An hp-Analysis of the Loal Disontinuous
Galerkin Method for Diusion Problems
Ilaria Perugia
Dominik Shotzauy
Journal of Sienti Computing, Vol. 17, pp. 561{571, 2002
Abstrat
We present an hp-error analysis of the loal disontinuous Galerkin method for diusion problems, onsidering unstrutured meshes
with hanging nodes and two- and three-dimensional domains. Our
estimates are optimal in the meshsize h and slightly suboptimal in
the polynomial approximation order p. Optimality in p is ahieved
for mathing grids and polynomial boundary onditions.
Key words: hp-FEM, loal disontinuous Galerkin methods
1 Introdution
The aim of this paper is to present an hp-error analysis of the loal disontinuous Galerkin (LDG) method, introdued by Cokburn and Shu (1998),
for the diusion problem
r ( ru) = f in u = gD on ;
(1)
where is a bounded polygonal or polyhedral domain in Rd , d = 2; 3, and
2 L1 (
)dd a symmetri, uniformly positive denite diusion tensor.
Department of Mathematis, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy.
E-mail: perugiadimat.unipv.it. Supported in part by the NSF (DMS-9807491) and
by the University of Minnesota Superomputing Institute. This work was arried out
when
the author was visiting the Shool of Mathematis, University of Minnesota.
y Department of Mathematis, University of Basel, Rheinsprung 21, 4051 Basel,
Switzerland. E-mail: shotzaumath.unibas.h. Supported in part by NSF grant DMS0107609 and by the University of Minnesota Superomputing Institute. This work was
arried out when the author was aÆliated with the Shool of Mathematis, University
of Minnesota.
1
An hp-Analysis of the LDG Method for Diusion Problems
2
The1 right-hand side f belongs to L2 (
), and the Dirihlet datum gD to
H 2 ( ). Deriving error bounds that take into aount both the elemental
meshsize and approximation order ompletes previous work by Castillo,
Cokburn, Perugia and Shotzau (2000) on the LDG method applied to
purely ellipti problems. This is relevant sine the LDG method, being
based on disontinuous spaes, is ideally suited for hp-adaptivity.
The analysis in this paper follows the same lines as the one developed in Perugia and Shotzau (2001) in the more ompliated situation
of the low-frequeny time-harmoni Maxwell equations, and uses the general framework introdued in Arnold, Brezzi, Cokburn and Marini (2001).
On meshes without hanging nodes and for polynomial boundary onditions,
our setting immediately gives optimal hp-onvergene rates. Furthermore,
on unstrutured meshes with hanging nodes, we prove error estimates that
are optimal in the mesh-size h and slightly suboptimal in the polynomial degree p (half a power of p is lost). We point out that, for multi-dimensional
ellipti problems on general unstrutured grids, no better p-bounds an
be found in the literature (see, e.g., Riviere, Wheeler and Girault (1999),
Prudhomme, Pasal, Oden and Romkes (2000), Houston, Shwab and Suli
(2000), where the same bounds have been obtained for dierent disontinuous Galerkin methods and with dierent analysis tehniques). Optimal hpbounds have also been proved in Castillo, Cokburn, Shotzau and Shwab
(2000) for one-dimensional onvetion-diusion problems and, reently, in
Georgoulis and Suli (2001) for two-dimensional reation-diusion problems
on aÆne quadrilateral meshes ontaining hanging nodes.
The outline of the paper is as follows. In setion 2, we dene the LDG
method for the diusion problem (1). We arry out the error analysis in
setion 3, obtaining hp-error estimates in a problem-related energy norm,
as well as in the L2 -norm. We end our presentation in setion 4 with
onluding remarks.
2 LDG Disretization
2.1
Meshes and Finite Element Spaes
We onsider shape-regular meshes Th that partition the domain into
triangles and/or parallelograms, if d = 2, and tetrahedra and/or parallelepipeds, if d = 3, with possible hanging nodes and aligned with the
possible disontinuities of the diusion tensor , so that is smooth within
eah element of Th . We denote by hK the diameter of the element K 2 Th .
