Mathematial Models and Methods in Applied Sienes
fWorld Sienti Publishing Company
Vol. 11 (2001) 301{337
THE HP-VERSION OF THE STREAMLINE DIFFUSION
FINITE ELEMENT METHOD IN TWO SPACE DIMENSIONS
K. GERDES1
1
3

D. SCHOTZAU
J.M. MELENK2
C. SCHWAB2
Dept. of Mathematis, Chalmers University of Tehnology,
SE-412 96 Goteborg, Sweden.
2 Seminar for Applied Mathematis, ETH Z
urih,
ETH Zentrum, CH-8092 Zurih, Switzerland.
3 Shool of Mathematis, University of Minnesota,
Vinent Hall, Minneapolis MN 55455, USA.
The Streamline Diusion Finite Element Method (SDFEM) for a two dimensional
onvetion-diusion problem is analyzed in the ontext of the hp-version of the Finite
Element Method (FEM). It is proved that the appropriate hoie of the SDFEM parameters leads to stable methods on the lass of \boundary layer meshes", whih may
ontain anisotropi needle elements of arbitrarily high aspet ratio. Consisteny results
show that the use of suh meshes an resolve layer omponents present in the solutions
at robust exponential rates of onvergene. We onrm these theoretial results in a
series of numerial examples.
Keywords : Streamline Diusion, hp-FEM, Robust Exponential Convergene.
1. Introdution
Standard Galerkin Finite Element Methods (FEM) for onvetion-dominated problems are known to often produe wildly osillatory solutions due to stability problems intrinsi in the shemes. To irumvent these stability problems the Streamline
Diusion Finite Element Method (SDFEM) was introdued in pioneering work by
Hughes, Johnson and their oworkers.5;7;8;9;11;12;13;14;15 In the meantime, numerous papers have appeared in whih the SDFEM and related Streamline Upwind
Petrov Galerkin (SUPG) tehniques have been applied suessfully to inompressible uid ow problems and to solid mehanis problems with analogous mathematial struture.9;4;33;34
However, all these approahes were mainly onerned with the h-version FEM where
onvergene is ahieved by rening the mesh T at a xed, low polynomial degree
p. The onvergene rates were onsequently at best algebrai. In the 1980s, the pand hp-FEM were introdued by Babuska, Szabo and their oworkers, and it was
shown that the hp-FEM ahieves exponential onvergene for ellipti problems with
pieewise analyti solutions (f. the survey paper by Babuska and Suri1 and the
referenes therein).
A omplete analysis of the hp-version of the SDFEM for onvetion-dominated
problems is available in one spae dimension.20 In partiular, robust exponential
1
2 The hp Streamline Diusion FEM
onvergene of the hp-SDFEM (and of the hp-Galerkin FEM) is shown in global
norms (mainly the L2 norm and the energy norm) provided that boundary layers
and fronts in the solution are resolved. In the h-version FEM sale resolution an
be ahieved with anisotropi, so-alled Shishkin meshes23;24;25 ; in the ontext of the
hp-version sale resolution an be obtained very eÆiently by inserting anisotropi
needle elements of the proper width into the layer.17;19;20;21;27;29 Furthermore, the
behavior of the hp-SDFEM in one dimension was investigated under the assumption
that layers are not resolved, whih may happen, for example, when the preise
loation of the layer is unknown. In this ase, the hp-SDFEM an lead to robust
exponential onvergene on ompat subsets upstream of the layer/shok while the
performane of the hp-Galerkin FEM is poor throughout the domain.20 This is
important for adaptive shemes that try to loate and resolve the layers. In the onedimensional ase, a omplete onvergene analysis of the hp-SDFEM was possible
beause preise regularity results for the exat solution were available.20
In two spae dimensions, however, the regularity theory is onsiderably more involved and sharp analyti regularity required for proving robust exponential onvergene does not seem to be available at present. Usually, the regularity of the
exat solution is desribed by means of asymptoti expansions. There, the solution
is deomposed into a smooth (pieewise analyti) part, a layer part, and a (small)
remainder. While the regularity of the smooth and layer parts are well understood,
it is the analyti regularity of the remainder, ontaining, for example, orner layers,
that is not very well understood. Nevertheless, mesh design priniples for the robust
exponential approximation of the dominant solution omponents, the smooth part
and the layer part, are well understood.21
In the present work we extend the hp-SDFEM to onvetion-diusion problems in
two spae dimensions and show that:
1. The appropriate hoie of the SDFEM parameters leads to stable methods
on boundary layer meshes that may ontain anisotropi needle elements of
arbitrarily high aspet ratio.
2. Boundary layer meshes an resolve the layer omponents present in the exat
solution; the SDFEM based on suh boundary layer meshes an lead to exponential rates of onvergene where the onstants depend only very weakly on
the perturbation parameter.
3. The SDFEM on shape regular meshes yields robust hp-approximation results
for smooth solutions.
The performane of the hp-SDFEM is studied in a series of numerial examples.
We illustrate that our hp-version mesh design priniples indeed allow for resolving
loalized small sale features suh as boundary layers and lead to robust exponential
onvergene in the global energy norm. If the layers are not resolved, we still observe
exponential rates of onvergene upstream of the layer, in agreement with the orresponding results in one dimension.20 We point out, however, that this onvergene
behavior upstream is fairly sensitive to the hoie of the SDFEM parameters.
The outline of the paper is as follows: In Se. 2 we present our model onvetiondiusion problem and review the SDFEM formulation. In Se. 3 we introdue the
lass of boundary layer meshes and disuss the hp-approximation of layer omponents on suh meshes. Se. 4 is devoted to the stability analysis of the SDFEM on
boundary layer meshes. In Se. 5 we prove onsisteny results for the hp-SDFEM
and show that smooth and layer omponents an be resolved at robust exponential
rates of onvergene. On shape regular meshes, optimal (in the sense made preise
in Remark 5.10) hp error bounds are derived for smooth solutions. We onlude
our presentation with numerial results in Se. 6.
The hp Streamline Diusion FEM 3
Standard notations and onventions are followed in this paper. For a domain D
the Lebesgue spae of square integrable funtions is denoted L2(D). We write
(; )D for the L2 (D) inner produt where the index D is omitted if lear from
the ontext. The orresponding norm is k kL2 (D) . L1 (D) is the spae of all
bounded funtions equipped with the supremum norm k k1 . The Sobolev spaes
of order k 0 are denoted by H k (D). We write k kH k (D) and j jH k (D) for the
orresponding norms and semi-norms. H01 (D) is the spae of H 1 (D)-funtions
with
vanishing trae on the boundary D of D. The Sobolev spaes of L1 -funtions are
denoted by W k;1 (D). We use the notation Qp (D) for the set of all polynomials
of degree p in eah variable and P p(D) for the set of all polynomials of total
degree p. In the following we denote by ; C; C1 ; C2 ; : : : generi onstants not
neessarily idential at dierent plaes but always independent of the meshwidths,
the polynomial degrees, and the singular perturbation parameter ". Similarly, the
onstant arising in expressions suh as e p is independent of the polynomial
degree p and the perturbation parameter " but not neessarily the same in dierent
instanes.
2. The Streamline Diusion Finite Element Method (SDFEM)
In this setion we introdue the SDFEM disretization of onvetion-diusion problems based on hp Finite Element spaes.
2.1. The Convetion-Diusion Problem
In a urvilinear Lipshitzian polygon R2 we onsider the onvetion-diusion
problem
"u(x) + ~a(x) ru(x) + b(x)u(x) = f (x)
in ,
u = 0
on :
(2.1)
(2.2)
Here, the parameter " 2 (0; 1℄ may approah zero, the right-hand side f is in L2 (
),
and the oeÆients ~a = (a1 ; a2 ) and b are assumed to be bounded, dierentiable,
and to satisfy
1
r ~a(x) > 0; x 2 :
(2.3)
b(x)
2
Condition (2.3) guarantees stability and unique solvability of (2.1), (2.2) for all
" 2 (0; 1℄. The standard weak formulation of (2.1), (2.2) is: Find u 2 H01 (
) suh
that
"
Z
ru rv dx +
Z
(~a ru + bu)v dx =
Z
fv dx
8v 2 H01 (
):
(2.4)
In order to solve (2.1), (2.2) numerially by a Finite Element Method, the innite
dimensional spae H01 (
) in (2.4) is replaed by a nite dimensional FE-spae of
pieewise mapped polynomials on a mesh T .
2.2. Meshes and Finite Element Spaes
We onsider meshes T on R2 that onsist of urvilinear quadrilateral and/or
triangular elements fK g that satisfy the following standard assumptions:
(i) The elements fK g partition the domain, i.e., they are open, pairwise disjoint,
and = [K 2T K .
4 The hp Streamline Diusion FEM
(ii) Eah element K is the image of the generi referene element K^ , whih is
either the referene triangle T^ = f(x; y) : 0 < x < 1; 0 < y < xg or the
referene square Q^ = (0; 1)2; i.e., with eah K 2 T there is an assoiated
element mapping FK : K^ ! K . FK is an analyti dieomorphism in a
neighborhood of K^ with det DFK > 0 on K^ .
(iii) The intersetion K \K 0 of two elements K and K 0 is either empty, one ommon
vertex or one entire side. (Verties and sides are the images of the verties
and sides of the referene element K^ under FK .)
(iv) The parameterization is the same \from both sides": Let = K \ K 0 be the
ommon side of K and K 0 with endpoints P1 and P2 . Then for any point
P on we have dist(FK 1 (P ); FK 1 (Pi ))=lK = dist(FK 1 (P ); FK 1 (Pi ))=lK for
i = 1; 2 where lK and lK denote the lengths of the orresponding edges of the
referene elements of K and K 0, respetively.
