Max-Planck-Institut
fur Mathematik
in den Naturwissenschaften
Leipzig
Inverse inequalities on non-quasiuniform
meshes and application to the Mortar
element method
by
W. Dahmen, B. Faermann, I. G. Graham,
W. Hackbusch, and S. A. Sauter
Preprint no.:
24
2001
Inverse Inequalities on Non-Quasiuniform Meshes and Application to
the Mortar Element Method
W. Dahmen
B. Faermanny
I.G. Grahamz
W. Hackbuschx
S.A. Sauter{
Abstract
We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise
linear nite element functions u dened on locally rened shape-regular (but possibly non-quasiuniform)
meshes. These inequalities involve norms of the form kh ukW s p () for positive and negative s and , where
h is a function which reects the local mesh diameter in an appropriate way. The only global parameter
involved is N , the total number of degrees of freedom in the nite element space, and we avoid estimates
involving either the global maximum or minimum mesh diameter. Our inequalities include new variants
of inverse inequalities as well as trace and extension theorems. They can be used in several areas of nite
element analysis to extend results - previously known only for quasiuniform meshes - to the locally rened
case. Here we describe applications to: (i) the theory of nonlinear approximation and (ii) the stability of
the mortar element method for locally rened meshes.
AMS-Classication: 65N12, 65N30, 65N38, 65N55, 41A17, 46E35
Keywords: Inequalities, mesh-dependent norms, inverse estimates, nonlinear approximation theory, nonmatching grids, mortar element method, boundary element method
Acknowledgement This work was supported in part by British Council/DAAD ARC Grant 869 and in part
by the Max-Planck-Institut Mathematik in den Naturwissenschaften, Leipzig through a visiting position for
I.G. Graham. This support is gratefully acknowledged.
1 Introduction
Many classical a priori estimates for the error in a nite element approximation u to a solution v of an
operator equation take the form kv ; ukH t () C (hmax )s;t kvkH s () , for some s > t, where hmax is the
global maximum mesh diameter and H s () is the usual Sobolev space of index s on a domain . Since
modern applications produce meshes with signicant local variation of mesh size, many authors avoid the
introduction of hmax and use instead localised estimates, such as
kv ; ukH t () C
(X
2T
h2(s;t) kvk2 s
H()
)1=2
(1.1)
with denoting a typical element of the mesh T and h the diameter of . Other authors work with estimates
of the form
;(s;t)
(1.2)
h (v ; u) t C kvkHs () H ()
([email protected]), Institut fur Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55,
D-52062 Aachen, Germany
y ([email protected]), Mathematisches Seminar II, Christian-Albrechts-Universit
at Kiel, Olshausenstr. 40, D-24098 Kiel,
Germany
z ([email protected]) Dept. of Mathematical Sciences, University of Bath, Bath, BA2 7AY, U.K.
x ([email protected]), Max-Planck-Institut Mathematik in den Naturwissenschaften, D-04103 Leipzig, Inselstr. 22-26, Germany
{ ([email protected]), Institut f
ur Mathematik, Universitat Zurich, Winterthurerstr 190, CH-8057 Zurich, Switzerland
1
where h(x) is some positive \mesh-size" function, designed to reect the local mesh diameter near each point
x . Estimates such as these allow quite general mesh renement procedures including some produced by
adaptive algorithms (e.g. 21]).
Some nite element error analyses require, as an additional ingredient, certain fundamental inequalities
which have to be satised by the nite element functions. Examples are inverse estimates, which bound the
Sobolev norm of a nite element function in terms of its Sobolev norm of lower index, multiplied by a meshdependent constant. These are used in many classical analyses, for example uniform norm error estimates for
nite elements (8]) and stiness matrix conditioning analysis in boundary elements (e.g. 24]). More recently,
inverse estimates have appeared also in the analysis of the mortar element method for PDEs, where negative
index spaces on lower dimensional manifolds appear naturally (4]).
Classical inverse estimates (like classical a priori error estimates) are usually global: The mesh-dependent
constant in the bound grows as an inverse power of hmin (the global minimum mesh diameter). If the mesh
is strongly locally rened this bound is large and may not be useful. Thus analyses which make use of inverse
estimates often make the additional mesh assumption of global quasiuniformity (i.e. hmin hmax), and so
most interesting adaptive procedures are then ruled out.
In this paper we prove a range of inequalities, including localised inverse estimates which apply to both
piecewise constant and continuous piecewise linear functions dened with respect to meshes of simplices on
domains in Rd . The meshes are assumed shape-regular but need not be quasi-uniform. Our results include
estimates of the form
kh ukW s p() C kh;s ukLp() for a range of nonnegative s
(1.3)
khs+ ukLq () C khukW ;s q () for all nonnegative s (1.4)
and
where 2 R and h is the mesh-size function mentioned above. If the mesh is quasiuniform then h is bounded
above and below in terms of either hmin or hmax and these bounds reduce to standard inverse estimates. In
the locally rened case they represent localised inverse inequalities which may be considerably less pessimistic
than the classical ones.
While some particular localised inverse estimates have been developed in the literature in connection with
special applications (e.g. 1], proved special cases of (1.3) and 18] proved special cases of (1.4)), we know
of no source in the literature for the general inequalities presented here. In fact (1.3) and (1.4) are only
examples of a range of inequalities which we prove. Some of these are elementary and others depend on a
rather more delicate analysis. A recurring elementary tool throughout the analysis is the use of mesh-size
dependent discrete `p norms dened on the degrees of freedom of the nite element functions. Preliminary
manipulations using such `2 norms can be found in 18].
A more sophisticated tool which we need is the scale of Besov spaces Bqs (Lp ()) of smoothness s and
primary index p, in which are embedded the Sobolev spaces W sp (). In fact we prove (1.3) by obtaining its
more general analogue in the Besov scale, which allows even p < 1 and a range of s up to the regularity limit
of u. On the other hand, (1.4) is proved using a separate and non-standard duality argument and holds for
all negative s and all 1 < q 1.
A range of elementary inequalities (involving relations between dierent Lp and `p norms) are given in
xx2, 3, while the inverse estimates (1.3), (1.4) are proved in x4. In x5 we give localised versions of trace and
extension theorems for nite element functions using the same Lp norm on the ambient space and on the
lower-dimension manifold.
The rest of the paper is devoted to applications of these inequalities. As mentioned above, one application
(in particular of (1.4)) is in the analysis of boundary element methods for operators on negative order spaces.
Since this is described in some detail in 18, 19], here we restrict ourselves instead to describing two other
applications.
Our rst application is to the Jackson and Bernstein inequalities which arise in the theory of adaptive nite
element approximation. Recall that in a quasiuniform mesh we have h N ;1=d, where N is the number of
degrees of freedom in the nite element space. The estimate (1.1) then reads (e.g. in the case t = 0):
kv ; ukL2 () CN ;s=d kvkH s () (1.5)
When v fails to be in H s (), but still possesses enough Besov regularity, then it is known (e.g. 13]) that
adaptive processes exist which ensure that the rate of convergence N ;s=d of best nite element approximation
2
to v is still attained, but with kvkH s () on the right-hand side of the error estimate replaced by kvkBqs(L2 ()) .
This variant of the classical \Jackson-type" approximation estimate is given as part of the Appendix (in
fact for primary index 2 replaced by general p). The Jackson inequality is classically accompanied by the
\Bernstein-type" inverse inequality:
kukBqs(Lp()) CN s=d kukLp() for nite element functions u :
This is proved as the rst application of our inverse inequalities in x6.
Our second application (x7) concerns the stability of the mortar element method in the case of nonquasiuniform meshes on the subdomains. The mortar method seems to be particularly well suited for problems
with strong jumps in coecients. Since one therefore expects to deal with possibly irregular solutions, the
use of nonuniform meshes appears to be very desirable. The so called dual basis mortar method has indeed
recently been shown in 22] to lead to stable and accurate discretisations for the much more exible class of
shape regular meshes provided that a certain weak matching condition on adjacent meshes hold along interface
boundaries. Our objective here is to establish stability for this version of the mortar method without requiring
this matching condition. Aside from the inverse estimates from x3 we also make essential use of the extension
theorem in x5.