We dene the (d 1)-dimensional faes of Th as follows. An interior fae of
Th is the (non-empty) interior of K + \ K , where K + and K are two
adjaent elements of Th , not neessarily mathing. Similarly, a boundary
An hp-Analysis of the LDG Method for Diusion Problems
3
fae of Th
is the (non-empty) interior of K \ , where K is a boundary
element of Th . We denote by EI the union of all interior faes of Th , by ED
the union of all boundary faes, and set E = EI [ ED .
Let p = fpK gK 2Th be a degree vetor that assigns to eah element K 2
Th a polynomial approximation order pK 1. The generi hp-nite element
spae of pieewise polynomials is given by S p (Th ) := fu 2 L2 (
) : ujK 2
S pK (K ); 8K 2 Th g, where S pK (K ) is the spae P pK (K ) of polynomials of
degree at most pK in K , if K is a triangle or a tetrahedron, and the spae
QpK (K ) of polynomials of degree at most pK in eah variable in K , if K
is a quadrilateral or a parallelepiped.
2.2
The Flux Formulation of the LDG Method
By introduing the auxiliary variables q = s and s = ru as in Cokburn
and Dawson (2000), the diusion problem (1) an be rewritten as
q = s in r q = f in s = ru in u = gD on :
We approximate the variables (q,s,u) by disrete funtions (q h ; sh ; uh) in
the hp-nite element spae Qh Qh Vh , where Qh = S p (Th )d and Vh =
S p (Th ), for a given degree distribution p.
The LDG method then onsists in nding (qh ; sh ; uh) 2 Qh Qh Vh
suh that for any (r; t; v) 2 Qh Qh Vh and for any element K 2 Th
Z
ZK
Z
K
K
q h r dx =
sh t d x +
q h rv d x
Z
Z
K
K
Z
sh r d x
uh r t dx
K
Z
K
qbh nK v ds =
ubh t nK ds = 0
Z
K
(2)
f v dx:
Here, nK denotes the outward unit normal vetor to K . The quantities
ubh and qbh are the so-alled numerial uxes, whih are approximations to
the traes of u and q on K , and are dened as follows.
Consider an interior fae e shared by two elements K + and K . Denoting by v and r the traes on K of funtions v and r that are smooth
in K , we dene the averages and jumps of v and r aross e by
ffvgg = (v+ + v )=2
ffrgg = (r+ + r )=2
[ v ℄ = v + n K + + v nK
[ r ℄ = r + nK + + r nK :
If v 2 H 1 (
), then [ v℄ = 0 on EI . Similarly, if r 2 H (div; ), then [ r℄ = 0
on EI . In partiular, for the exat solution u, we have [ ru℄ = 0 on EI .
4
An hp-Analysis of the LDG Method for Diusion Problems
The numerial uxes are dened by
(
(
f
f
ugg + b [ u℄ if e EI
ffqgg a[ u℄ b[ q℄ if e EI
b
ubje =
q
j
e =
gD
if e ED
q a(u gD )n if e ED ;
with parameters a and b to be properly hosen. This ompletes the denition of the LDG method. Notie that the ux in u is independent of q.
This allows for an element-by-element elimination of the auxiliary variables
q and s, giving rise to the so-alled primal formulation of the method in the
variable u only. This loal solvability gives the name to the LDG method.
We refer to Castillo (2001) for a disussion of this elimination proess from
a omputational point of view. Let us also point out that the LDG method
dened above is onsistent and, if a is stritly positive, denes a unique
disrete solution (qh ; sh ; v) 2 Qh Qh Vh ; see Castillo et al. (2000).
2.3
The Primal Formulation
We develop the error analysis of the LDG method in the framework introdued in Arnold et al. (2001), that is, by onsidering its primal formulation.