0
0
0
0
We denote by hK;max and hK;min the maximal and minimal lengths of the sides of
K 2 T . The mesh T is alled shape regular if
(i) There is a onstant > 0 independent of the element and the partition suh
that hK;max hK;min .
(ii) On K^ we have kD FK k1 ChK;max for multi-indies with 1 j j 2 and
C1 h2K;min det DFK C2 h2K;max with onstants C , C1 and C2 independent
of the elements.
The mesh T is alled aÆne if the element mappings FK are aÆne transformations.
Let now p = fpK : K 2 T g be a degree vetor on T that assigns to eah element
K 2 T a polynomial degree pK . The spae S p (T ) of pieewise mapped polynomials
is then dened as follows:
S p (T ) := fu 2 H 1 (
) : ujK Æ FK 2 S pK (K^ ) 8K 2 T g:
(2.5)
Here, the generi polynomial spae S p (K^ ) is to be understood as Qp (Q^ ) if K^ = Q^
p
and as P p (T^) if K^ = T^. Further, we dene S0 (T ) := S p (T ) \ H01 (
). If pK = p for
p
all K 2 T , we simply write S p (T ) and S0 (T ), respetively.
2.3. SDFEM Disretization
The standard Galerkin Finite Element Method for (2.1), (2.2) is:
p
Find U 2 S0 (T ) suh that
B (U; V ) := "
p
Z
rU rV dx +
Z
(~a rU + bU )V dx = F (V ) :=
Z
fV dx (2.6)
for all V 2 S0 (T ).
To improve the stability of the sheme (2.6), the test funtion V is replaed by
a test funtion \upwinded" in stream diretion ~a given on eah element K by
V jK + ÆK K (~a rV )jK where K 0 is a mesh-dependent parameter (depending
on hK;max, hK;min and pK ) and ÆK 0 is a user-speied parameter to be seleted
later on. This yields the SDFEM formulation:
Find U
2 S0p (T ) suh that BSD (U; V ) = FSD (V ) for all V 2 S0p (T )
(2.7)
The hp Streamline Diusion FEM 5
where
X
BSD (U; V ) := B (U; V )
FSD (U; V ) := F (V ) +
K 2T
X
K 2T
Æ K K
ÆK K
Z
Z
K
K
("U
~a rU
bU )(~a rV )dx;
f (~a rV )dx:
If f 2 L2(
), then the exat solution u of (2.1), (2.2) satises BSD (u; V ) = FSD (V )
p
for all V 2 S0 (T ), and hene we have the orthogonality property
8V 2 S0p (T ):
We assume the SDFEM parameters fÆK g and fK g to be given by
BSD (U
u; V ) = 0
(2.8)
h2
1 h2
(2.9)
2K = 2 2 K;min K;max
pK hK;max + h2K;min
for some onstants Æ0 > 0 and 0. We will make use of the following short-hand
notation
h2
h2
;
K 2T:
(2.10)
h2K := 2 K;min K;max
hK;max + h2K;min
We will ome bak to the hoie of the parameter in our numerial experiments
ahead. We note that the parameter ÆK = 0 is not exluded in (2.9) and leads to the
standard Galerkin FEM (2.6). In order to be able to analyze the \true" SDFEM, it
will be onvenient to assume oasionally the following non-degeneray ondition:
0 ÆK Æ0 ;
9Æ > 0
suh that ÆK Æ
8K 2 T :
(2.11)
Remark 2.1 Non-degeneray onditions of the form (2.11) are implemented in
pratie slightly dierently: Typially, ÆK Æ is stipulated only for element K with
K > ".
Our analysis will be performed in the framework of the \energy norm" k kE and
the mesh-dependent SDFEM norm k kSD given on H01 (
) as:
kuk2E := "kruk2L2(
) + kuk2L2(
) ;
kuk2SD := kuk2E +
X
K 2T
ÆK K k~a ruk2L2(K ) :
(2.12)
Remark 2.2 In the subsequent analysis the parameters fK g in (2.9) an equiv-
alently be hosen as 2K = h2K;min =p2K beause there holds hK hK;min by the
standard two-sided bound 12 min fa; bg aab
+b min fa; bg, valid for all a, b > 0.
3. Approximation on Boundary Layer Meshes
Due to the singular perturbation parameter ", solutions of (2.1), (2.2) exhibit boundary layer phenomena. The numerial approximation of these layers requires arefully
designed meshes with anisotropi needle elements. In Se. 3.2 the lass of boundary
layer meshes is introdued, and in Se. 3.3 it is shown that layers an be resolved
on suh meshes at robust exponential rates of onvergene.
3.1. Properties of the Solutions
6 The hp Streamline Diusion FEM
For the design of hp methods for (2.1), (2.2), it is important to have preise information about the solution behavior, that is, to have some regularity theory. The best
tool available at present for desribing the regularity are asymptoti expansions.
There, the solution u is deomposed in a number of omponents, typially in the
form
u = usmooth + ulayer + urem:
(3.1)
Eah solution omponent aounts for a dierent feature of the solution. In pratie,
the \smooth" part is pieewise analyti. The \layer" part onsists of (possibly
several) omponents whih have a typial layer behavior: With respet to a speial,
tted oordinate system (r; s) the layer omponent ulayer = ulayer (r; s) behaves
smoothly in the variable s but deays sharply in the other variable r. From an
analysis of the simpler one-dimensional ase19;20 and the related reation-diusion
equation in two dimensions21 , we an expet the layer parts to satisfy the following
regularity properties:
Denition 3.1 An analyti funtion u = u(r; s) is said to be of layer type with
length sale l > 0 if there are onstants C , , d > 0 suh that
jrm sn u(r; s)j C m+n n! maxfm; l 1gm e
dr=l ;
m; n 2 N 0 ; r > 0:
(3.2)
For the solutions of (2.1), (2.2), the two most important length sales l are l = O(")
giving exponential layers and l = O("1=2 ) yielding paraboli layers. We mention
that further length sales of the form l = O("1=m ) with m 2 N 0 , m 3 an arise
in ertain ases.31;32 In the present work, however, we will restrit our attention to
the two dominant layers, namely, exponential and paraboli layers.
For suh small length sales, we see that Denition 3.1 reets the typial, anisotropi behavior of layers, namely, the rapid deay in one diretion (here: for r ! 1)
oupled with the fat that dierentiation with respet to the variable r produes
negative powers of the length sale and a smooth (here: analyti) behavior in the
other diretion.
Layers appear as boundary layers near the outow boundary + = fx 2 j ~a(x) ~n(x) > 0g. There, the tted oordinate system (r; s) is given by r = dist (x; + ) and
s is the ar-length parameter for + . The layers near + are of exponential type.
A similar situation holds for the harateristi part 0 = fx 2 j ~a(x) ~n(x) = 0g,
where paraboli layers are present. Another instane of paraboli layers is given
by internal layers, aused by unsmoothness of the data near the inow boundary
= fx 2 j ~a(x) ~n(x) < 0g.
The remainder urem in (3.1) is dened suh that (3.2) holds true one the smooth
part usmooth and the layer part ulayer are dened by means of asymptoti analysis.
Little an be found in the literature about the regularity of urem (we hint at some
of the diÆulties by pointing out that in the ase of a polygonal domain , the
solution has to exhibit orner singularities, whih an interat with boundary layers
near some of the verties30). Often in pratie, the omponents usmooth, ulayer are
the dominant features of the solution. In our analysis, therefore, we will neglet the
eets of urem .
3.2. Boundary Layer Meshes
We introdue the lass of boundary layer meshes that are designed to resolve layer
omponents arising in solutions of (2.1), (2.2):
Denition 3.2 A mesh T is alled a boundary layer mesh if for eah element
K 2 T there exists a triangle R^K = f(; ) : 0 < < hx; 0 < < hhxy g or a
retangle R^K = (0; hx ) (0; hy ) suh that the mapping F~K (; ) := FK ( hx ; hy ),
The hp Streamline Diusion FEM 7
(; ) 2 R^K , satises on R^K
kD F~K k1 C2 CFj~j j j!
C1 1 det DF~K C1 ;
8 multi-indies 2 N 20
(3.3)
with onstants C1 ; C2 and CF~ independent of the elements.
Denition 3.2 formalizes the idea that for boundary layer meshes the element mappings FK an be fatorized into two transformations F1 and F~K ,
FK (^x; y^) = F~K Æ F1 (^x; y^);
F1 (^x; y^) =
hx 0
0 hy
x^
y^
:
(3.4)
This is indiated in Figure 1. The saling and anisotropy properties of the element
K are oded into the map F1 , while F~K is ompletely independent of these issues
due to (3.3).
K
omposemesh.eps
109 53 mm
FK
y^
F1
K^
x^
F~K
R^K
Figure 1: The fatorization of the element mapping in boundary layer meshes.
The rather tehnial assumptions in (3.3) may be diÆult to hek in pratie. A
possible onstrution of boundary layer meshes is based on a maro-element tehnique: We start with a marosopi shape regular mesh Tm = fM g. Some of these
maro-element pathes M are now further partitioned by mapping a orresponding renement T^ of the referene element into M with the element transformation
FM . We will be interested in the following referene pathes T^ that are dened
in terms of additional parameters suh as polynomial degrees p, length sales l,
grading fators q, and number of layers L (f. Fig. 2):
(P1) The trivial path: T^ = K^ .
(P2) The two-element path: T^pl = f(0; pl) (0; 1); (pl; 1) (0; 1)g.