2 Preliminaries
2.1 Meshes and nite elements
Throughout this paper denotes either an open polyhedron or (a connected subset of) the surface of a
polyhedron. In both cases, the dimension/surface dimension of is denoted by d 2 N . Although most of our
results extend to general dimension d, we give the proofs for the cases d 2 f1 2 3g.
A mesh T on is a decomposition
= f : 2 T g where the elements are either intervals, triangles or tetrahedra. The elements have nodes and edges and
also (when they are tetrahedra) they have faces. For convenience we will always consider the elements to be
closed sets. We assume throughout that our meshes are conforming, i.e. if 2 T , with 6= , then \ is either a vertex, an edge or a face. We identify two sets of points in which are useful as index sets.
Let N0 denote the set of centroids of elements of T . We identify cj 2 N0 with its counting index j and we
write j 2 N0 as well as cj 2 N0 . The sets T and N0 are in one-one correspondence and for j 2 N0 , we denote
the element of T which contains cj by j .
Also we let N1 denote the set of nodal points of T . Similarly, we
R write i 2 N1 as well as xi 2 N1.
Each element has a diameter denoted h and a volume j j = dx.
We are concerned with inequalities for piecewise polynomial functions on T in the two most important
cases:
S0 (T ) = v : ! R : vjint( ) is constant, 2 T
(2.1)
S1 (T ) = fu : ! R : u is continuous and uj is ane 2 T g :
(2.2)
For each j 2 N0 , let j 2 S0 (T ) denote the characteristic function of j and for each i 2 N1 , we dene
i 2 S1 (T ) to be the \hat" function with values i (xj ) = ij , for i j 2 N1 . Each u 2 Sk (T ) then has the
familiar expansion:
u=
X
j 2N0
X
uj j with uj = u(cj ) u 2 S0 (T ) ui i with ui = u(xi ) u 2 S1 (T ) :
i2N1
If is a d;dimensional surface in Rd+1 , the surface derivatives of a suciently smooth function
u=
(2.3)
(2.4)
u:!R
on (plane) surface elements 2 T are dened by introducing local (d + 1)-dimensional Cartesian coordinates
so that the rst d coordinates lie in the tangential plane through . Let denote the mapping from local to
global coordinates and put u^ := u . Then, for any 2 Nd0 , we dene
@ uj := (@ u^ ) ;1 : ! R
3
Similarly, we put
ru := (ru^ ) ;1 : ! Rd :
Note that, by using this denition, Leibniz' rule for dierentation of products of functions holds as usual. At
various places in the text, we consider polynomials on elements 2 T . In the case of surfaces this notation
has always to be understood in the sense that the function, in local coordinates is a polynomial.
2.2 Mesh regularity
Denition 2.1 Two vertices xi xj are called neighbouring if there is an element 2 T such that xi xj 2 (i.e. xi and xj are connected by an edge of the mesh). Two elements 2 T are called neighbouring if
\ =
6 :
We shall consider classes of meshes which satisfy the following weak regularity assumptions.
Denition 2.2 For K 1 and " > 0, MK" denotes the class of meshes T which satisfy
h Kh for all neighbouring elements 2 T
(2.5)
d
and j j "h for all 2 T :
(2.6)
When d 2, it may be shown that for the conforming meshes considered here, the \shape regularity"
assumption (2.6) implies the \local quasiuniformity" or \K-mesh" condition (2.5), and so in this case the
meshes could be characterised by the single parameter ". However in other situations it is of interest to
consider locally quasiuniform meshes which are not shape regular, and so we choose to keep the parameters
K and " separate in our analysis.
Notation 2.3 Throughout the paper, if A(T ) and B(T ) are two mesh-dependent properties, then the inequality
A ( T ) . B (T )
will mean that there is a constant C depending on K and " such that
A(T ) CB (T ) for all T 2 MK" :
Also the notation
A(T ) B (T )
will mean that A(T ) . B (T ) and B (T ) . A(T ).
In some situations we will be considering other parameters as well. If the constants in the estimates are
independent of another parameter, say 2 ], we shall write explicitly, \A(T ) . B (T ) uniformly in 2
]" or \A(T ) B (T ) uniformly in 2 ]", as appropriate. This means that the equivalence constants
may depend on and but not on 2 ].
For our later estimates we need to introduce a mesh dependent function h on , such that the value of h
on any 2 T is proportional to the size of elements of T near . To this end, for i 2 N1 , we introduce the
subsets of T :
T (xi ) := f 2 T : has a corner at xi g :
(2.7)
Then we set
hi := maxfh : 2 T (xi )g and we dene h 2 S1 (T ) to be the interpolant of these values, i.e.
h=
X
i2N1
4
hi i :
(2.8)
Our aim in this paper is to create methods of analysis which are relevant to locally rened meshes and to
exploit them in several applications. To this end, information about the mesh being used will be contained
in the function h dened above and this plays a key role in most of our estimates. In some situations we also
need to include a global mesh parameter. For this we avoid using the maximum or minimum mesh diameters,
given by hmax = maxfh : 2 T g and hmin = minfh : 2 T g, but choose instead to use the cardinality
of the mesh N := #N1 . Note that asymptotically it is unimportant whether N is dened as the number of
nodes or elements since it follows from the conformity of the meshes that
#N0 #N1 :
(2.9)
Since adaptive techniques try to construct a good approximation for a minimal N , estimates involving N are
more natural in the context of adaptivity than those involving hmin or hmax .
Remark 2.4 Since min
h(x) = imin
h > 0 any power hs is well-dened, for s 2 R.
x2
2N i
1
Throughout the paper we will frequently use the estimates in the following proposition without further
reference.
Proposition 2.5 Let T 2 MK" . Then
(a) h hi Kh for all 2 T (xi ) i 2 N1
(b) K ;1 hi0 hi Khi0 , for all pairs of neighbouring vertices xi and xi0 2 N1 .
(c) For all j 2 N0 ,
h j i2min
h h(cj ) i2max
h Kh j :
\N i
\N i
j
(d) For all j 2 N0 ,
j
1
1
hdj jj j hdj :
(e) For any two points x y 2 , let j (x y)j denote the length of the minimal path in connecting x and y
and C denote the minimal constant such that, for all x y 2 , j (x y)j C jx ; yj. The function h is
Lipschitz continuous with Lipschitz constant satisfying
L C (K ; 1) =" uniformly in K 1
where C depends only on C and d.
Proof. The proofs of (a) - (d) are trivial. To obtain (e), rst observe that by the denition of h we have, for
any j 2 N0 and any x 2 j ,
rh(x) = rh(cj ) = r(h ; h(cj ))(cj ) =
X
i2 j \N1
(hi ; h(cj ))ri (cj )
Now using the shape regularity property (2.6), it follows that jri (cj )j C (h j );1 where C only depends
on d. Also, for i 2 j \ N1 , we have
(1 ; K )h j i2min
h ; max h hi ; h(cj ) i2max
h ; min h (K ; 1)h j \N i i2 \N i
\N i i2 \N i
j
1
and thus
j
j
1
1
j
1
krhkL1() C (K ; 1)=
uniformly in K .
Now take any x y 2 and recall that is connected. Let denote a shortest path in connecting x and
y. Since the element in Tm are simplices the restriction of to any is either empty or a straight line. Choose
a minimal sequence (i )i=1 in T such that i \ =: Ai Ai+1 for 1 i m ; 1 and A1 = x, Am = y. Then,
jh (x) ; h (y)j mX
;1
i=1
jh (Ai ) ; h (Ai+1 )j mX
;1
i=1
krhkL1( i ) jAi ; Ai+1 j krhkL1() jj
C krhkL1() jx ; yj CC K "; 1 jx ; yj :
5
2.3 The Sobolev norms
For 1 p 1, we introduce the usual space Lp() with norm kkLp() . Extending this denition to 0 < p < 1
we obtain a quasi-norm which satises the modied triangle inequality ku + vkLp () C kukLp() + kvkLp() ]
with C := 21=p;1 : For further details on Lp spaces with p < 1, see 13, 14] and the references therein. The
case p < 1 will become important in x6.