For v belonging to V (h) := Vh + H 1 (
), we dene L(v) 2 Qh by
Z
L(v) r dx =
Z
EI
(ffrgg b [ r℄ ) [ v℄ ds +
Z
ED
v r n ds
8r 2 Qh :
Similarly, we dene the lifting G D 2 Qh of the boundary datum gD by
Z
G D r dx =
Z
ED
gD r n ds
8r 2 Qh :
Notie that, for the exat solution u, we have L(u) = G D . Adding the rst
and seond equations in (2) over all elements and simple alulations yield
(3)
qh = rhuh L(uh) + GD ;
2
with denoting the L -projetion onto Qh and rh the elementwise gradient. Inserting this expression in the third equation of (2) yields the primal
form of the LDG method: nd uh 2 Vh suh that
Ah (uh ; v) + Ih (uh ; v) = Fh (v)
8v 2 Vh ;
(4)
where
Z
Ah (u; v) = rh u L(u) rh v L(v) dx
Ih (u; v) =
Fh (v) =
Z
E
Z I
a [ u℄ [ v ℄ ds +
f v dx +
Z
Z
a u v ds
ED
Z
a gD v ds
GD rh v L(v) dx:
An hp-Analysis of the LDG Method for Diusion Problems
5
For disrete trial and test funtions, the primal form (4), together with
identity (3), is equivalent to the original ux form (2) of the LDG method.
However, unlike (2), the formulation (4) is no longer onsistent. Nevertheless, the form Ah + Ih has the ontinuity and oerivity properties that
allow us to arry out an error analysis in a straightforward way by using
Strang's lemma.
3 Error Analysis
In this setion, we develop the hp-error analysis of the LDG method. We
start by speifying the parameters a and b in the denition of the method.
Then we prove abstrat error estimates in a broken norm and derive our
atual error bounds. Finally, we address the issue of the stability of the
LDG formulation.
3.1
The Disontinuity Stabilization Parameter
We introdue the funtions h and p in L1 (E ) related to the loal meshsize
and approximation degree as
(
fhK ; hK 0 g if x in the interior of K \ K 0
x) := hmin
if x in the interior of K \ K
(
maxfpK ; pK 0 g if x in the interior of K \ K 0
p = p(x) :=
pK
if x in the interior of K \ :
h = h(
Regarding the diusivity, we assume to be Lipshitz ontinuous in K ,
for any K 2 Th . This implies that jK an be extended up to K , and we
denote this extension by K . Therefore, for any K 2 Th , there are positive
onstants nK and NK suh that nK i (K (x)) NK for all x 2 K ,
where i (K (x)), i = 1; 2; 3, are the eigenvalues of K (x). For any K 2 Th ,
the onstants nK and NK are assumed to satisfy
NK n K ;
8K 2 Th ;
with a onstant > 0. Whenever is a pieewise onstant salar funtion,
this assumption holds true with = 1. We set
(
fjK (x)j; jK 0 (x)jg if x is in the interior of K \ K 0
x) := jmax
K (x)j
if x is in the interior of K \ ;
where j (x)j is the spetral norm of the tensor (x).
n = n(
6
An hp-Analysis of the LDG Method for Diusion Problems
We dene the disontinuity stabilization parameter a 2 L1 (E ) in terms
of h, p and n by
p2 n
;
a=
h
with > 0 independent of meshsize, approximation order and diusion.
Moreover, the parameter b is taken suh that
kbk1;EI ;
with 0 independent of meshsize and approximation order. In partiular, the hoie b = 0 is possible.
3.2
Continuity and Stability
We introdue the energy norm
jj u jj2h = k 12 rh uk20;
+ kh 12 p n 21 [ u℄ k20;EI + kh 12 p n 21 uk20;ED :
We have the following ontinuity and oerivity properties.
Proposition 3.1 Assume the above hypotheses on and on the oeÆients in the denition of the numerial uxes. Then
jAh (w; v) + Ih (w; v)j Cont jj w jjh jj v jjh 8w; v 2 V (h)
Ah (v; v) + Ih (v; v) Coerjj v jj2h
8v 2 Vh ;
with Cont and Coer only depending on , , and the shape-regularity
of the mesh.