(P3) The geometri path: The referene element K^ is rened geometrially and
anisotropially towards the line x = 0 with grading fator q 2 (0; 1) and a
number of layers L 2 N 0 suh that qL l. That is, the mesh is given as
T^ = f(xi+1 ; xi ) (0; 1) j i = 0; : : : ; L + 1g, where xL+1 = 0 and xi = qi for
i 2 f1; : : : ; Lg.
8 The hp Streamline Diusion FEM
(P4) The tensor produt path: For two length sales l1 , l2 and grading fator
q 2 (0; 1) let L1 , L2 2 N 0 suh that qL1 l1 , qL2 l2 . Then the referene
element K^ is rened geometrially towards the lines x = 0 and y = 0 with L1
layers in the x-diretion and L2 layers in the y-diretion. That is, the mesh is
given as T^ = f(xi+1 ; xi ) (yj+1 ; yj ) j i = 0; : : : ; L1 +1; j = 0; : : : ; L2 +1g where
xL1 +1 = 0, yL2 +1 = 0, and xi = qi , i = 0; : : : ; L1, yj = qj , j = 0; : : : ; L2 .
(P1)
1
0
(P2)
1
(P3)
l
pl
pathes.eps
84 73 mm
(P4)
l1
l2
Figure 2: Referene pathes (P1){(P4).
Remark 3.3 We restrited ourselves to quadrilateral mesh pathes. However, analogous renement strategies an also be dened on triangular pathes.
3.3. Approximation of Layers on Boundary Layer Meshes
The purpose of the present setion is to illustrate that the mesh pathes presented
above are well-suited for the approximation of layer funtions at robust exponential
rates. The approximation results presented here over not only the approximation
in \energy norms" but also in stronger norms in order to enable us to analyze the
SDFEM below. As the onstrution of the interpolants is fairly tehnial, the atual
proofs are olleted in the Appendix.
We start at this point by showing that the property of being a funtion of boundary
layer type in the sense of Denition 3.1 is invariant under analyti hanges of variables. This will be our tool to be able to restrit our attention to very few referene
ongurations.
Throughout, let S denote the losed referene square, i.e., S = [0; 1℄2 .
Lemma 3.4 Let S = [0; 1℄2 and F : S ! R2 , (x; y) 7! (r; s) = F (x; y) be an
analyti dieomorphism in a neighborhood of S . Assume that r = r(x; y) satises
r(0; y) = 0 (i.e., F maps the line x = 0 into the line r = 0). Let u be of boundary
layer type and satisfy (3.2) on F (S ). Then there are C 0 , 0 , d0 > 0 depending
only on the onstants C , , d of (3.2), and the mapping F suh that the funtion
The hp Streamline Diusion FEM 9
u~ := u Æ F satises on S :
jxm yn u~(x; y)j C 0 ( 0 )m+n n! max fm; l 1gme
d0 x=l
8(m; n) 2 N 20 ; (x; y) 2 S:
(3.5)
Proof : A very similar argument was arried out in detail by Melenk and Shwab22
(see Lemma 3.6).
2
The merit of Lemma 3.4 is that it allows us to perform approximation theory on
referene ongurations in the framework of the mesh pathes introdued above.
We may therefore assume for the purpose of the present setion that a funtion u
of boundary layer type is given on S and satises (3.5).
We an now formulate the following three approximation results for funtions satisfying (3.5). The proofs of Theorems 3.5, 3.7 and 3.8 are relegated to the Appendix.
We start with the two-element path (P2) (f. Fig. 2):
Theorem 3.5 For > 0, p 2 N let x := min f1=2; plg and dene the \twoelement mesh" T^pl = f(0; x) (0; 1); (x; 1) (0; 1)g. Let u satisfy (3.5). Then
there exist C , , 0 depending only on C 0 , 0 , d0 of (3.5) suh that for 2 (0; 0 )
there exists p 2 S p (T^pl ) with
plkr(u p )kL2(S) + ku p kL2(S)
plkr(u p )kL (S) + ku p kL (S)
1
1
Cl1=2 e p ;
Ce p :
Remark 3.6 The two-element path (P2) is essentially the minimal mesh that an
resolve funtions of layer type at a robust exponential rate; we refer to Shwab
and Suri27 where the neessity of a small element of size O(pl) in the layer was
demonstrated for a one dimensional model problem.
The next theorem irumvents the need to hoose by onsidering meshes that are
graded geometrially towards the line x = 0. These meshes are the ones presented
as type (P3) (f. Fig. 2):
Theorem 3.7 Let u satisfy (3.5) on S . Let q 2 (0; 1) be a xed grading fator and
let L 2 N 0 be suh that qL l. Let T^x be a mesh on (0; 1) given by the points
xL+1 = 0, xi = qi for i = 0; : : : ; L. Let T^y be an arbitrary mesh on (0; 1) and
T^ = T^x T^y . Then there exist onstants C , > 0 depending only on C 0 , 0, d0 , of
(3.5), and an interpolant p 2 S p (T^ ) suh that
lkr(u p )kL2(S) + ku p kL2 (S)
lkr(u p )kL (S) + ku p kL (S)
X hK;min hK;max
l2 kr(u p )k2L (K )
l
K 2T^
X
hK;min kr(u p )k2L2 (K )
K 2T^
1
1
1
Cl1=2 e p ;
Ce p ;
Ce p ;
Ce
p :
It should be noted that the number of elements in the meshes onsidered in Theorem 3.7 is not independent of the length sale l of the layer: From the ondition
qL l we immediately get L = O(j ln lj). Nevertheless, this is a rather weak side
ondition. The preeding two theorems were onerned with the approximation of
a single layer. Let us now turn to the problem of approximating two layers on S
10 The hp Streamline Diusion FEM
loated at the lines x = 0 and y = 0. Speially, let u1 , u2 be two funtions of
boundary layer type satisfying for all (m; n) 2 N 20 and (x; y) 2 S
jxm yn u1 (x; y)j C m+n n! max fl1 1; mgme
jym xn u2 (x; y)j C m+n n! max fl2 1; mgme
dx=l1 ;
dy=l2 :
(3.6)
(3.7)
The simultaneous approximation of these two dierent layers on the same domain
S an be handled by tensor produt meshes of the type (P4):
Theorem 3.8 Let u1 , u2 satisfy (3.6), (3.7), q 2 (0; 1) be a xed grading fator
and let L1 , L2 2 N 0 be suh that qL1 l1 , qL2 l2 . Let T^1 , T^2 be two meshes on
(0; 1) given by the points
xL1 +1 = 0; xi = qi ; i = 0; : : : ; L1
yL2 +1 = 0; yj = qj ; j = 0; : : : ; L2 ;
respetively. Dene the tensor produt mesh T^ as T^ := T^1 T^2 . Then there exist
onstants C , > 0 independent of p, l1 , l2 , and interpolants i 2 S p (T^ ), i = 1; 2,
suh that for i = 1; 2 there holds:
li kr(ui i )kL2 (S) + kui i kL2(S)
li kr(ui i )kL (S) + kui i kL (S)
X hK;min hK;max
li2 kr(ui i )k2L (K )
l
i
K 2T^
X
hK;min kr(ui i )k2L2 (K )
K 2T^
1
1
1
Cli1=2 e p ;
Ce p ;
Ce p ;
Ce p :
Remark 3.9 The meshes of Theorem 3.5, 3.7, 3.8 onsist of quadrilaterals only.
This is due to onvenient tensor produt onstrutions of the interpolants. Interpolants on meshes inluding triangles an also be onstruted (with additional
eort, however).
4. Stability of the SDFEM
In this setion we address the stability of the SDFEM on boundary layer meshes.
This problem is losely related to inverse inequalities on anisotropi elements, whih
are presented in the following.
4.1. Inverse Inequalities
We reall basi inverse estimates26 valid on the referene element K^ (= Q^ or T^):
Lemma 4.1 There exists a onstant C independent of p suh that
krp kL2 (K^ ) Cp2 kp kL2(K^ ) ;
kp kL1( K^ ) Cpkp kL2 (K^ )
for all polynomials p 2 S p (K^ ).
On a boundary layer mesh T we have:
Proposition 4.2 Let T be boundary layer mesh in the sense of Denition 3.2.