We introduce the usual Sobolev spaces W sp () and we note that for s 2 (0 1), these are equivalently
dened using the Slobodeckij norm :
8
Z Z jv(x) ; v(y)jp 9
< p
=1=p
kvkW s p () := :kvkLp () +
jx ; yjd+ps dxdy
(see, e.g., 20, Section 6.2.4]). In the special case p = 2 we write kkH s () := kkW s 2 () . For negative ;s,
W ;sq () is the dual space W ;sq () := (W sp ())0 with p1 + 1q = 1 p < 1 endowed with the dual norm.
The norms can also be used when is a d-dimensional manifold in Rd+1 , d = 1 2 (see, e.g., 18], 19]).
2.4 The ` norms
p
Let v = (vi )i2I 2 RI be a vector with I denoting its index set. Then, as usual, we write
kvk`p(I ) =
X
jv jp
i2I i
1=p
for p 2 (0 1) and kvk`1(I ) = maxfjvi j : i 2 Ig:
If w = (wi )i2I , then we dene the `2 inner product and the pointwise product, respectively, by:
(v w)`2 (I ) =
X
i2I
vi wi vw = (vi wi )i2I :
If f is any function on , we introduce the discrete norm of f on N0 and on N1 dened by
8
91=p
(X
)1=p
<X
=
kf k`p(N0 ) = : jf (cj )jp kf k`p(N1 ) =
jf (xi )jp
j 2N0
i2N1
when these quantities are well-dened.
3 Estimates in `p and Lp norms
3.1 Relations between discrete and continuous norms
Proposition 3.1 Let p0 > 0 and < be given. Then, for i = 0 or 1,
kh ukLp() kh+d=p uk`p(Ni ) uniformly in u 2 Si (T ) p 2 p0 1] and 2 ] :
(3.1)
Proof. Throughout this proof the relations . and will hold uniformly in p 2 p0 1] and 2 ]. First
consider p0 p < 1, observe that for any f 2 Lp (), we can write
8
Z 9
<
=1=p
X
p
p
kh f kLp() : h j jf j :
j
j 2N0
Thus if we consider any u 2 S0 (T ), we have,
91=p
91=p 8
8
=
=
<
<
X
X
kh ukLp() : hpj jj j ju(cj )jp : h(cj )p+d ju(cj )jp = kh+d=p uk`p(N0 ) :
j 2N0
j 2N0
6
(3.2)
This estimate is uniform in u 2 S0 (T ), p 2 p0 1) and 2 ].
On the other hand, suppose u 2 S1 (T ). Since, for all j 2 N0 , uj j is a polynomial of degree 1, it follows
by a simple scaling argument that
(Z
)1=p
j
ju(x)jp dx
8
91=p
<
=
X
jj j1=p :
ju(xi )jp uniformly in j p 2 p0 1) :
i2 j \N1
(Note that the usual scaling argument is still valid even for p < 1, since the function u 7! kukLp() , although
not a norm, is still a continuous function of u .) Hence, inserting this in (3.2), we get, uniformly in u 2 S1 (T ),
8
91=p (
)1=p
<
=
X
X
X p+d
p
p
p
kh ukLp() : h j jj j
ju(xi )j hi ju(xi )j
= kh+d=p uk`p(N1 ) j 2N0
i2 j \N1
i2N1
as required. (In the second last step we have used the fact that the mesh is conforming and shape regular and
so the number of elements attached to any given node is bounded over all meshes in the class MK" .)
The remaining case of p = 1 follows by similar arguments.
The following corollary identies two simple special cases of Proposition 3.1.
Corollary 3.2 There holds
X
j 2N0
h(cj )d ;d=p
h Lp() N 1=p X
i2N1
hdi 1
uniformly in p 2 p0 1) :
3.2 Estimates between dierent ` norms
p
First we recall some inequalities satised by `p norms.
Proposition 3.3 a) Let 0 < p p0 1: Then, for any index set I ,
kuk`p0 (I ) kuk`p (I ) :
b) Let 0 < 1 and p . Then
kuk`p(I ) kuk`(I ) kuk1`;(I )
with =
(3.3)
;p = 1 ; p; if
p ;
p ;
=p
< 1
if = 1:
(3.4)
Proof. To prove (3.4), we use Holder's inequality to obtain (for nite ):
p;
;p
;p
p;
kukp`p(I ) = (juj ; juj ; )`2 (I ) kjuj ; k` ;;p (I ) kjuj ; k` p;; (I ) :
On the other hand, if = 1 we can write
1;=p
kuk`p(I ) kuk=p
` (I ) kuk`1 (I ) (3.5)
and together these two estimates prove (3.4).
It can be easily checked that kuk`1(I ) kuk` (I ) for any 0 < 1: Inserting this inequality into (3.5),
one obtains kuk`p (I ) kuk` (I ) and hence (3.3) follows.
The next two propositions contain inverses of inequality (3.3). These can only be obtained at a cost of
either an N -dependent
factor (Proposition 3.4) or a weighting by a negative power of h (Proposition 3.5).
0
The exponent ppp;0p appearing below should be understood as 1=p if p0 = 1:
Proposition 3.4 Let 0 < p p0 1: Then, if i = 0 or 1,
kuk`p(Ni ) N
p0 ;0p
pp kuk`p0 (N ) :
i
7
(3.6)
Proof. Recalling (2.9) we have
kukp`p(Ni ) = (1 jujp )`2 (Ni ) k1k p0p;0 p
`
(Ni )
kjujp k pp0
` (Ni )
=N
p0 ;0 p
p kukpp0
` (Ni ) :
Proposition 3.5 For i = 0 1,
kuk`p(Ni ) . kh;
d(p0 ;0 p)
pp uk`p0 (N ) i
uniformly in u 2 RNi and p0 p p0 1:
(3.7)
Proof. We give the proof for i = 0. The case i = 1 is analogous. Take any u 2 RN0 and dene u 2 S0 (T )
by requiring u(cj ) = uj j 2 N0 . Then, by Proposition 3.7 below, we have kukLp() . kukLp0 () : Using
Proposition 3.1 we obtain khd=puk`p (N0 ) . khd=p0 uk`p0 (N0 ) : Then (3.7) follows by replacing u by h;d=p u.
From this we have the immediate corollary:
Corollary 3.6
kh uk`p(Ni ) . kh;
d(p0 ;0 p)
pp uk`p0 (N )
i
uniformly in u 2 RNi 2 R and p0 p p0 1 :
3.3 Estimates between dierent L norms
p
The following result is obtained directly from Holder's inequality.
Proposition 3.7
kukLp() . kukLp0 () uniformly in p0 p p0 1 and u 2 Lp0 ():
(3.8)
In the following generalisation of Proposition 3.7, we balance powers of h inside the right-hand norm with
an appropriate power of the global parameter N outside.
Proposition 3.8 For i = 0 1, the estimate
kukLp() . N khd ukLp0 () (3.9)
0
holds uniformly in u 2 Si (T ) p0 p p0 1 and 0 ppp;0p :
Proof. We give the proof for i = 0. The case i = 1 is very similar.
(a) Let u 2 S0 (T ). Then using (3.6)
with i = 0 and with u replaced by hd=p u and then (3.1), we obtain
0 ;p
p
the required result in the case = pp0 :
0 00
0
(b) More generally, consider 0 < ppp;0p : Then we can choose p00 2 (p p0 ] such that = pp;00 pp0 : Then by
(3.8) and part (a), we have kukLp() . kukLp00 () N khd ukLp0 () , as required.
Finally we obtain an inverse to the inequality in Proposition 3.8. As in Proposition 3.5, we pay the penalty
of a negative power of h on the right-hand side.
Proposition 3.9 For i = 0 1 the estimates
kh ukLp0 () . h;
d(p0 ;0 p)
pp u
hold uniformly in u 2 Si (T ) 2 ] and p0 p p0 1 Proof. Combine (3.1) and (3.3).
8
Lp ()
4 Inverse estimates in Sobolev norms
In this section we prove two types of inverse estimate in Sobolev norms. The rst two subsections concern
upper bounds for the W sp norm of a function (s > 0) in Si (T ), i = 0 1 in terms of the Lp norm of an
appropriately weighted function. The range of s is naturally restricted by the regularity of the spaces Si (T ).
In the case of S1 (T ) our results are a generalisation of those already given in 18].