Proof. With arguments similar to the ones in Perugia and Shotzau (2001,
Proposition 4.2), we have
i
h
k 12 L(v)k0;
Clift ( + 1) kh 21 p n 21 [ v℄ k0;EI + kh 12 p n 21 vk0;ED (5)
for all v 2 V (h). The onstant Clift > 0 only depends on the shaperegularity of the mesh. The ontinuity and oerivity of Ah + Ih are now
easy onsequenes of estimate (5).
2
From Proposition 3.1 and Strang's lemma, we immediately have the
following abstrat error bound.
Theorem 3.1 Assume the above hypotheses on and on the oeÆients
in the denition of the numerial uxes. Then we have
jj u v jjh + C 1 sup jRjhj (wu;jjw)j ;
jj u uh jjh 1 + CCont vinf
2V
oer
h
oer w2Vh
with the residual Rh (u; w) = Ah (u; w) + Ih (u; w) Fh (w).
h
7
An hp-Analysis of the LDG Method for Diusion Problems
Remark 3.1 For regular meshes without hanging nodes, onstant approximation orders pK = p and boundary onditions that are restritions to of nite element funtions, we an hoose v in the estimate of Theorem 3.1
as an optimal H 1 -onforming approximant of u, see, e.g., Babuska and Suri
(1987, Theorem 4.6), and obtain estimates for jj u uh jjh that are optimal
in h and p (the residual is optimally onvergent, see Lemma 3.2 below).
Next, we derive hp-error estimates on general unstrutured meshes with
hanging nodes.
3.3
hp-Error Estimates
We assume that there exists a onstant ` > 0 suh that ` 1 hK hK 0 `hK
and ` 1 pK pK 0 `pK for all K and K 0 sharing a (d 1)-dimensional fae.
For K 2 Th , we set NÆK := maxfNK 0 : K and K 0 share at least one faeg.
We have the following error bound.
Theorem 3.2 Assume the above hypotheses on , the oeÆients in the
denition of the numerial uxes and the meshes and polynomial degree
distributions. Let the exat solution u satisfy ujK 2 H sK +1 (K ) and
rujK 2 H sK (K ), for all K 2 Th , with sK 1. Then
jj u uh jj2h C
2 min(pK ;sK ) h
1 k ruk2 i;
hK
2
N
k
u
k
+
ÆK
sK +1;K
sK ;K
nK
p2KsK 1
K 2Th
X
with C independent of hK and pK . Moreover,
2 min(pK ;sK ) h
i
hK
NÆK kuk2sK +1;K +(1+ nK1)k ruk2sK ;K :
2
sK 1
pK
K 2Th
kq qh k20;
C
X
Remark 3.2 The bounds in Theorem 3.2 are slightly suboptimal in the
orders pK . On aÆne quadrilateral meshes in two dimensions with hanging
nodes, the p-bounds an be improved by using the projetor of Georgoulis
and Suli (2001), provided that u belongs to an augmented Sobolev spae.
In order to prove Theorem 3.2, we need the following hp-approximation
result (see Babuska and Suri (1987, Lemma 4.5)).
Lemma 3.1 Let K 2 T and v 2 H K (K ), t 1. Then there exists a
h
t
K
sequene of polynomials v in S (K ), pK = 1; 2; : : :, satisfying
1
2
kv vkq;K + hK12
pK
hK
pK
q
hK
pK
pK
min(pK +1;tK )
ptK q
kv phKK vk0;K C hK
q
K
q
kvktK ;K
8
An hp-Analysis of the LDG Method for Diusion Problems
for all q with 0 q tK . The onstant C is independent of v, hK and pK ,
but depends on the shape-regularity of the mesh and on tK .
Let now hp v be given by hp vjK = phKK (vjK ), for any K 2 Th , with phKK
from Lemma 3.1. For r = (r1 ; : : : ; rd ), we set hp r = (hp r1 ; : : : ; h rd ).