Then there exists Cinv > 0 depending only on the onstants of Denition 3.2 suh
The hp Streamline Diusion FEM 11
that for all p 2 S p (T ) and all elements K 2 T
h2
Cinv K4 krp k2L2 (K )
p
2
h
Cinv K4 kD2 p k2L2 (K )
p
Cinv hK;min
k k2
p2 hK;max p L1 (K )
kp k2L2(K ) ;
(4.1)
krp k2L2(K ) ;
(4.2)
kp k2L2(K ) :
(4.3)
Proof : The element mapping FK : K^ ! K an be fatorized into FK = F~K Æ F1 ,
where F1 is the saling = hx x^, = hy y^ from K^ onto R^K with saling fators hx ,
hy of Denition 3.2 and (3.4). Taking into aount (3.3) we see that
C
C
1 hK;max
1 hK;min
maxfhx; hy g ChK;max;
minfhx ; hy g ChK;min :
(4.4)
~ p (; ) =
Fix now a mapped polynomial p on K , i.e., p Æ FK 2 S p (K^ ). We set p Æ F~K (; ). Note that ~ p is a polynomial of degree p on R^K . Using again (3.3)
we have
krp kL2(K ) C kr~ p kL2 (R^K ) ;
kr~ p kL2(R^K ) C krp kL2(K ) ;
kD2 p kL2(K ) C k~ p kH 2 (R^K ) :
k~ p kL2 (R^K ) C kp kL2 (K );
(4.5)
(4.6)
From a Poinare inequality for onvex sets10 there holds (writing jR^K j for the area
of R^K )
~ kL2(R^ ) inf k
K
2R p
s
2
2
2
^K )2 kr
~ K kL2 (R^ ) C hpx + hy kr
~ p kL2(R^ ) :
(diam
R
K
K
hx hy
jR^K j
Thus, observing that the left-hand side of (4.6) does not hange if p is replaed
with p + , 2 R, we obtain
#
"
2
2
kD2 p k2L2(K ) C kD2 ~ p k2L2(R^K ) + hxh +hhy kr~ p k2L2 (R^K ) ;
x y
(4.7)
where we additionally exploited the fat that (h2x + h2y )=(hxhy ) 1. The inverse
~ p Æ F1 yield
estimates of Lemma 4.1 and a saling argument applied to ~
p
k kL2 (R^K ) Chx 1 p2 k~ p kL2 (R^ K ) ;
~
p
k kL2(R^K ) Chy 1 p2 k~ p kL2 (R^K ) :
(4.8)
12 The hp Streamline Diusion FEM
Combining (4.4), (4.5), and (4.8) results in
h2K;maxh2K;min
h2xh2y
22
C
kr
k
kr~ p k2L2 (R^ K )
p
L (K )
h2K;max + h2K;min
h2x + h2y
h2 h2
~
~
= C 2 x y 2 fk p k2L2(R^K ) + k p k2L2(R^K ) g
hx + hy 2
2
Cp4 hh2y ++ hhx2 k~ p k2L2 (R^K ) Cp4 kp k2L2 (K ) :
x
y
Analogously, using (4.7) and (4.8),
h2K;maxh2K;min
h2 h2
k
D2 p k2L2(K ) C 2 x y 2 kD2 ~ p k2L2 (R^K ) + C kr~ p k2L2 (R^ )
2
hx + hy
K;max + hK;min
2
2
x hy
2~ 2
hCh
2x + h2y fk p kL2 (R^ K )
~ p k2 2 ^
+2k ~ p k2L2 (R^ K ) + +k2~ p k2L2(R^K ) g + C kr
L (R)
4
fh2y k ~ p k2L2 (R^ K ) + h2x k ~ p k2L2(R^K )
h2Cp
2
+
h
x
y
~ p k2 2 ^
+h2xk ~ p k2 2 ^ g + C kr
h2
L (RK )
L (R)
2
2
C hhy2 ++ hhx2 p4 kr~ p k2L2(R^K ) Cp4 krp k2L2(K ) :
x
y
This shows (4.1) and (4.2).
To prove (4.3), we alulate similarly by writing ^ p := p Æ FK :
kp kL1 (K ) C k~ p kL1( R^K ) ChK;max k^ p kL1 ( K^ ) ;
k^ p kL2 (K^ )
s
=
s
1
k~ k
C h 1h
kp kL2 (K ) :
hK;max hK;min p L2 (R^K )
K;max K;min
(4.3) an now be inferred from these last two estimates together with Lemma 4.1.2
4.2. Stability in the SDFEM Norm
Let T be a boundary layer
mesh.p We show that the bilinear form BSD of the
p
SDFEM is oerive on S0 (T ) S0 (T ).
Proposition 4.3 Let the weights K be given by
h
K = K if p7K hK3 "2 C for some arbitrary, xed onstant C > 0; (4.9a)
pK
hK
K = 2 else:
(4.9b)
pK
Then there exists a onstant Æ0 depending only on Cinv in Lemma 4.2, the onstant
C of (4.9a), and on the data ~a, b suh that for parameters ÆK satisfying 0 ÆK Æ0
The hp Streamline Diusion FEM 13
the SDFEM (2.7) is oerive, i.e.,
1
p
k
U k2SD BSD (U; U )
8
U 2 S0 (T ):
2
Proof : Assume rst that 0 ÆK Æ00 with Æ00 to be hosen later on. We have
BSD (U; U ) = B (U; U ) +
X
K 2T
ÆK K
Z
K
( "U + ~a rU + bU )(~a rU )dx:
Due to assumption (2.3) we have B (U; U ) kU k2E , and it remains to estimate the
ritial terms
A1 :=
A2 :=
X
K 2T
X
K 2T
A1;K :=
ÆK K
Z
X
K 2T
K
ÆK K
Z
K
"U (~a rU )dx;
bU (~a rU )dx:
We rst bound A1 : Fix K 2 T and onsider the ase (4.9b) where K = hp2KK . We
use Cauhy-Shwarz and the inverse estimates in Proposition 4.2 to get
jA1;K j ÆK K "kU kL2(K ) k~a rU kL2(K )
2
CÆK K hpK "krU k2L2(K ) CÆ00 "krU k2L2(K ) :
K
In the ase (4.9a) we get by a twofold appliation of Proposition 4.2:
jA1;K j ÆK K "2 kU k2L2(K ) + 14 ÆK K k~a rU k2L2(K )
8
CÆ00 K "2 hpK4 kU k2L2(K ) + 14 ÆK K k~a rU k2L2(K )
K
CÆ00 kU k2L2(K ) + 14 ÆK K k~a rU k2L2 (K ) :
Hene, we get for A1 the bound
X
jA1 j CÆ00 kU k2E + 41 ÆK K k~a rU k2L2(K ) :
K 2T
The seond term A2 is estimated as follows:
jA2 j C
C
C
X
K 2T
ÆK K kU kL2(K ) k~a rU kL2(K )
X p
K 2T
X
K 2T
p
ÆK K kU kL2(K ) ÆK K k~a rU kL2(K )
ÆK K kU k2L2(K ) +
CÆ00 kU k2L2(
) + 41
X
K 2T
1 X
Æ k~a rU k2L2(K )
4 K 2T K K
ÆK K k~a rU k2L2(K ) :
14 The hp Streamline Diusion FEM
Combining the above estimates gives
BSD (U; U ) (1 CÆ00 )kU k2E +
1 X
Æ k~a rU k2L2(K ) :
2 K 2T K K
2
Seleting now Æ00 Æ0 := 21C nishes the proof.
Remark 4.4 In Proposition 4.3 we hose K = hK =pK when " is small ompared
with hK and 1=pK in the sense of (4.9a). This partiular hoie is motivated by our
analysis on quasiuniform meshes in Remark 5.9 to give hp-error bounds. By similar
tehniques stability of the SDFEM an also be obtained for K = hK =pK for 0
when a ondition analogous to (4.9a) is satised. The performane of the SDFEM
in dependene on is investigated numerially in Se. 6.
5. Consisteny of the SDFEM
In this setion the hp-approximation results of Se. 3 and the stability properties
in Se. 4 are ombined into our main result: We prove in Se. 5.1{5.3 that robust
exponential rates of onvergene an be ahieved in the hp-SDFEM provided that
all layer omponents present in the solutions are resolved. Moreover, in Se. 5.4 we
derive on shape regular meshes optimal hp onvergene results for smooth solutions.
5.1. Exponential Convergene
p
p
By Proposition 4.3 we may assume that the SDFEM is oerive on S0 (T ) S0 (T ),
i.e., there holds for some 0 > 0
0 kU k2SD BSD (U; U )
8 U 2 S0p (T ):
We use the Galerkin orthogonality in (2.8) and obtain for every interpolant Iu 2
p
S0 (T )
0 kU
= "
Z
Iuk2SD BSD (u Iu; U
r(u Iu)r(U Iu)dx +
X
K 2T
ÆK K
=: S1 (; U
Z
K
Z
Iu)
(~a r(u Iu) + b(u Iu))(U
("(u Iu) ~a r(u Iu) b(u Iu))(~a r(U
Iu) + S2 (; U
Iu) + S3 (; U
Iu)dx
Iu))dx
Iu);
where we wrote = u Iu. Next, we introdue the semi-norms
Ti () :=
Si (; )
supp
;
06=2S0 (T ) k kSD
i 2 f1; 2; 3g
(5.1)
and then get
ku U kE ku IukE + kU IukSD kkE + 0 1 [T1 () + T2 () + T3 ()℄ : (5.2)
p
Note that the interpolant Iu 2 S0 (T ) was so far not speied and is still at our
p
disposal. Let us write Iu 2 S0 (T ) in the form Iu = Iusmooth + Iulayer + Iurem
and introdue smooth = usmooth
Iusmooth, layer = ulayer
Iulayer , rem =
The hp Streamline Diusion FEM 15
urem Iurem. As the expressions Ti are semi-norms on H 1 (
), we get the a priori
bound
ku U kE ksmooth kE + klayer kE + krem kE
3
X
+0 1(
i=1
Ti (smooth ) + Ti (layer ) + Ti (rem )):
(5.3)
The smooth omponent usmooth and the layer omponent ulayer are (pieewise)
analyti, and they an be approximated at robust exponential rates of onvergene
using, for example, the mesh pathes introdued in Se. 3. This approximability
result holds for the SDFEM as well. As no regularity theory is available for the
remainder urem we will restrit ourselves to the assumption that urem vanishes (or
at least is negligible). In this ontext, we an formulate the following main result
of this paper:
Theorem 5.1 Let u be the solution of (2.1), (2.2). Assume that u is of the form
u = usmooth + ulayer with usmooth being (pieewise) analyti and ulayer onsisting of
nitely many layer omponents in the sense of Denition 3.1. Let U 2 S0p (T ) be the
SDFEM solution where we assume that the method is stable aording to Proposition
4.3 and satises (2.9), (2.11). Let T be a boundary layer mesh on generated by
mesh pathes of the form (P1), (P3), or (P4) suh that all layer omponents an
be resolved. Then there exist C , > 0 suh that for the error u U there holds
ku U kE C jT j1=2 e
p ;
where jT j stands for the number of elements in the triangulation T .