In the third subsection we obtain lower bounds for the W ;sq norm (for s > 0) of a function in Si (T )
in terms of the Lq norm of an appropriately weighted function. These are obtained by direct estimation
of negative norms. The range of negative ;s which can be reached is unlimited and again the argument
generalises that in 18], in which only the cases u 2 S1 (T ) and 0 s 1 were covered.
4.1 Inverse estimates for u 2 S1 (T ) in W () s 0
s p
.
Theorem 4.1 Suppose that 1 p 1 and that 0 s < 1 + 1=p. Then the estimate
kh ukW s p () . h;s uLp()
(4.1)
holds uniformly in u 2 S1 (T ), 2 ] and 1 p 1.
Proof. We give an elementary proof for range 0 s 1. The proof for the extended range of s requires the
introduction of Besov norms and follows from Theorem A.1. In fact Theorem A.1 even allows an extension
of (4.1) to the case p < 1, provided the Sobolev norm appearing on the left-hand side is replaced by an
appropriate Besov norm.
Throughout this proof the inequality . will be uniform in u 2 S1 (T ), 2 ] and 1 p 1.
Suppose s = 1 and 2 T . The product ruleyields r (hu) = h;1 urh+ h ru on . Since Proposition
h;1uLp( ). Moreover a simple scaling
2.5 (e) shows that j (rh)j j . 1, it follows that h;1 urhLp ( ) .
argument shows that kh rukLp ( ) . h;1 kh ukLp ( ) . h;1 uLp( ). Summing the pth powers of these
inequalities over all 2 T and taking the pth root, we obtain
kh ukW 1 p () . h;1 uLp () :
(4.2)
Interpolating this result with the trivial estimate kh ukLp() . kh ukLp() (for s = 0), we obtain (4.1)
for general s 2 0 1]. (Note that here we have used the fact that the norm interpolating kh;1ukLp() and
khukLp() is kh;s ukLp() (see Triebel 23, (1.15.2/4)]).
4.2 Inverse estimates for u 2 S0 (T ) in W (), s > 0
s p
Analogously to Theorem 4.1, we have the following estimate for piecewise constant functions.
Theorem 4.2 Suppose that p0 p 1 and 0 s < 1=p. Then
kh ukW s p() . kh;s ukLp() uniformly in u 2 S0 (T ), 2 ], and p0 p 1.
(4.3)
We give the proof for p = 2 only since it provides explicit constants based on the localisation of the
Slobodeckij norm given in the following lemma. Remarks on the proof in the case p 6= 2 are given at the end
of this subsection.
Lemma 4.3 Let Rd be a bounded domain and let T be any conforming mesh on . Any function
v 2 H s () s 2 (0 1) satises
i
X h ;2s 2
X Z Z jv(x) ; v(y)j2
2
dx dy (4.4)
kv k
c kvk
+
H s ()
2T
L2 ( )
S
0 2T
0 \ 6=
0
jx ; yjd+2s
where := dist( D ) and D := f 0 2 T : 0 \ = g . For a domain Rd , the constant c is
explicitly given by c = 1 + 4s for d = 1 and by c = 1 + 4s
for d 2 f2 3g provided that 1: The constant
c is more involved for a d-dimensional surface , but still independent of v T K " .
9
The proof of Lemma 4.3 can be found in 16] for d = 1 and in 17] for d = 2. The 3-dimensional case can
be shown analogously to the 2-dimensional one. Note that (4.4) holds for any conforming triangulation and
the requirement that T 2 MK" is not needed for Lemma 4.3.
Proof of Theorem 4.2 (restricted case). As mentioned above, we restrict to the case p = 2. In addition we
describe here only the case d = 2. The proofs for d = 1 and d = 3 are similar. So, let u 2 S0 (T ). Since we
are considering meshes from the class MK" , (4.4) implies that
kh uk2H s()
.
kh;s uk2L2() +
X Z Z j(h u)(x) ; (h u)(y)j2
dxdy :
j j0 2N
j0 \j 6=0
jx ; yj2+2s
j0
j
(4.5)
Observe that by elementary arguments j(h u) (y)j . hj ju (cj )j, for y 2 j . Using this in (4.5), we obtain
kh uk2H s () . kh;s uk2L2() +
+
with
J 0 :=
ZZ
X
j 2N0
ju (cj )j2 H j
X n
j j0 2N0 j6=j0
j0 \j 6=
o
h2j0 ju(cj0 )j2 + h2j ju(cj )j2 J j j0
jx ; yj;2;2s dx dy and H :=
0
Z Z jh (x) ; h (y)j
dxdy:
jx ; yj2+2s
Since the summand in (4.6) is symmetric with respect to j j 0 , we may write
kh uk2H s () . kh;s uk2L2() +
X
j 2N0
h2j ju(cj )j2
We begin with the estimate
H kr (h )k2L1()
ZZ
X
j0 2N0 j6=j0
j0 \j 6=
J j j0 +
X
j 2N0
ju (cj )j2 H j :
(4.6)
(4.7)
(4.8)
jx ; yj;2s dxdy:
R
Now, using polar coordinates with respect to y 2 , it follows that jx ; yj;2s dx . h2;2s . This, together
with kr (h )kL1( ) . h;1 yields H . h2(;s) j j and the last term in (4.8) satises
X
j 2N0
ju (cj )j2 H j . kh;s uk2L2() :
We nish the argument by showing that
J 0 . h;2s j j for all 0 2 T with \ 0 6= and 6= 0 :
(4.9)
If (4.9) holds then, since the number of triangles 0 with 0 \ 6= is bounded independently of h, (4.8)
implies that
kh uk2H s () . kh;s uk2L2() +
X
h2j;2s ju(cj )j2 jj j
j 2N0
;
s
2
. kh ukL2() + kh;s+1 uk2`2(N0 )
kh;s uk2L2() by Proposition 3.1 yielding the required result.
It remains to show (4.9). Let y be an interior point of : Then 0 R2 n Br (y) with r := dist(y @ ) > 0 .
Using polar coordinates with respect to y shows
Z
jx ; yj;2;2s dx 0
Z
R2nBr (y)
jx ; yj;2;2s dx = s r;2s = s dist(y @ );2s 10
p1
p
j
1
2
p
pj;1
p3 = p0
3
q
pj
p2
Figure 1: Subdivision of into j j = 1 2 3
hence
J0 .
Z
dist(y @ );2s dy 0 2 T with \ 0 6= and 6= 0 :
(4.10)
Let pj , j = 1 2 3, be the vertices of and let p be the centre of the largest circle inscribed inside . This
circle has radius which, by the non-degeneracy condition (2.6), satises h . Also, each of the sides of are tangent to this circle. For the further estimation of J 0 , we split as in Fig. 1 (left) into three triangles
j , j = 1 2 3, with vertices pj;1 pj and p . Then there is one and only one q 2 pj;1 pj ] (here pj;1 pj ]
is the line connecting pj;1 and pj ) satisfying jp ; qj = % and p ; q ? pj ; pj;1 (see Fig. 1 (right)). For
y 2 j , we have dist(y @ ) = fj (y) with the function fj (y) := dist(y pj;1 pj ]) , i.e., dist(y @ ) is constant
on the level lines parallel to pj;1 pj ] . Therefore, we introduce the following transformation of coordinates
T () := pj;1 + 1 (p ; q) + 2 (pj ; pj;1 )
for = (1 2 ) 2 0 1]2 :
Since p ; q ? pj ; pj;1 , we get for 2 0 1]2 that
fj (T ()) = dist(T () pj;1 pj ]) = 1 jp ; qj = 1 %
and
j det(T 0())j = j det(p ; q pj ; pj;1 )j = jp ; qj jpj ; pj;1 j = 2 jj j :
Since dist( @ ) = fj on j and j T (0 1]2) , we obtain
Z
j
dist(y @ );2s dy Z
T (
01]2 )
fj (y);2s dy =
= 2 jj j %;2s
Z1
0
Z
01]2
fj (T ());2s j det(T 0 ())j d
1;2s d1 = 1 ;2 2s %;2s jj j :
Hence, substituting this in (4.10), we obtain
J0 .
3 Z
X
j =1 j
dist(y @ );2s dy . %;2s j j
which, by the non-degeneracy assumption, yields the estimate (4.9).