Lemma 3.2 Let u be the exat solution. Assume ( ru)j 2 H K (K ) ,
s
K
K 2 Th , with loal regularity exponents sK 1. Then, for w 2 V (h),
jRh (u; w)j C
Proof.
d
2 min(pK +1;sK ) 1
i 21
hK
k
ruk2sK ;K jj w jjh :
2
sK
nK
pK
K 2Th
h X
Simple alulations lead to
Rh (u; w) =
Z
EIZ
+
ff ru ( ru)gg b [ ru ( ru)℄℄ [ w℄ ds
ED
w ru
( ru) n ds;
with being the L2-projetion onto Qh . By writing ru ( ru) =
[ ru hp ( ru)℄ [ ru hp ( ru)℄, the estimate follows as in Perugia
and Shotzau (2001, Lemma 4.11) from the triangle inequality, the CauhyShwarz inequality, inverse estimates for [ ru hp ( ru)℄ from K to
K , K 2 Th , the L2-stability of and the approximation properties in
Lemma 3.1.
2
We are now ready to prove Theorem 3.2.
Proof of Theorem 3.2. We start by estimating jj u hp u jjh . Our assumptions on , meshes and polynomial degree distributions and the approximation properties in Lemma 3.1 yield
jj u hp u jj2h C
2 min(pK ;sK )
hK
2
2sK 1 NÆK kuksK +1;K :
p
K
K 2Th
X
By inserting this and the result of Lemma 3.2 in the bound of Theorem 3.1,
we obtain the estimate of jj u uh jjh .
To estimate kq qh k0;
, we use (3), the triangle inequality and L(u) =
G D to obtain kq qh k0;
k ru ( rh uh)k0;
+ k( L(u uh))k0;
.
From the L2 -stability of and (5), k( L(u uh))k0;
C jj u uh jjh .
By the triangle inequality, the identity hp ( ru) = hp ( ru) , and
again the L2 -stability of , we get
k ru ( rh uh)k0;
2k ru hp ( ru)k0;
+ jj u uh jjh :
9
An hp-Analysis of the LDG Method for Diusion Problems
Therefore, the desired result follows from the bound for jj u uh jjh and
Lemma 3.1.
2
2
An estimate for the L -error in u an be obtained by using a standard
duality argument. We assume that and are suh that the following
ellipti regularity result holds true: for any 2 L2 (
), the solution z to
the problem
r ( rz ) = in z = 0 on ;
(6)
satises z 2 H 2 (
), rz 2 H 1 (
)d and kz k2;
C kk0;
, k rz k1;
C kk0;
, with a onstant C > 0.
Theorem 3.3 With the same assumptions as in Theorem 3.2 and the
above hypothesis on and , we have
min(p;s)+1
ps+ 12
ku uh k0;
C h
kuks+1;
+ k ruks;
;
with h = maxK 2Th hK , p = minK 2Th p 1 and s = minK 2Th sK 1.
Proof. Let z be the solution to problem (6) with = u uh . Simple
alulations give ku uhk20;
= Ah (z; u uh)+ Ih (z; u uh) Rh(z; u uh).
Sine Ah (zh; u uh) + Ih (zh ; u uh ) = Rh (u; zh), for any zh 2 Vh and
Rh (u; zh) = Rh (u; z zh ), we obtain
ku uh k20;
=Ah (z zh; u uh ) + Ih (z zh ; u uh)
Rh (u; z zh ) Rh (z; u uh ):
Therefore, from Proposition 3.1, Lemma 3.2 and the regularity of z ,
i
h
ku uhk20;
Cont jj z zh jjh + C hp k rz k1;
jj u uh jjh
min(p+1;s)
+ C h ps k ruks;
jj z zh jjh :
By hoosing zh = hp z , from the estimates in Lemma 3.1 and the ellipti
regularity assumption, jj z zh jjh C hp 1 kz k2;
C hp 1 ku uh k0;
,
in addition to k rz k1;
C ku uhk0;
. The result then follows from the
estimate of jj u uh jjh in Theorem 3.2.
2
The stability of the LDG formulation with respet to the right-hand
side, under mild smoothness assumptions, is implied by the following result.