Remark 5.2 We exluded the use of the \two-element" path in Theorem 5.1 as
we wanted to be able to handle the ase of the simultaneous approximation of two
dierent layers on the same path. The mesh pathes of type (P2) an be used if only
one type of layer has to be approximated on eah path. A areful inspetion of the
proof of Proposition 5.7 shows that the fator jT j1=2 stems from the approximation
of \paraboli" layers on meshes of type (P3) and (P4). Hene, the fator jT j1=2
ould be avoided if only \exponential" layers our.
If mesh pathes of type (P3) or (P4) are used in the generation of the mesh, then
the number of elements (and hene also the number of degrees of freedom) depends
weakly on ", i.e., jT j C (ln ")2 and DOF Cp2 (ln ")2 .
Se. 5.2 and Se. 5.3 are devoted to the proof of Theorem 5.1, i.e., to obtaining
bounds for the terms Ti (smooth ) and Ti (layer ) in (5.3). The proof of Theorem
5.1 will then follow from (5.3), the fat that the maro element maps are analyti
dieomorphisms and from Propositions 5.4, 5.7 below. Stritly speaking, one has
to hek thatp the pathwise dened interpolants of these propositions lead to an
element of S0 (T ); this is indeed the ase.
5.2. hp-Approximation of the Smooth Part
For the smooth part usmooth we have the following lemma.
Lemma 5.3 Let T be a boundary layer mesh in the sense of Denition 3.2 and let
2 H01 (
) \ K 2T H 2 (K ). Then there exists C > 0 depending only on and the
16 The hp Streamline Diusion FEM
oeÆient funtions ~a, b suh that
jT1 ()j " 1 =2
X
K 2T
!1=2
kk2 1
H (K )
8
<
;
!1=2 9
=
1
k
k2L2(K )
;
jT2 ()j C min :kkH 1 (
) ;
;
K 2T " + ÆK K
X
jT3 ()j C ( ÆK K f"2 kk2H 2 (K ) + kk2H 1 (K ) + kk2L2(K ) g) 21 :
K 2T
Proof
: The bounds for T1 () and T3 () are obvious. To estimate T2 (), let p
S0 (T ) and integrate the expression S2 (; ) in (5.1) by parts to get
Z
Hene, we have
Z
(~a r
X
Z
(~a r) dx =
) dx
X
K 2T
~a r dx
Z
(div ~a) dx:
kkL2 (K ) k~a rkL2 (K ) + C kkL2 (
) kkL2(
) :
As by the hoie of the parameters ", ÆK K there holds " + ÆK K
C > 0, and we an estimate
(
1
kkL2(
) kkL2 (
) C
kk2L2(K )
"
+
Æ
K
K
K 2T
Next, the Cauhy-Shwarz inequality for sums gives
X
K 2T
2
X
) 1 =2
C
for some
kkSD :
kkL2(K ) k~a rkL2 (K )
(
1
kk2L2(K )
"
+
Æ
K
K
K 2T
X
(
) 1 =2 (
) 1 =2
X
K 2T
(" + ÆK K )k~a rk2 2
)1=2
L (K )
1
C
kk2L2(K ) kkSD :
"
+
Æ
K
K
K 2T
The desired bounds for T2 () now follow.
2
For the smooth part usmooth , we an now formulate:
Proposition 5.4 Let T be a boundary layer mesh in the sense of Denition 3.2
onsisting of quadrilaterals only. Assume that (2.9) and (2.11) hold. Let usmooth
be analyti on . Let Iusmooth 2 S p (T ) be the pieewise Gauss-Lobatto interpolant
of usmooth . Then there exist C , > 0 suh that, upon writing smooth = usmooth
Iusmooth, there holds
X
ksmoothkE +
3
X
i=1
jTi (smooth )j Ce
p :
The hp Streamline Diusion FEM 17
Proof : The proof follows diretly from
the observation that for eah element K
p
there holds ksmooth kW 2; (K ) Ce
2
.
1
5.3. hp-Approximation of the Layer Part
For the approximation of exponential boundary layers, Lemma 5.3 is not appropriate
as it annot lead to robust estimates. For estimates of the layer part, we have to
treat the term T3 dierently. This is aomplished in the next lemma.
Lemma 5.5 Let T be an arbitrary boundary layer mesh in the sense of Denition 3.2 and let 2 W 1;1 (
). Then there exists C > 0 depending only on , the
oeÆient funtions ~a, b, and the onstants of Denition 3.2 suh that
X
kk2 1
! 1 =2
jT1()j " 1 =2
jT2()j 1
C
kk2L2(K ) ;
"
+
Æ
K
K
K 2T
C E1 () + E2 () + E3 () + kkL2(
)
jT3()j K 2T
H (K )
;
!1=2
X
where
(
E1 () :=
K 2T
(
E2 () :=
(ÆK K )2
i
p4 " h
p4
22
"
kr
k
L
(
K
)
p4 " + h2K;min h2K;min
i
(ÆK K )2 p4 p2 hK;max h 2
2
"
kr
k
L (K )
2
4
K 2T p " + hK;min hK;min
X
) 1 =2
)1=2
1
(
E3 () :=
X
X
K 2T
min fF1;K (); F2;K ()g
) 1 =2
; (5.4)
;
(5.5)
;
(5.6)
"
#
p4
p2 h
(Æ )2 p4
k
k2L2(K ) + K;max kk2L (K ) ; (5.7)
F1;K () := 4 K K2
2
p " + hK;min hK;min
hK;min
(5.8)
F2;K () := ÆK K k~a rk2L2 (K ) :
1
Remark 5.6 On polygons, the exat solution u has orner singularities suh that
estimates that ontain terms like krkL (K ) are not diretly appliable. However,
the proof of Lemma 5.5 shows that these L1 bounds are not neessary in all elements. Nevertheless, to get meaningful bounds avoiding these L1 estimates, more
information about the regularity of the exat solution in the viinity of the orners
is neessary. Suh regularity issues must be addressed in future work.
Proof of Lemma 5.5 : The bounds on the terms T1 and T2 are thosep of Lemma 5.3.
We an therefore turn diretly to bounding T3 (). For any 2 S0 (T ), the term
S3 (; ) is a sum of integrals over elements K . Eah term of this sum an be written
as ÆK K (t1 (K ) + t2 (K ) + t3 (K )) with
1
t1 (K ) = "
Z
K
~ar dx; t2 (K ) =
Z
K
(~ar) (~ar) dx; t3 (K ) =
Z
K
b ~ar dx:
18 The hp Streamline Diusion FEM
To estimate t1 (K ), t2 (K ), t3 (K ), we start by showing the existene of C > 0 suh
p
that for all 2 S0 (T )
s
4
kkH 1 (K ) C p4 " +ph2 kk1;";K ;
K;min
s
p2
(5.9)
4
p
kkH 2 (K ) C h
kk ;
(5.10)
4
p
"
+
h2K;min 1;";K
K;min
where we wrote kk21;";K := "krk2L2(K ) + kk2L2(K ) . The estimate for kkH 2 (K )
follows immediately from that for kkH 1 (K ) by Proposition 4.2. For the former one,
we use
2
krkL2(K ) C h p kkL2(K )
1=2 "1=2 kr k 2
L (K ) ;
krkL2(K ) "
and thus arrive at
2 h
1 g
minf" 1=2; p2 hK;min
p4
kk2 :
C 4 2
"p + hK;min 1;";K
krk2L2(K ) C
K;min
"krk2L2(K ) + kk2L2(K )
i
We estimate now ti (K ) and start with t1 (K ): An integration by parts yields
t1 (K ) = "
Z
K
n (~a r) "
Z
r (D~ar) dx "
K
Z
K
r (~aD2 ) dx;
where D2 here stands for the Hessian of . Eah of these three terms is estimated
separately. The rst term an be bounded by
Z
"
K
n (~a r
)
C"krkL
1
s
(K ) kr kL1 (K )
Cp hhK;max "krkL
K;min
s
h
Cp K;max
hK;min
s
1
(K ) kr kL2 (K )
p4
4
p " + h2
K;min
"krkL (K ) kk1;";K ;
1
using Proposition 4.2 and (5.9). For the remaining two omponents of t1 we have
Z
"
K
r (D~ar) dx C"krkL2(K ) krkL2(K )
s
Z
"
K
r~a D2 dx
4
C "p4 +ph2 "krkL2(K ) krk1;";K ;
K;min
C"krkL2(K ) kkH 2 (K )
p2
s
4
p
"krkL2(K ) kk1;";K :
Ch
4
p
"
+
h2K;min
K;min
The hp Streamline Diusion FEM 19
Therefore, we get for t1 (K ):
jt1 (K )j s
Cp4
"p2
krkL2(K )
2
4
p " + hK;min hK;min
s
h
+p K;max "krkL (K )
hK;min
1
#
kk1;";K :
The Cauhy-Shwarz inequality for sums now gives
X
K 2T
ÆK K jt1 (K )j C [E1 () + E2 ()℄ kkSD :
We now turn to t2 (K ). We have
Z
jt2 (K )j = K
(~a r)(~a r) dx k~a rkL2 (K ) k~a rkL2 (K ) ;
and this gives immediately the term F2;K () in the minimum ourring in E3 ().