Remark 4.4 The full assertion of Theorem 4.1 for the extended range s < 1 + 1=p and even for p < 1, as
well as the proof of Theorem 4.2 for p =
6 2 follows from a more general class of inverse inequalities for certain
Besov norms that will be formulated and proved later in Appendix A.1.
11
4.3 Inverse estimates in W ; , s > 0
s q
In this section we extend the estimates of xx4.1, 4.2 to negative norms. Our main result is Theorem 4.7, which,
for example, can be used to estimate the W ;sq norm of a nite element function u from below by the Lq
norm of hs u. Unlike the previous section, where p < 1 was allowed, the argument in this section is restricted
to 1 < q 1.
Our result in this section is a generalisation of one obtained in 18], where we considered only the case
q = 2, ;s 2 ;1 0] and u 2 S1 (T ). For our proof in 18] we used a duality argument, with test functions
chosen to be suitably weighted functions from S1 (T ). The range of negative ;s which can be reached using
this argument is restricted. As we shall see such restrictions are articial and we can obtain our inverse
estimates for ;s indenitely negative and for functions u 2 S0 (T ) as well as u 2 S1 (T ). The key to the
proof is to consider more general test functions in the duality argument. These are built from the \bubble
functions" introduced in the following preliminary lemma.
Lemma 4.5 Let t be any closed simplex in , not necessarily an element of the mesh T . Let ht denote its
diameter and jtj denote its volume. Then, for any s 0, there exists a function Pst on with the following
properties:
(a) Pst 0 on (b) supp Pst = t
(c) There exist constants C1 C2 such that, for all 1 p 1,
C1 jtj1=p kPst kLp(t) C2 jtj1=p !
(d) There exists a constant C3 such that, for all 1 p 1, 0 s0 s,
kPst kW s0 p (t) C3 kh;t s0 Pst kLp(t) :
The constants C1 C2 C3 are independent of p and C1 and C2 are also independent of t. Moreover C3 can be
chosen independent of t (but dependent on "), provided t is restricted to the set of all simplices satisfying the
shape-regularity assumption (2.6) for some " > 0 .
Remark 4.6 Note that putting p = 1 in (c) and using (a) implies
Z
C1 jtj Pst C2 jtj :
t
Proof of Lemma 4.5. The proof employs the Bernstein representation of polynomials. Let i0 i1 : : : id denote
the vertices of t and introduce the barycentric coordinates = (x t) = (0 1 : : : d ) dened by
x=
Xd
Xd
ij j =0 j
= 1:
j =0 j
It is clear that each j (x t) is a polynomial of total degree 1 in x. Also, since t is the convex hull of the
points i0 i1 : : : id, it follows that
j (x t) 0 x 2 t :
(4.11)
Now, for given s > 0, let r denote the smallest integer satisfying r s and dene the multi-index 2 N d+1
by i = r + 1, i = 0 : : : d. Then dene
(x t) x 2 t
Pst (x) =
d
x 2 R nt :
0
(Here we have used the usual multiindex notation, i.e. = 0 0 1 1 dd .)
Property (a) now follows from (4.11) and, since t is closed, property (b) is trivial. Properties (c) and (d)
follow from standard scaling arguments (see, e.g. 10, Thms. 3.1.2 and 3.2.6]).
Now we have the main theorem of this section.
12
Theorem 4.7 Let s 0 be given. Then
hs+u . khuk
W ;s q ()
Lq ()
(4.12)
holds uniformly for u 2 Si (T ) i = 0 1 , 1 < q 1 and 2 ] :
Proof. The result is clear for s = 0. So, let s > 0, let 1 < q 1 and let p denote the conjugate index of q.
We will consider meshes T 2 MK" and throughout the proof the constant involved in the relations . and may depend on d, s and , as well as on K and " (see Notation 2.3), but will always be independent of q
and u 2 Si (T ) i = 0 1.
For any 2 T , dene
= hs+ uL1( ) :
(4.13)
Then, simple geometric considerations show that it is always possible to construct a simplex t = t( ) with
the properties
(A) u does not change sign on t( ).
(B) There exists a constant 2 (0 1) depending only on d, s, , and " such that
;
min hs+ u (x) (4.14)
jt( )j j j :
(4.15)
x2t( )
and
Note that (4.15), together with T 2 MK" implies
hdt( ) jt( )j j j "hd "hdt( ):
(4.16)
Now consider rst q in the range q qmax < 1, so that p qmax =(qmax ; 1) > 1. Then dene coecients
b 2 R by
q=p
;hs+u (x)
b = sign(ujt( )) xmin
2t( )
(4.17)
and then set (with Pst( ) as dened in Lemma 4.5):
w=
X
2T
b hs Pst( ) :
We shall use w as test function to estimate the negative norm of h u, i.e. we write
kh uk
j(h u w)j :
W ;s q ()
kwkW s p ()
(4.18)
(4.19)
Now, to estimate the numerator in (4.19), we use Lemma 4.5(a), (4.17) and Remark 4.6 to obtain
Z
X ;hs+ u (x)q jt( )j s
+
h
u
b
P
&
min
st( ) 2T t( )
2T x2t( )
;
X
j(h u w)j = hs+ u h;s w = and combining this with (4.14) and (4.15), we obtain
j(h u w)j & q+1
X
2T
q j j & q+1 khs+ ukqLq () :
(4.20)
Turning to the denominator in (4.19), we rst establish that
kwkW s p () . kh;swkLp () :
13
(4.21)
To obtain this, consider rst integer s and, for 2 N d0 , let @ := @11 @22 : : : @dd . Then,
jwjpW s p () :=
X
2Nd0
jj=s
k@ wkpLp() =
X
2T
jb jp hs Pst( )pW s p (t( )) :
(4.22)
Now recalling Proposition 2.5(e), which shows (rh)j is constant and jrhj . 1, and applying the Leibniz
formula, it is easy to show that
hsP p
st( ) W s p (t( )) .
sup hjp Pst( )pW j p (t( )) j =01::: s
where the constant of equivalence
only depends on s and jrhj. By Lemma 4.5 and (4.16) this supremum
may be bounded by Pst( ) pLp(t( )) and substituting in (4.22) we obtain (4.21). Arguing as in the proof of
Theorem 4.1 we obtain (4.21) for all s 2 R0 by interpolation.
Furthermore, by denition of w and by Lemma 4.5(c), we have
h;sw ( X jb jp kP kp )1=p . ( X jb jp jt( )j)1=p
st( ) Lp (t( ))
Lp ()
2T
(X
)1=p 2T( X
=
2T
j xmin
(hs+ u)(x)jq jt( )j
2t( )
khs+ ukq=p
Lq ()
khs+ ukqLq (t( ))
2T
:
)1=p
(4.23)
Combining (4.20), (4.21) and (4.23) in (4.19), we obtain the required result (4.12) when q qmax < 1.
To complete the proof, we consider the case q = 1, p = 1. Let 0 2 T satisfy 0 = max 2T .
Then with the simplex t( 0 ) as dened above (and satisfying properties (A) and (B)), we dene w =
sign(ujt( 0 ) )hs Pst( 0 ) : Then, using analogous arguments to those above, we obtain
j(h u w)j = j(hs+ u h;sw)j x2min
j(hs+ u)(x)j
t( 0 )
Z
t( 0 )
Pst(
0)
& hs+ uL1 ()
Z
t( 0 )
Pst( 0 ) :
On the other hand, one can easily check that (4.21) remains true for p = 1, and so
kwkW s 1 ()
. kh;s wkL1 () =
Z
0
Pst(
0)
:
These last two estimates imply the result (4.12) for q = 1. The result for q 2 qmax 1] follows by interpolation.
Remark 4.8 We emphasise that the estimates (4.12) hold uniformly as q ! 1. However this does not prove
(4.12) for q = 1. Indeed, since L1 is not the dual of L1 , the proof given here fails when q = 1.
