees ppp, and the local error indicatorsηK, K∈Th, are defined in (3.11).
presence of hanging nodes inTh, a nonconforming interpolant is used in the proof of Theorem 3.1.
Indeed, in the absence of hanging nodes, the factor of hKpK can be improved to hKp−1K . We point out
that the energy norm a posteriori error indicators derived in Houston et al. (2007, 2008), for example, suffer from a similar suboptimality with respect to the spectral order.
3.4 Numerical Example
On the rectangleΩ = (−1,1)×(0,1), we consider the PDE problem: find u such that
−∆u=0 inΩ,
u=g onΓ.
We choose the Dirichlet boundary data g in such a way that the analytical solution is given by u(r,θ) = 1
πθ,
where(r,θ)denote polar coordinates inR2. Note that g is smooth on∂Ω, except at the point(0,0). Indeed, in Cartesian coordinates we have that
g(x,y=0) =
(
1 for x<0
0 for x>0, (x,y)∈∂Ω.
In addition, we remark that u6∈H1(Ω). However, there holds u∈Hα1,1(Ω)for anyα∈(0,1), where the weight function for this problem is given byΦα(xxx) =|xxx|α. Furthermore, u is analytic away from(0,0) and belongs to the Babuˇska-Guo space (see, e.g., Babuˇska & Guo (1988))
B1α(Ω) =nv∈L2Ω):|v|Hk,1
ααα (Ω)6Cd
kk! ∀k>1,and constants C,d∈Ro.
With this in mind, we might therefore be able to achieve exponential convergence when hp–mesh re-finement is employed; cf. Sch ¨otzau & Schwab (2001).
Firstly, however, we investigate the practical performance of the a posteriori error estimate derived in Theorem 3.1 within an automatic h–version adaptive refinement procedure which is based on 1-irregular quadrilateral elements. The h–adaptive meshes are constructed by marking the elements for refinement/derefinement according to the size of the local error indicatorsηK; this is done by employing
−1 −0.5
0 0.5
1 0 0.5
1 −0.2
0 0.2 0.4 0.6 0.8 1 1.2
y x
(a)
−1 −0.5
0 0.5
1 0 0.5
1 −0.2
0 0.2 0.4 0.6 0.8 1 1.2
y x
(b)
−1 −0.5
0 0.5
1 0 0.5 1 −0.2
0 0.2 0.4 0.6 0.8 1 1.2
y x
(c)
102 103 104 105
10−6
10−5
10−4
10−3
10−2
10−1
100
Error Bound True Error
Degrees of Freedom
0 2 4 6 8 10 12 14 16
0 5 10 15 20 25
E
ff
ec
ti
v
it
y
In
d
ex
Mesh Number
(a) (b)
FIG. 2. h–Refinement. (a) Comparison of the actual and estimated L2(Ω)–norm of the error with respect to the number of degrees of freedom; (b) Effectivity indices.
0 5 10 15 20 25
10−8
10−6
10−4
10−2
100
Error Bound True Error
(Degrees of Freedom)31
0 5 10 15 20
0 5 10 15 20 25
E
ff
ec
ti
v
it
y
In
d
ex
Mesh Number
(a) (b)
0 10 20 30 40 50 60
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
h−Refinement hp−Refinement
(Degrees of Freedom)13
k
eD
G
k0,
Ω
FIG. 4. Comparison between h– and hp–refinement.
We now turn our attention to hp–mesh adaptation. Here, we again mark elements for refine-ment/derefinement according to the size of the local error indicatorsηK based on employing the fixed
fraction strategy, with refinement and derefinement fractions set to 25% and 10%, respectively. Once an element K∈Thhas been flagged for refinement or derefinement, a decision must be made whether the local mesh size hK or the local degree pK of the approximating polynomial should be adjusted
accord-ingly. The choice to perform either h–refinement/derefinement or p–refinement/derefinement is based on estimating the local smoothness of the (unknown) analytical solution. To this end, we employ the hp–adaptive strategy developed in Houston & S¨uli (2005), where the local regularity of the analytical solution is estimated from truncated local Legendre expansions of the computed numerical solution; see, also, Houston et al. (2003).