We therefore have to see that t2 (K ) an also be bounded by F1;K (). To that end,
we start just as in the treatment of t1 (K ) by an integration by parts to arrive at
t2 (K ) =
Z
K
Z
(~a r)(~a ~n) ds
K
(~a r) div ~a dx
Z
K
~a r(~a r) dx:
Proeeding along the same lines as above, we get
Z
(~a
K
Z
(~a
r
K
Z
K
)(~a ~n) ds
) div ~a dx
r
~a r(~a r) dx
s
s
h
p4
kk
kk ;
Cp K;max 4
hK;min p " + h2K;min L (K ) 1;";K
1
C kkL2(K ) k~a rkL2 (K ) C kkL2(K ) kkH 1 (K )
s
4
C kkL2(K ) p4 " +ph2 kk1;";K ;
K;min
C kkL2(K ) kkH 2 (K )
C kkL2(K ) h
p2
K;min
s
p4
4
p " + h2
K;min
kk1;";K :
Therefore,
s
s
"
4
p2
jt2 (K )j C p4 " +ph2
kkL2(K ) + p hhK;max kkL (K )
h
K;min
K;min
K;min
1
#
kk1;";K :
We reognize that this bound leads diretly to F1;K (). Finally, bounding
jt3 (K )j kbkL
1
(
) k kL2 (
) k~a r kL2 (
)
20 The hp Streamline Diusion FEM
and using the Cauhy-Shwarz inequality for sums, we get
X
K 2T
ÆK K jt3 (K )j
kbkL
1
(
(
)
X
ÆK K kk2L2(K )
)1=2
K 2T
C kkL2(
) kkSD ;
kkSD
whih nishes the proof.
2
The terms Ti (layer ) are exponentially small for meshes that do resolve loalized
small sale features:
Proposition 5.7 Let = (0; 1)2 and let u be a funtion of boundary layer type
dened on satisfying (3.5) for l = " or l = "1=2 . Assume that the SDFEM
parameters fK g satisfy (2.9) and the non-degeneray ondition (2.11). Let T^ be
one of the pathes of type (P2), (P3), or (P4) (in the ase (P2), the onstant is
hosen suitably depending on u; in the ase of (P4) the number of layers in the ydiretion is arbitrary). Then there exists an approximant 2 S p (T ) and onstants
C , > 0 independent of p, l suh that the error := u satises on kkE +
3
X
i=1
jTi ()j C jT^ j1=2 e
p :
jT^ j denotes the number of elements of T^ .
Proof : Theorems 3.5, 3.7 and 3.8 immediately give that for all pathes of type
(P2){(P4) that resolve the layer there holds
kkE Ce
p ;
"1=2 kkH 1 (K^ ) l1=2 kkH 1 (K^ ) Ce p :
In order to estimate T2 (), we observe that
1
kk2 C " + Æh 1 p hK;min hK;maxkk2L (K ) Ce p
" + ÆK K L2(K )
K;min
1
for some appropriate onstants C , > 0. Hene,
X
1
kk2L2(K ) jT^ je p :
"
+
Æ
K K
K 2T^
It remains to bound T3 (). To that end, we have to estimate E1 , E2 , E3 , and
kkL2(
) of Lemma 5.5. The reader may easily onvine himself that E1 () and
E2 () satisfy the desired bounds (even with onstants independent of the number
of elements of T^ ). For bounds on E3 (), we see that for the ases (P3) and (P4)
Theorems 3.7 and 3.8 yield immediately the desired bound via
E32 () X
K 2T^
F2;K () Ce p :
For the two-element path (P2) we see that we an bound
F1;K
F2;K
Ce p
Cplkk2H 1 (K ) Ce
p
if hK;min hK;max ,
if hK;min = pl.
The hp Streamline Diusion FEM 21
2
This onludes the argument.
5.4. hp-Approximation on Shape Regular and Quasiuniform Meshes for
Smooth Solutions
As a onsequene of Lemma 5.3, we derive in this setion optimal onvergene
results in h and p on shape regular meshes. To do so, we assume throughout the
setion that the hypothesis in (4.9a) is satised and that the weights K = hK =pK
on all elements.
Let T be a shape regular mesh on and let the solution u of (2.1), (2.2) be
in H s (
) \ H01 (
) for some s 2. Partiularly, we assume that usmooth = u,
p
ulayer = urem = 0. U is the SDFEM solution in S0 (T ). We denote by Iu a
p
suitable hp-interpolant of u in S0 (T ) to be speied in (5.11) ahead. From (5.3)
and Lemma 5.3 we see that kU IukSD an be bounded by
C(
X
K 2T
(" + ÆK K )kk2H 1 (K ) + ÆK K "2 kk2H 2 (K )
+(ÆK K + 1 +
1
1
)kk2L2 (K ) ) 2 :
" + ÆK K
p
We an now hoose the interpolant Iu 2 S0 (T ) in suh a way that the following
hp-approximation properties hold.26
K r
ku IukH r (K ) C hpKs r kukH s(K ) ;
0 r pK ; K = min(pK + 1; s): (5.11)
K
Inserting this into the bound (5.11) we get
kU Iuk2SD C
with
h2K 2
aK K2s 2 kuk2H s(K )
pK
K 2T
X
(5.12)
p2
h2K
1
aK = f(" + ÆK K ) + ÆK K "2 K
)
+
(
Æ
+
1
+
g:
K
K
h2K
" + ÆK K p2K
(5.13)
Proposition 5.8 Let T be a shape regular mesh on and u 2 H s (
) \ H01 (
),
s 2. Assume (2.11). Let U be the SDFEM solution of (2.7). In the preasymptoti
ase (4.9a) we then have
(
h2KK 2 2
hK
ku U kSD C
2s 2 kukH s (K ) " + pK
p
K 2T K
with K = min(pK + 1; s).
X
) 21
Proof : Due to assumption (4.9a) we an bound the term ÆK K "2 hpK2K in (5.13) by
2
C hp6KK . Hene, from (5.12) and (5.13) we get that
2
kU
hK
h2KK 2 2
SD C
2s 2 kukH s (K ) " + pK :
p
K 2T K
Iuk2
X
22 The hp Streamline Diusion FEM
The laim follows now by an appliation of the triangle inequality.
2
Remark 5.9 If the mesh T is quasiuniform, i.e., hK h, and if we use a uniform
polynomial degree p on all elements, i.e., pK = p, we get
1=2
h
h(min(p+1;s) 1) 12
" +
ku U kSD C p(s 1)
p
!
kukH s(
) :
Remark 5.10 The error estimate for the SDFEM in Proposition 5.8 and Remark
5.9 is half a power of h=p away from being optimal.26 For the h-version of the
SDFEM this is a well known fat, whih is extended in Proposition 5.8 to the
p-version of the SDFEM. In this sense the hp-estimates are optimal.
6. Numerial Examples
In this setion we onrm our theoretial results in a series of numerial examples.
6.1. Model Problems
We onsider two onvetion-diusion model problems of the form (2.1), M1 and
M2, with known exat solution:
Model Problem M1: Here, = ( 1; 0) ( 1; 1), ~a(x) = (1; 0)t , b(x) = 1
and the right-hand
side is hosen as f (x) = 1 (")x1 + 1 + 2 with 1 (") =
p
exp( (1+2"1+4") ) 1 and 2 = 1. The exat solution is
p
1 + 1 + 4"
u(x) = exp(
x1 ) + 1 (")x1 + 2 :
2"
This solution satises (2.1) with zero Dirihlet onditions at the boundaries
1g, fx = 0g and symmetry onditions at fy = 1g, fy = 1g. It has
an exponential boundary layer along the right side of , is essentially onedimensional and thus suited to onrm the one-dimensional numerial results
of Melenk and Shwab20.
fx =
Model Problem M2: Here, = (0; 1)2 , a(x) = (1; 1)t , b(x) = 1 and
f (x) = 1 (x)x2 1 (x)2 (x)x2 + x1 2 (x) x1 1 (x)2 (x) + x2 x2 2 (x)
+x1 x1 1 (x) + x1 x2 x1 x2 2 (x) x1 1 (x)x2 + x1 1 (x)2 (x)x2
with 1 (x) = exp ((x1 1)=") and 2 (x) = exp ((x2 1)="). The exat solution is
u(x) = x1 x2 (1 exp ( (1 x1 )="))(1 exp ( (1 x2 )=")):
It satises (2.1), the zero Dirihlet boundary onditions in (2.2) and has two
exponential boundary layers along the top and right side of .
To disretize these equations by the hp-SDFEM in (2.7) we use the Fortran 90 ode
HP903, a general hp-FEM framework for Finite Element implementations. The
SDFEM parameters are hosen as in (2.9), i.e., K = hK =pK with the onstant still at our disposal. We are mainly interested in = 1 and = 2. In addition,
we will also ompare the Galerkin approah (2.6) with the SDFEM (2.7) whih an
easily be done by setting ÆK = 0 for all K 2 T .
The hp Streamline Diusion FEM 23
6.2. Results for M1
We present the results for the model problem M1: We onsider rst the SDFEM and
Galerkin performane for " = 0:1 on a four-element mesh given by an equidistant
partition in x-diretion. For this large value of " the layer is rather weak and an
be resolved by the p-version SDFEM on this four-element mesh. We measure the
relative H 1 error on the element on the left side, i.e., on an element away from
the boundary layer. From Figure 3 we see that the Galerkin method performs best
followed by the SDFEM with K = hK =p2K . We observe that K = hK =pK is not
the orret hoie for the SDFEM parameter, if " is large ompared to hK and
1=pK ; this is in agreement with Proposition 4.3.
uniform mesh, ε = 1.E−1
−1
10
galerkin
sdfem h/p2
sdfem h/p
−2
10
−3
rel H1 error first element
10
−4
10
−5
10
−6
10
−7
10
−8
10
−9
10
2
3
4
5
approx. order
6
7
8
Figure 3: M1: Loal H 1 error upstream on uniform 4-element mesh.
uniform mesh, ε = 1.E−8
2
10
0
10
rel H1 error first element
−2
10
−4
10
−6
10
−8
10
galerkin
2
sdfem h/p
sdfem h/p
−10
10
2
3
4
5
approx. order
6
7
Figure 4: M1: Loal H 1 error upstream on uniform 4-element mesh.