5 Traces and extensions
Suppose that is a (d + 1)-dimensional domain (d = 1 2) which is triangulated with a mesh T 2 MK" and
that ~ is a d-dimensional surface consisting of boundary faces of elements in T . Thus, the restriction of T to
~ K and "~ ". We can dene the function h 2 S1 (T )
~ denes a mesh T~ on ~ . Then T~ 2 MK
~ "~ with K
exactly as in (2.8). Analogously we can dene h~ 2 S1 (T~ ). The mesh regularity implies the estimate
h (x) . h~ (x) h (x) 8x 2 ~ :
Suppose that the index sets of the elements and nodes of the meshes T and T~ are denoted, respectively, by
Ni , N~i , i = 0 1.
14
5.1 Traces
Every function u 2 S1 (T ) has an obvious restriction u~ 2 S1 (T~ ), dened by requiring the nodal values of u~ to
coincide with those of u on the surface ~ , i.e. u~ is just the trace of u. For discontinuous functions u 2 S0 (T )
we can dene a restriction u~ by requiring that the values of u~ on each element of ~ 2 T~ should be the average
of the values of u over each 2 T with ~ . In each case, using Proposition 3.1 (twice) we obtain,
Lemma 5.1 For i = 0 1, the estimates
kh~ u~kLp(~ ) kh~ + dp u~ k`p(N~i ) . kh+ pd uk`p(Ni ) kh; p1 ukLp()
hold uniformly in u 2 Si (T ) p 2 p0 1] and 2 ].
(5.1)
Note also that the constants in (5.1) are not dependent on the d-dimensional measure of ~ :
5.2 Extensions
With the same notation as above, suppose that u~ 2 S1 (T~ ) is given. We can again dene the obvious extension
u 2 S1 (T ) by dening u to be zero at nodes of T nT~ . For u~ 2 S0 (T~ ), a suitable extension would be to dene
uj , for each 2 T , to be the average of the value(s) of u~ on the elements of T~ which intersect and zero if
there are no such intersections. Then, analogously to (5.1), we have
Lemma 5.2 For i = 0 1 the estimates
; 1 h p uLp() h+ pd u`p(Ni) he+ dp u~`p(Nei ) ehu~Lp(e)
hold uniformly in u 2 Si (T ), p 2 p0 1] and 2 ].
Notation 5.3 For future reference, we denote the extension operator from S1 (T~ ) to S1 (T ) described above
by #0 .
6 Mixed norm estimates
It is well-known that when mth order nite elements u are used to approximate the solution v (of some
dierential equation, say) then, on quasi-uniform meshes Th of mesh size h, we obtain estimates of the form
s
kv ; ukLp() <
(6.1)
h kvkW s p() for s m and v 2 W sp (). In the quasi-uniform case we have h N ;1=d (where N is the number of nodes
in the mesh) and so we have the \Jackson estimate"
;s=d
kv ; ukLp() <
N kvkW s p() (6.2)
s=d
kukW s p() <
N kukLp() (6.3)
which measures the accuracy in terms of the number of degrees of freedom used in the approximation. The
Jackson estimate often is accompanied by its companion \Bernstein estimate"
an inverse estimate holding for all u in the relevant nite element space.
If we replace the right-hand of the estimate (6.1) by a sum of local error bounds hs kvkW s p( ), then
quantitatively improved accuracy can be expected by choosing the mesh to equilibrate these local errors.
However, such a (theoretical) adaptive process still gives rise to meshes that stay asymptotically quasi-uniform
(with possibly large but uniformly bounded mesh size ratios) and therefore does not reect what one often
sees in practice when the O(N ;s=d ) convergence rate is attained even for quite badly-behaved functions
v 62 W sp ().
Thus, to see under which circumstances non-quasiuniform meshes oer a better asymptotic accuracy, one
has to abandon measuring error and smoothness in the same Lp metric. This is the point of view taken in the
eld of nonlinear or best N -term approximation : See 13] for a recent excellent survey of relevant concepts.
One of the key results shows that the estimate (6.2) remains true even when the degree of smoothness s is
15
measured not in the Sobolev norm, but in an appropriate (weaker) Besov norm, dened in Appendix A.1.
This explains observed convergence rates for adaptive methods, for example in nite element approximation
of elliptic problems with jumping coecients or on domains with re-entrant corners. Here, the solution does
not have W 2p () regularity, but it still has regularity of order 2 provided we measure regularity in the weaker
Besov scale, see e.g. 11].
The r^ole of the Besov spaces can be illustrated with reference to the Sobolev embedding theorem, which
states that if p 1, r 1 and s > 0 satisfy the equality
1 = s + 1:
(6.4)
r
d
p
then the spaces W sr () are embedded in Lp (). In fact, given the integrability index p and the smoothness
s, decreasing r so as to satisfy (6.4) yields the largest Sobolev space of smoothness s that is still embedded
in Lp (). This can be illustrated with reference to a (1=r s)-diagram, with the vertical axis corresponding
to smoothness s and the horizontal axis corresponding to integrability 1=r. Let p and d be xed we identify
W sr () with the corresponding point in the rst quadrant of the (1=r s) plane. The critical line represented
by (6.4) is then a line of slope d passing through (1=p 0) and (1=r s). In this diagram the spaces W sp ()
(for non-integer s) lie on the vertical line 1=r = 1=p. For xed s > 0, larger spaces which are still embedded
in Lp() can be found between this vertical line and the critical line when moving to the right, decreasing
r. In particular the spaces W sr with r s and p given by (6.4) slide up the critical line with increasing s.
They are much larger than W sp (with increasing discrepancy for growing s) and permit singularities which
are prohibited in W sp ().
Note also that as s gets bigger then along the critical line r ! 0 and so eventually we encounter r < 1.
This picture can be rened somewhat by invoking the announced concept of Besov spaces (see, for example
13, p.92]). In fact, W sr () agrees for s 62 N with the Besov space Brs (Lr ()). With each point in the (1=r s)
diagram one actually identies the class of Besov spaces Bqs (Lr ()), where r is the primary index and the
secondary index q 2 (0 1] is (essentially) arbitrary. For points (1=r s) strictly above the critical line, the
corresponding Besov spaces are compactly embedded in Lp () (irrespective of the secondary index q) and
moreover p and r (like s) are now required merely to be positive numbers (14]). For points on the critical
line the embedding is only continuous and q may be constrained depending on the value of p.
A more general Jackson estimate than (6.2) is obtained by replacing the norm in W sp () by the norm
in Bqs (Lr ()) where r p and s satisfy (6.4). Such estimates for one-dimensional problems can be found in
13, p.102]. In the present multidimensional case, following the arguments of 12], one can prove a Jackson
estimate of the following type.
Theorem 6.1 Suppose 1 p 1 and i 2 f0 1g. Let 0 < s i + 1 and
1 <s+1 :
r0 d p
Then, for any v 2 Bqs (Lr0 ()) where 0 < q 1 there exists a mesh T 2 MK" and a projector QT : Lp () !
Si (T ) such that
where N := #T .
kv ; QT vkLp() . N ;s=dkvkBqs(Lr0 ())
(6.5)
Some comments on the type of argument leading to the above result (for the more dicult of the two cases
i = 1) along with an idealized algorithm for generating appropriate meshes (for a model domain) are given in
Appendix A.2. The main reason for including this result here explicitly is to motivate the use of the inverse
estimates developed in the earlier sections of this paper to prove the companion Bernstein estimate for (6.5),
i.e. the analogy of (6.3). This is given in the following theorem.
Theorem 6.2 Let 1 p 1, and assume that p r s satisfy (6.4) and in addition that for i 2 f0 1g one has
0 s < i + 1=r. Then
uniformly in u 2 Si (T ), p and r.
s=d
kukBrs(Lr ()) <
N kukLp() 16
(6.6)
Proof. By Theorem A.1 and Proposition 3.1, one has
;s
;s+d=r uk`r (N ) :
kukBrs(Lr ()) <
i
kh ukLr () <
kh
Using this and restricting to i = 1, we write
kukBrs(Lr ()) .
X
i2N1
d
d d
jh;i s+ r ; p jr jhip ui jr
X ! ppr;r X
i2N1
1
i2N1
!1=r X
=
d
jh p u jp
i i
!1=p
i2N1
d
jh p u jr
(6.7)
!1=r
i i
s=d
<
N kukLp() where we have made use of (6.4), Holder's inequality and again Proposition 3.1. The proof for i = 0 follows
entirely analogous lines.