Proposition 3.2 Assume that and are suh that the
solution z of (6)
d and k rz k
with right-hand side 2 V (h) satises rz 2 H s (
)
s;
C kk0;
for s > 12 . Then, jFh (v)j C [kf k20;
+ kh 12 p n 21 gD k20;ED ℄ 12 jj v jjh ,
for all v 2 V (h).
An hp-Analysis of the LDG Method for Diusion Problems
10
The assertion follows from the broken Poinare inequality kvk0;
2 V (h), that an be proved following Arnold (1982), and the
estimate k 12 G D k0;
Cliftkh 21 p n 21 gD k0;ED , obtained as in Perugia and
Shotzau (2001, Proposition 4.2).
2
Proof.
C jj v jjh , v
4 Conlusions
In this paper, we presented the rst hp-error analysis of the LDG method
for diusion problems in several spae dimensions and extended the previous h-analysis in Castillo et al. (2000). Although we used the setting of
Arnold et al. (2001) to ast the method in its primal form, we proposed a
new tehnique to atually derive error estimates based on Strang's lemma.
Referenes
Arnold, D.N.: 1982, An interior penalty nite element method with disontinuous
elements, SIAM J. Numer. Anal. 19, 742{760.
Arnold, D.N., Brezzi, F., Cokburn, B. and Marini, L.D.: 2001, Unied analysis
of disontinuous Galerkin methods for ellipti problems, SIAM J. Numer.
Anal. 39, 1749{1779.
Babuska, I. and Suri, M.: 1987, The hp-version of the nite element method with
quasiuniform meshes, Model. Math. Anal. Numer. 21, 199{238.
Castillo, P.: 2001, Performane of disontinuous Galerkin methods for ellipti
partial dierential equations, Tehnial Report 1764, IMA, University of
Minnesota, submitted.
Castillo, P., Cokburn, B., Perugia, I. and Shotzau, D.: 2000, An a priori error
analysis of the loal disontinuous Galerkin method for ellipti problems,
SIAM J. Numer. Anal. 38, 1676{1706.
Castillo, P., Cokburn, B., Shotzau, D. and Shwab, C.: 2000, Optimal a priori
error estimates for the hp-version of the loal disontinuous Galerkin method
for onvetion{diusion problems, Tehnial Report 1689, IMA, University
of Minnesota, in press in Math. Comp.
Cokburn, B. and Shu, C.-W.: 1998, The loal disontinuous Galerkin method
for time{dependent onvetion{diusion systems, SIAM J. Numer. Anal.
35, 2440{2463.
Cokburn, B. and Dawson, C.: 2000, Some extensions of the loal disontinuous
Galerkin method for onvetion-diusion equations in multidimensions, in
J. Whitemann (ed.), The Proeedings of the 10th Conferene on the Mathematis of Finite Elements and Appliations, Elsevier, pp. 225{238.
Georgoulis, E.H. and Suli, E.: 2001, hp-DGFEM on shape{irregular meshes:
reation{diusion, Tehnial Report NA 01{09, Oxford University Computing Laboratory.
An hp-Analysis of the LDG Method for Diusion Problems
11
Houston, P., Shwab, C. and Suli, E.: 2000, Disontinuous hp nite element methods for advetion{diusion problems, Tehnial Report NA 00{15, Oxford
University Computing Laboratory, in press in SIAM J. Numer. Anal.
Perugia, I. and Shotzau, D.: 2001, The hp-loal disontinuous Galerkin method
for low{frequeny time{harmoni Maxwell's equations, Tehnial Report
1774, IMA, University of Minnesota, in press in Math. Comp.
Prudhomme, S., Pasal, F., Oden, J. and Romkes, A.: 2000, Review of a priori
error estimation for disontinuous Galerkin methods, Tehnial Report 200027, TICAM, University of Texas at Austin.
Riviere, B., Wheeler, M. and Girault, V.: 1999, Improved energy estimates for
interior penalty, onstrained and disontinuous Galerkin methods for ellipti
problems, Part I, Computational Geosienes 3 (4), 337{360.