In Figure 3(a) we present a comparison of the actual and estimated L2(Ω)–norm of the error versus the third root of the number of degrees of freedom in the finite element space VDG(Th,ppp)on a linear-log scale, for the sequence of meshes generated by our hp–adaptive algorithm. We remark that the third root of the number of degrees of freedom is chosen on the basis of the a priori error analysis carried out in Wihler et al. (2003); cf., also, Sch ¨otzau & Wihler (2003). Here, we observe that the error bound over-estimates the true error by a (reasonably) consistent factor; indeed, from Figure 3(b), we see that the computed effectivity indices are in the range 15–19 as the mesh is refined. Moreover, from Figure 3(a) we observe that the convergence lines using hp–refinement are (roughly) straight on a linear-log scale, which indicates that exponential convergence is attained for this problem. We point out that the slight suboptimality with respect to the polynomial degree in the last two terms of the local error indicatorηK defined in (3.11) does not adversely affect the quality of the local indicators,
cf. Remark 3.2. Indeed, computations based on employing a modified local indicator ˆηK, where ˆηK is
defined in an analogous fashion toηK with the factor of hKpK in the last two terms in (3.11) replaced
by hKp−1K , leads to quantitatively similar behaviour of the L2(Ω)–norm of the error as the finite element
(a) 3 3 2 2 2 3 2 3 3 3 2 2 2 3 3 2 2 2 2 3
3 3 3 3 3 3 3 4 4 4 4 4 4 4 3 3 3 3 3 222 2 3 3 3 33223 43 3 2 3 2 3 2 2 2 2 2 2 2 222222222222232222223222232
3 2 2 22 3 3 4 3 4 4 4 4 3 4 34343
3232334222233 4 4 4 5 4 5 5 5 4 (b) 4 3 3 2 3 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 2 3 2 3 2 2 2 2 3 3 3 4 (c)
(a) 5 5 4 4 4 4 4 4 4 4 4 4 4 5 5 5 4 4 4 4 4 4 5 5 6 6 6 5 5 6 6 5 5 5 6 6 4 4 5 5 55 4 4 5 54 3 4 4 4 322 2 3 3 4 4 3 3 3 2 4 3 3 3 2 2 3 3 3 3 3 32222 3 2 3 2 3 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 22223342222222222222223222232323343233454334445443344
5 5 5 5 5 5 6 6 6 7 6 7 66565
5446425544333322424232323343 6 7 7 7 6 7 7 7 6 (b) 3 2 3 2 3 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 3 2 3 2 2 2 2 3 3 3 3 (c)
FIG. 6. hp–Mesh distribution after 14 adaptive refinements, with 206 elements 4904 degrees of freedom: (a) h–mesh alone; (b)
when ˆηK is employed, in contrast to those computed usingηK. For brevity, these results have been
omitted.
In Figure 4, we present a comparison between the actual L2(Ω)–norm of the error employing both h– and hp–mesh refinement. Here, we clearly observe the superiority of employing a grid adaptation strategy based on exploiting hp–adaptive refinement: on the final mesh, the L2(Ω)–norm of the er-ror using hp–refinement is around three orders of magnitude smaller than the corresponding quantity computed when h-refinement is employed alone.
Finally, in Figures 5 & 6 we show the mesh generated using the proposed hp-version a posteriori error indicator stated in Theorem 3.1 after 9 and 14 hp-adaptive refinement steps, respectively. For clarity, we also show the h–mesh alone, as well as a zoom of the mesh in the vicinity of the origin. Here, we observe that h–refinement of the mesh has been performed in the vicinity of the discontinuity present in g, cf. above. Within this region, the polynomial degree has been kept at 2. Away from this region, the hp–adaptive algorithm increases the degree of the approximating piecewise polynomials where the analytical solution is smooth.
4. Conclusions
In this work, we have introduced a new variational framework for linear second-order elliptic PDE with discontinuous Dirichlet boundary conditions based on locally weighted Sobolev spaces. In particular, we have proved the well-posedness of the new setting by means of an inf-sup condition. In addition, we have proposed the use of symmetric hp–version interior penalty discontinuous Galerkin methods for the numerical approximation of such problems. For this discretisation scheme, we have derived an L2– norm a posteriori error estimate whose performance within h– and hp–adaptive refinement procedures has been displayed with a model numerical experiment. Future work will deal with some extensions of the present setting to systems such as, e.g., the Stokes equations for cavity flow problems.
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