Remark 6.3 (i) The Besov norm on the left-hand side of (6.6) may be replaced by the Sobolev norm kukW s r(),
when r 1. The proof is analogous, but uses Theorems 4.1 and 4.2 instead of Theorem A.1.
(ii) In the proof we have made use of various results from Section 3 in the norms in `r and Lr . Here the
importance that these results hold for r < 1 becomes apparent.
Let us nally obtain a Bernstein companion to the \almost optimal" Jackson estimate (6.5). Choose p0 > p
such that r10 = ds + p10 . Then (6.6) combined with Proposition 3.9 provides
0
s=d
s=d ; d(p ;p)
s=d s0 ;s
kukBrs0 (Lr0 ()) <
N kukLp0 () <
N kh pp0 ukLp() = N kh ukLp() (6.8)
0
where r10 = sd + 1p .
7 Application to non-quasiuniform mortar elements
In this section we shall apply the inequalities derived above in the context of the mortar nite element method,
see, e.g., 2, 5, 6, 25]. The mortar method seems to be particularly well suited for problems with strong jumps in
coecients. Since one therefore expects to deal with possibly irregular solutions, the use of highly nonuniform
meshes appears to be very desirable. When dealing with quasiuniform meshes certain mesh dependent norms
provide a convenient basis for the stability and accuracy analysis. Thus our main objective here is to extend
the stability analysis for the mortar method to the much more exible class MK" introduced above where
appropriate mesh dependent norms involve now mesh functions. Specically, we will focus on the dual basis
mortar method 22, 26] which has been shown in 22] to yield stable and accurate discretisations also in
the three dimensional case provided that certain weak matching conditions along the boundary of interfaces
between adjacent subdomains hold. Here we employ concepts from the previous sections to establish stability
without any such matching conditions.
7.1 The continuous problem
Consider the second order elliptic boundary value problem
; div a(x) grad u(x) = f (x)
a(x)@u=@n (x) = g(x)
u = 0
in on ;N @ on ;D := @ n ;N (7.1)
where a(x) is a uniformly positive denite matrix with coecients in L1 (), the domain Rd is bounded,
;D is a subset of the boundary ; of (with positive measure relative to ;) and ;N := ; n ;D . H01D ()
denotes the closure in H 1 () = W 12 () of all C 1 -functions vanishing on ;D .
Suppose that is decomposed into non-overlapping subdomains k k = 1 : : : kmax , i.e.,
% =
k
max
k=1
% k k \ l = 17
for k 6= l:
(7.2)
For simplicity we will assume throughout the rest of the paper that the domain Rd and that the
subdomains k in (7.2) are polyhedral. If k and l share a common interface, we set ;% kl := % k \ % l . The
interior faces form the skeleton
S :=
kl
;kl :
;kl , ;N , and ;D will always be assumed to be the union of polyhedral subsets of the boundaries of the k .
The mortar method is based on a variational formulation of (7.1) with respect to the product space
X := fv 2 L2() : vjk 2 H 1 (k ) k = 1 : : : kmax vj;D = 0g
endowed with the norm
kvk1 :=
Xkmax
k=1
kvk2H 1 (k )
1=2
:
The space H01D () is characterised as a subspace of X determined by appropriate constraints on jumps
across interfaces.
This suggests the following weak formulation of (7.1): For a suitable pair of spaces X M nd (u ) 2
X M such that
a(u v) + b(v ) = (f v)0 + (g v)0;N for all v 2 X
(7.3)
b(u )
= 0
for all 2 M
where (u v)0 and (g v)0;N denote the L2 inner products on and ;N , respectively and
PR
a(u v) := k k (a(x)ru(x)) rv(x)dx
P
b(v ) := ;kl S ( v])0;kl :
The jump v] of a function v 2 X is dened on S by
v] := vjk ; vjl
on ;kl
(see 5] for further background information). It is important to note that therefore each interface ;kl appears
only once 1 in the sum over S .
7.2 Discretisation
In order to describe next the mortar method as a discrete version of (7.3), we choose for each subdomain k a
family of (conforming) simplicial meshes Tk independently of the neighbouring subdomains. This means that
the nodes in Tk which belong to ;kl need not match with the nodes of Tl . Throughout the rest of this section
each family Tk will be assumed to lie in the class MK" , for some xed K and ". (Note that the mesh sizes in
the adjacent subdomains are not required to satisfy any compatibility condition). The corresponding spaces
of piecewise linear nite elements on Tk are denoted as before by S1 (Tk ). We set
Xh := X \
kY
max
k=1
S1 (Tk )
(7.4)
where the index h is used to signify the mesh dependence. The crucial step is to x the Lagrange multipliers
for each ;kl (i.e. the discrete analogue of the space M in (7.3)). The corresponding domain k is called the
non-mortar side, while l is the mortar side. It is important to stress here the following implicit notational
convention to be used throughout the rest of the section. The indexing of the interface ;kl (as opposed to
;lk ) always expresses that k has been chosen as the non-mortar side. We also emphasise that the choice of
the mortar side is actually arbitrary.
A common strategy is to choose the Lagrange multipliers also as continuous piecewise linear nite elements
to keep them as close as possible to the traces on the non-mortar side. Here we consider an interesting
1
Note that the indices
k l
in ;kl have a dierent meaning. The rst index will refer to the non-mortar side, as explained later.
18
alternative that has been recently proposed in 26] for the case d = 2 and in 22] for d = 3. In these papers
the Lagrange multipliers are allowed to be discontinuous in favour of an additional practically very desirable
feature, namely the fact that the Lagrange multiplier spaces are spanned by a basis which is dual to those of
the corresponding trace spaces on the non-mortar sides. Let us briey recall the construction from 22] for
d = 3 and refer to 26] for d = 2. Dene the space
Skl0 := S1 (T~kl ) \ H01 (;kl )
where T~kl denotes the restriction of the mesh Tk on the non-mortar side to ;kl . The corresponding multiplier
space Mkl is most conveniently dened with the aid of the following mapping Fkl . Let be any triangle in
T~kl and, for any v 2 Skl0 , let the values of v at the nodes xi of be denoted by vi . Then Fkl v = w is dened
as the unique piecewise linear function on T~kl whose restriction to is determined by its nodal values wi as
follows:
(i) wi := 3vi ; vr ; vs for all pairwise dierent vertices xi xr xs of when none of these vertices belongs
to @ ;kl !
(ii) If exactly one vertex, say xi lies on @ ;kl set wi := (vr + vs )=2, wr := (5vr ; 3vs )=2, ws := (5vs ; 3vr )=2!
(iii) If exactly two vertices xr xs belong to @ ;kl let wi = wr = ws := vi !
(iv) If all vertices of belong to @ ;kl set wi = wr = ws = vq where xq is a nearest interior node to .
Of course, since w = Fkl v is generally discontinuous the nodal values wi are to be understood as limit
values obtained when approaching the respective node from the interiour of the triangle under consideration.
Let Nkl denote the set of interiour nodes of T~kl and let i denote the standard piecewise linear basis for
Skl0 normalised by i (xr ) = ir for xr 2 Nkl . Dening
i := Fkl i xi 2 Nkl it is not hard to show that
(i j )0;kl = 0 i 6= j (i i )0 = j3j
holds for all 2 T~kl and xi 2 . Hence setting Mkl := span fi : xi 2 Nkl g, it follows that
dim Mkl = dim Skl0 :
(7.5)
(7.6)
The
P following further facts will be needed later. Given any 2 Mkl it has a unique representation
= xi 2Nkl i i and it follows from (7.5) that
kk2L2( ) hd;1
X
xj 2
2j 2 T~kl kk2L2(;kl ) X
xi 2Nkl
hdi ;1 2i = kh 2d ; 12 k2`2 (Nkl ) (7.7)
Clearly the second estimate follows from the rst one which has been established in 22, Eq. (4.2)]. Moreover,
as a consequence one has (cf. 22, Eq. (4.3)])
kvkL2 (;kl ) kFkl vkL2(;kl ) v 2 Skl0 :
The space of discrete multipliers is now dened as
Mh :=
Y
;kl S
Mkl :
(7.8)
(7.9)
where, again, the index h indicate mesh dependence.