8
24 The hp Streamline Diusion FEM
In Figure 4 we perform the same experiment on exatly the same mesh with " =
10 8. This time the behavior of the three urves is dierent. We learly see a poor
performane of the Galerkin method. The SDFEM with K = hK =pK performs
best and indeed onverges exponentially, whereas the SDFEM with K = hK =p2K
diverges, and the error is several orders of magnitude worse. Therefore, for " small
ompared to hK and 1=pK the stabilization parameter should be hosen as K =
hK =pK .
The pointwise error along the line y = 0 through for the three methods is shown
in Figure 5, and we learly see that the SDFEM error with K = hK =pK deays
exponentially in the upstream diretion. The deay with K = hK =p2K is still
exponential outside the boundary layer but again several orders of magnitude worse.
uniform mesh, p=8, ε = 1.E−8
2
10
0
10
−2
10
−4
10
−6
L∞ error
10
−8
10
−10
10
−12
10
−14
10
sdfem h/p2
sdfem h/p
galerkin
−16
10
−18
10
−1
−0.9
−0.8
−0.7
−0.6
−0.5
x−axis
−0.4
−0.3
−0.2
−0.1
0
Figure 5: M1: Pointwise error on uniform 4 element mesh on mesh line ( 1; 0) f0g.
We now turn to a study of the eet of the parameter in the fator h=p appearing
in the denition of the SDFEM. For " = 10 8 we investigate this on a quasiuniform
mesh that is not aligned with the oordinate axes. Again, we measure the relative
H 1 error on the element farthest away from the boundary layer. The results in
Figure 6 indiate the superior performane of the SDFEM for = 1, whih is
onsistent with one-dimensional numerial results and with the theoretial analysis
for the limiting ase " = 0.20;6 In partiular, we note again that = 2 is not the
orret hoie for small values of ".
In Figure 7 we show the performane for M1 on geometri boundary layer meshes.
We start with p = 2 on a four-element mesh that is not aligned with the ow. We
insert geometrially graded layers towards the boundary fx = 0g and also uniformly
inrease p. We show the results for the Galerkin method, whih does not onverge
in the range of p = 1; : : : ; 8, for K = hK =p2K and for K = hK =pK . In the last ase
we see again a superior performane.
6.3. Results for M2
The results for M1 are now onrmed for M2. We use here a uniform mesh with 16
elements that are either aligned or slightly perturbed. These meshes are shown in
Figure 8.
We again perform similar experiments and present in Figures 9 and 10 the loal
relative H 1 error in the element that has largest distane to the boundary layers.
The hp Streamline Diusion FEM 25
ε = 1.0e−8, mesh not aligned
2
10
α = 0.5
α = 0.75
α= 1
α = 1.5
α= 2
Galerkin
0
10
−2
−4
10
−6
10
1
rel H error on first element
10
−8
10
−10
10
−12
10
2
3
4
5
polynomial degree p
6
7
8
Figure 6: M1: Loal H 1 error upstream on quasiuniform non-aligned 16 element
mesh.
The results are for " = 10 8 and, as expeted from the previous results, the SDFEM
with K = hK =pK performs best and onverges exponentially. It is remarkable that
the SDFEM with K = hK =p2K diverges in the pratial range of p.
Finally, we present the global relative H 1 error on a boundary layer mesh with
anisotropi needle elements of width 10" (this mesh is essentially a tensor produt
onstrution orresponding to the path (P2) in order to resolve the layers at both
outow boundaries x = 1 and y = 1). From Figure 11 we see that the performane
of the Galerkin method is superior in this ase, whereas the SDFEM does perform
rather poorly for K = hK =pK . There is no need for the SDFEM stabilization if
the mesh already resolves the layers. However, we get exponential onvergene in
all ases as predited in Se. 5.
26 The hp Streamline Diusion FEM
ε = 1.0e−8, geometric not aligned mesh
2
10
α= 1
α= 2
Galerkin
0
10
−2
−4
10
−6
10
1
rel H error on first element
10
−8
10
−10
10
−12
10
−14
10
2
3
4
5
6
polynomial degree p and number of layers
7
8
Figure 7: M1: Loal H 1 error upstream on geometrially graded non-aligned mesh.
Figure 8: Uniform aligned mesh and non-aligned mesh.
uniform aligned mesh, ε = 1.E−8
3
10
2
10
1
rel H1 error first element
10
galerkin
sdfem h/p2
sdfem h/p
0
10
−1
10
−2
10
−3
10
−4
10
−5
10
2
3
4
5
approx. order
6
7
Figure 9: M2: Loal H 1 error upstream on aligned mesh.
8
The hp Streamline Diusion FEM 27
uniform non aligned mesh, ε = 1.E−8
3
10
2
10
1
rel H1 error first element
10
galerkin
sdfem h/p2
sdfem h/p
0
10
−1
10
−2
10
−3
10
−4
10
−5
10
2
3
4
5
approx. order
6
7
8
Figure 10: M2: Loal H 1 error upstream on non-aligned mesh.
boundary layer mesh, ε = 1.E−8
0
10
−1
global rel H1 error
10
−2
10
galerkin
sdfem h/p2
sdfem h/p
−3
10
2
3
4
5
approx. order
6
7
Figure 11: M2: Global H 1 error on boundary layer mesh.
8
28 The hp Streamline Diusion FEM
Appendix
Appendix A.1. Approximation in One Dimension
The present subsetion is onerned with pieewise polynomial approximation of
uni-variate funtions of boundary layer type. In analogy to ondition (3.5), funtions
of boundary layer type satisfy in one dimension:
jxn u(x)j C n max fl 1; ngne
dx=l
8n 2 N 0 ; x 2 (0; 1):
(A.1)
Our main tool for the approximation of funtions of boundary layer type are the
x dened in this subsetion.
operators ixp and Pp;x
On the referene element ( 1; 1), let ip be the Gauss-Lobatto interpolation operator.
For an arbitrary mesh T on the domain = (0; 1), we denote by ixp : C ([0; 1℄) !
S p (T ) the pieewise Gauss-Lobatto interpolation operator. The operator ixp has the
following well-known stability property:
Lemma A.1 There is C > 0 independent of p and the mesh T suh that
kixpukL ((0;1)) C (1 + ln p)kukL ((0;1)) 8u 2 C ([0; 1℄):
1
1
Proof : On the referene element ( 1; 1), this stability property of the operator ip
was proved by Sundermann28 . The ase of a general mesh follows easily.
2
We next onsider the approximation properties of the Gauss-Lobatto interpolation
operator on a single element:
Lemma A.2 Let I = ( 1; 1) and denote by ip : C (I ) ! Pp (I ) for p 2 N the
usual polynomial interpolation operator in the (p + 1) Gauss-Lobatto points. Let u
be analyti on I and satisfy there for some Cu , u > 0, and h 2 (0; 1℄
kxnukL (I ) Cu hn un n! 8n 2 N 0 :
Then there exist C , , > 0 depending only on u suh that
1
kxn (u ip u)kL
1
h
n
(I ) CCu n! h + p+1
8p 2 N ; n 2 N 0 :
Proof : The details may be found in Melenk's thesis18 (see Se. 6.2.4). The key
observation is that, using arguments as in Se. 12 of the book by Davis2 , one gets
polynomial approximants Pp of degree p suh that on an ellipse E C with I E
(and > 1 depending only on u ) there holds
ku Pp kL
1
h
(E ) CCu h + p+1
:
Next, as ip u Pp is a polynomial of degree p, we may apply Bernstein's estimate
(see, e.g., Se. III.15 of the book by Markushevih16) to get for all > 1
kip u Pp kL (E ) p kip u Pp kL (I ) p kip(u Pp )kL (I )
C (1 + ln p)p ku Pp kL (I ) ;
where we appealed to Lemma A.1 in the last step. Choosing now suÆiently lose
to 1 (depending on ) and then 0 < appropriately, we arrive at
1
1
1
1
kipu Pp kL
1
(E )
h p+1
:
CCu
h + 0
The hp Streamline Diusion FEM 29
The nal result onerning the n-th derivative of u ipu now follows from Cauhy's
integral representation of derivatives.
2
x
Furthermore, the following orollary onerning stability of ip applied to analyti
funtions will be useful.
Corollary A.3 Let T be an arbitrary mesh on (0; 1) and let ixp : C ([0; 1℄) ! S p (T )
be the pieewise Gauss-Lobatto interpolation operator. Let u be analyti on (0; 1)
and satisfy there
kxn ukL (0;1) Cu un n! 8n 2 N 0 :
Then there exist C , > 0 depending only on u suh that on every element Ii =
(xi ; xi+1 ) of T there holds
kxn(ixp u)kL (Ii ) CCu n n! 8n 2 N 0 ; p 2 N :
Proof : As ixpu is a polynomial of degree p on Ii , it suÆes to hek the bound
for n p. Next, by the triangle inequality, it suÆes to show the bound for
kxn(u ixp u)kL (Ii ) . The desired result now follows easily from Lemma A.2 after
an aÆne hange of variables to the referene element ( 1; 1).
2
For our analysis of funtions of boundary layer type, it is moreover useful to introx . On an arbitrary mesh T on (0; 1), the operator P x
due a seond operator, Pp;x
p;x
is dened as follows with the aid of an auxiliary operator Qxp;x :
1. For an arbitrary mesh point x of T , the auxiliary operator Qxp;x : C ([0; 1℄) !
S p (T ) is dened by
8
x
<(ix
u(x) on (0; x)
u)(x)
p
x
Qxp;xu(x) := x x
:
u(1)
on (x; 1);
1 x
x : C ([0; 1℄) ! S p (T ) is
2. For an arbitrary mesh point x of T , the operator Pp;x
dened as
(
ixu(x)
if x 1=2;
x
Pp;x u(x) := px
Qp;xu(x) if x < 1=2:
1
1
1
As will beome apparent shortly, the ase x << 1=2 is the ase of pratial interest
x approximates the funtion u by a
for us; note that in this ase, the operator Pp;x
linear funtion on the interval (x; 1).
Robust exponential approximations of suh funtions an be ahieved with projex provided that the mesh point x is hosen suh that x = O(pl).
tors Pp;x
Lemma A.4 Let u satisfy (A.1). Then there are C , , 0 > 0 independent of l and
p suh that the following holds: Let T be an arbitrary mesh on (0; 1) and assume
that for some 2 (0; 0 ) the point x := min f1; plg is a mesh point of T . Then, if
x 1=2, there holds
1
x 0
p
x uk
8 elements Ii :
ku Pp;x
L (Ii ) + plk(u Pp;x u) kL (Ii ) Cle
hi
On the other hand, if x < 1=2, then
x u)0 k 2
x
1=2 p ;
plk(u Pp;x
L ((0;1)) + ku Pp;x ukL2 ((0;1)) Cl e
x u)0 k
x
p
plk(u Pp;x
for Ii (0; x);
L (Ii ) + ku Pp;x ukL (Ii ) Ce
0
x
x
=l
x
i
plk(u Pp;x u) kL (Ii ) + ku Pp;x ukL (Ii ) Ce
for Ii = (xi ; xi+1 ) (x; 1).
1
1
1
1
1
1
30 The hp Streamline Diusion FEM
Proof : The essential ingredients for the proof an be found in Melenk and Shwab20
(see Lemma 2.4). Note that the speial ase of a two-element mesh given by T =
f(0; pl); (pl; 1)g is overed by Lemma A.4.
2
Appendix A.2. Two-Dimensional Approximation Results
Due to Lemma 3.4, we an restrit our attention in the two-dimensional setting to
a referene onguration, and for the remainder of this setion, we will assume that
the funtion u is dened on the losed referene square S = [0; 1℄2 and satises there
the growth onditions (3.5). We onsider tensor produt meshes on S in order to
be able to exploit our one-dimensional results. To that end, we denote by iyp the
one-dimensional (pieewise) Gauss-Lobatto interpolation operator as dened above
but ating on the y-variable instead of the x-variable. We have:
Lemma A.5 Let u satisfy (3.5). Then there are C , , 0 > 0 independent of
p and l suh that following holds: Let Tx , Ty be arbitrary meshes on (0; 1) and let
T := Tx Ty be the tensor produt mesh on S . Assume thaty forxsome 2 x(0; 0y) the
point x := min f1; plg is a mesh point of Tx . Set p := (ip Æ Pp;x)u = (Pp;x Æ ip )u 2
S p (T ). Then for x 1=2 there holds on all elements Ix Iy 2 T
1
jIx j + jIy j ku p kL
1
(Ix Iy ) + plkr(u
p )kL (Ix Iy ) Cle p :
1
If, on the other hand, x < 1=2, then for elements Ix Iy 2 T we have
plkr(u p )kL2 (S) + ku p kL2(S)
plkr(u p )kL (Ix Iy ) + ku p kL (Ix Iy )
1
1
Cl1=2 e p ;
Ce p
for Ix (0; x) and
plkr(u p )kL (Ix Iy ) + ku p kL (Ix Iy ) Ce xi =l
1
1
for Ix = (xi ; xi+1 ) (x; 1). In the last two estimates, Iy may be an arbitrary
element of Ty .
Proof : We will only show the L1 -bounds and restrit
our attention to the ase
x as well as x , iy ommute
x < 1=2. The denition of p and the fat that y , Pp;x
p
imply
ku p kL
1
(Ix Iy )
ku iyp ukL (Ix Iy )
x (iy u)k
+k(iyp u) Pp;x
p L (Ix Iy ) ;
y
ky (u ip u)kL (Ix Iy ) +
x ( iy u)k
k(y iyp u) Pp;x
y p L (Ix Iy ) ;
1
1
ky (u p )kL
1
(Ix Iy )
1
1
as well as
kx (u p )kL
1
(Ix Iy )
k(x u) iyp (x u)kL (Ix Iy ) +
kx iyp u Ppx(iyp u) kL (Ix Iy ) :
1
1
We now use one-dimensional approximation results to ontrol the terms on the righthand sides. We start by observing that (3.5) and Corollary A.3 together imply the
The hp Streamline Diusion FEM 31
existene of C , > 0 independent of Ty and p suh that for all Iy 2 Ty , n, m 2 N 0 ,
and x 2 (0; 1)
jynxm (x u)(x; y)j Cl 1 n+m n! max fm; l 1gme d x=l 8y 2 Iy ; (A.2)
jyn xm (iyp u)(x; y)j C n+m n! max fm; l 1gm e d x=l 8y 2 Iy ; (A.3)
jyn xm (y iyp u)(x; y)j C n+m n! max fm; l 1gm e d x=l 8y 2 Iy : (A.4)
0
0
0
x (iy u), x (iy u) P x (iy u) ,
Thus, Lemma A.4 an be applied to ontrol (iyp u) Pp;x
p
p
p;x p
y
x
y
and (y ip u) Pp;x (y ip u) in the desired fashion; we note in partiular that the
estimates obtained in Lemma A.4 are uniform in y 2 (0; 1). It remains to bound
the dierenes u iyp u, y (u iyp u), (x u) iyp (x u). For these, we note that an
appliation of Lemma A.2 gives the existene of C , > 0 depending only on C 0 , 0 ,
d0 of (3.5) suh that
ky (u iyp u)kL
1
ku iyp ukL (Ix Iy ) Ce d xi =l e p ;
yj
1
y
1 d xi =l e p :
jI j k(xu) ip (x u)kL (Ix Iy ) Cl e
1
(Ix Iy ) + jI
0
1
0
1
y
The result now follows. We remark that the proof shows that in fat a sharper
result (in its dependene on l) holds for ky (u p )kL (Ix Iy ) than laimed in the
statement of the lemma.
2
Proof of Theorem 3.5 : Theorem 3.5 follows immediately from Lemma A.5. 2
Proof of Theorem 3.7 : The proof of Theorem 3.7 is based on Lemma A.5. By
onstrution, the mesh Tx onsists of element Ii = (xi ; xi 1 ), i = 1; : : : ; L + 1 where
xL+1 = 0, xi = qi , i = 0; : : : ; L. Furthermore, we write hi = jIi j = xi 1 xi .
Let 0 be given by Lemma A.5. For simpliity of exposition, we assume that L is
hosen suh that qL+1 0 l. Next, we assume 0 pl 1 sine in the onverse ase,
Lemma A.5 implies the desired result straightforwardly. Thus, we may now hoose
k 2 f0; : : : ; Lg suh that qk+1 0 pl qk . This implies that there is always a mesh
point in Tx of the form x = min f1; plg with 2 (0 q; 0 ℄. In order to simplify the
notation for the remainder of this proof, we assume that qk = 0 pl. The rst two
estimates of Theorem 3.7 therefore follow immediately. It remains to see the last
two estimates of Theorem 3.7. We write hj for the length of the j -th element Ij of
Ty . For the element Ii (0; qk ), i.e., i k + 1, we get from Lemma A.5
1
LX
+1 X h h
i j 2
l kr(u
LX
+1
hi p
e
l
l
i=k+1
i=k+1 j
C 0lpl e p Ce p :
Now, for the elements that are \far" from x = 0, i.e., i k, we have by the
assumption that the mesh is graded geometrially towards x = 0, that there holds
hi xi with onstants independent of i and l. Hene, from Lemma A.5,
p )k2L (Ii Ij )
1
C
0
k X
X
hi hj
i=1 j
l
l2 kr(u
p )k2
L1 (Ii Ij )
C
C
k
X
xi
i=1 l
k
X
i=1
e xi =l
e xi =l Ce p :
0
00
32 The hp Streamline Diusion FEM
This ompletes the proof of the third estimate. For the last one, we proeed similarly. We have
LX
+1 X
i=k+1 j
min fhi ; hj gkr(u p )k2L2 (Ii Ij )
LX
+1
i=k+1
LX
+1
C
Finally, again as hi xi for i k, we get
k X
X
i=1 j
C
k
X
i=1
k
pl
hi l 1e p C 0 e p
l
i=k+1
Ce
min fhi ; hj gkr(u p )k2L2 (Ii Ij ) hi kr(u p )k2L2 (S)
0 p :
k X
X
i=1 j
h2i hj kr(u p )k2L (Ii Ij )
1
X
h2i l 2e xi =l C e xi =l Ce p :
0
00
i=1
This ompletes the proof of Theorem 3.7.
2
Proof of Theorem 3.8 : The proof of Theorem 3.8 is very similar to that of
Theorem 3.7. The key obsertion is that in Theorem 3.7 the mesh Ty is arbitrary;
in partiular, it may be a geometri mesh.
2
The hp Streamline Diusion FEM 33
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