Viewing the mortar method as a nonconforming discretisation, the above dual basis variant has been shown
in 22] to be stable for d = 3 even for shape regular locally quasiuniform meshes provided that the meshes on
adjacent subdomains match on the boundary @ ;kl of the respective interface, see condition (M.1) in 22]. This
19
assumption allows to establish the stability of the Mortar element method without employing mesh-depending
norms.
The objective of this section is to establish stability of the above dual basis mortar discretisation also for
any locally quasi-uniform meshes, i.e., for meshes in the class MK for arbitrary parameters K , but without
any additional constraints across the interfaces, thus retaining full mortar exibility.
Here we follow the lines in 5, 4, 25] and adopt the above formulation as a mixed method. Thus the central
issue is now to see under which circumstances
a(uh vh ) + b(vh h ) = (f vh )0 + (g vh )0;N vh 2 Xh (7.10)
b(uh h )
= 0
h 2 Mh
is a stable discretisation of (7.3). In other words, one has to show that the operator
A B| Lh := Bhh 0h : Xh Mh ! Xh0 Mh0
(7.11)
induced by (7.10) is uniformly bounded and has uniformly bounded inverses with respect to the underlying
meshes. Of course, this depends on the norms for Xh and Mh which have yet to be specied. At the rst
glance, kk1 seems to be a natural choice for Xh while Mh should be endowed with some sort of an H 1=2 norm
on the skeleton S . However, this turns out to be inappropriate and we refer for the details to 5].
7.3 Stability
Our ndings from the previous sections, in particular the inverse estimates allow us to handle mesh-dependent
norms that are suitable for the class of meshes MK" . To dene these norms one should note rst that the
mesh size function h is dened separately for each subdomain
k . More precisely, dene h(k) 2 S1 (Tk ) by
(
kl
)
(2.8) with T replaced by Tk . Also dene h 2 S1 T~kl in an analogous way. Since Tk belongs to the class
MK" the trace mesh T~kl belongs to the class MK~ "~ with K~ K~ and "~ ". The superscripts k kl in h will
be suppressed whenever the reference of h to an interface or subdomain is clear from the context.
Guided by 5, 4], we introduce the norms
kwk1=2h;kl := kh;1=2 wkL2 (;kl ) kwk;1=2h;kl := kh1=2 wkL2 (;kl ) where the mesh function h = h(kl) is induced by the non-mortar side. Moreover, for any vh 2 Xh dene
kvh k21h := kvh k21 +
X
;kl S
and nally, for 2 Mh ,
kk2;1=2h :=
kvh ]k21=2h;kl = kvh k21 +
X
;kl S
kk2;1=2h;kl =
X
;kl S
X
;kl S
kh;1=2vh ]k2L2(;kl ) kh1=2 k2L2(;kl ) :
(7.12)
(7.13)
Let us address rst the continuity of the bilinear forms a( ) b( ) with respect to these norms. Since
j(v )0;kl j = j(h;1=2 v h1=2 )0;kl j kvk1=2h;kl kk;1=2h;kl one has, in view of (7.12) and (7.13), for all u v 2 Xh and 2 Mh,
ja(u v)j . kuk1hkvk1h jb(v )j . kvk1hkk;1=2h:
(7.14)
The next step towards conrming stability of the discretisation is to conrm the ellipticity of the bilinear
form a( ) on
Vh := fv 2 Xh : b(v ) = 0 for 2 Mh g:
Proposition 7.1 The bilinear form a( ) is elliptic on Vh , i.e.,
a(v v) & kvk21h for all v 2 Vh :
20
(7.15)
(7.16)
Proof. The inequality
a(v v) & kvk21
for v 2 Vh
(7.17)
has been used frequently in the analysis of mortar elements and, in particular, in 22] to verify stability of
the nonconforming method. It follows from a compactness argument given in 2] as often found in proofs of
non-standard inequalities of Poincar&e-Friedrichs type. The argument covers a wide class of multiplier spaces
including the present version. So the desired ellipticity estimate (7.16) follows as soon as we have proved that
also
X
kv]k21=2h;kl . kvk21 for v 2 Vh :
(7.18)
;kl S
To this end, note that the quasi-interpolant
Qkl v :=
X
i2Nkl
v (1 )i
i 0;kl 0;kl
i (7.19)
takes L2 (;kl ) into Mkl and is uniformly bounded with respect to the L2 norm. The latter fact is a consequence
of (7.5) and (7.8).
Moreover, Qkl reproduces constants, i.e.,
Qkl (c) = c for all c 2 R:
(7.20)
To see this note that
X
!
X
X
X
Qkl (c) = c
1 (1 )i
i = c
i = c
Fkl i = cFkl
i = c
i 0;kl 0;kl
i2Nkl
i2Nkl
i2Nkl
i2Nkl
P
by denition of Fkl , because i2Nkl i takes the value one at each node i 2 Nkl . Furthermore, by denition
of Vh , one has Qkl (v]) = 0, for v 2 Vh . Thus, for v 2 Vh ,
kv]k1=2h;kl = k(id ; Qkl )v]k1=2h;kl = kh;1=2 (id ; Qkl )v]kL2 (;kl ) (7.21)
where id denotes the identity operator.
We now show that for any v 2 H 1=2 (;kl ) one has
kh;1=2(id ; Qkl )vkL2 (;kl ) . kvkH 1=2 (;kl ) :
(7.22)
To prove (7.22), consider any simplex in the triangulation of ;kl . Using (7.20), the denition of Qkl and the
fact that T 2 MK" , it is straightforward to show that
kh;1=2 (id ; Qkl )vk2L2 ( ) . h;1 cinf
kv ; ck2L2(^) . kvk2H 1=2 (^) 2R
S
where ^ := f 0 2 T~kl : 0 \ 6= g. (Here we have combined the standard estimate
inf c2R kv ; ckL2 (^) . h1=2 jvjH 1=2 (^) , valid since h diam^ , with an interpolation and density argument.) It
remains to observe that
P the ^ overlap only a nite number of times when runs through the triangulation of
;kl to conclude that kvk2H 1=2 (^) . kvk2H 1=2 (;kl ) (see 12]) which proves (7.22).
Now we combine (7.21) and (7.22) with the Trace Theorem to obtain
kv]k1=2h;kl . kvkH 1 (k ) + kvkH 1 (l ) for v 2 Vh (7.23)
which conrms (7.18) and proves the assertion.
It is well known that, once the continuity (7.14) and ellipticity (7.16) have been established, it remains to
verify the validity of the LBB-condition to ensure the stability of the discretisation (7.10), i.e., the uniform
bounded invertibility of the mappings Lh in (7.11), see, e.g. 9].
Theorem 7.2 Consider the induced meshes T~kl introduced in Section 7.2. Suppose they all belong to MK"
(with K " depending on ;kl ). Then there exists a constant > 0 depending only on the mesh parameters K "
so that the pairs of spaces Xh Mh dened above satisfy the LBB-condition
b(v )
:
(7.24)
inf sup
2Mh v2Xh kv k1h kk;1=2h
21
The core ingredient in the proof of Theorem 7.2 is the following result.
Lemma 7.3 For every 2 Mkl, there exists an element v 2 Skl0 such that
(v )0;kl c kvk21=2h;kl + kk2;1=2h;kl
(7.25)
for some constant c > 0 independent of v and .
P
P
Proof. Given = xi 2Nkl i i 2 Mkl dene v = xi 2Nkl vi i by
vi = hi i i 2 Nkl :
Then, by (7.5) and (7.26),
(v )0;kl =
X
(7.26)
h;i 1 vi2 (i i )0;kl kh 2d ;1 vk2`2 (Nkl )
(7.27)
xi 2Nkl
kh;1=2vk2L2 (;kl ) = kvk21=2h;kl where we have used Proposition 3.1, applied to the (d ; 1)-dimensional domain ;kl .
On the other hand, by (7.7),
kh d2 ;1 vk2`2 (Nkl ) = kh d2 k2`2 (Nkl ) X
2T~kl
h kk2L2( ) kh1=2 k2L2 (;kl ) = kk;1=2h;kl which together with (7.27) completes the proof.
We are now ready to complete the
Proof of Theorem 7.2. Given 2 Mh, let kl denote its component corresponding to ;kl S . We dene a
suitable v 2 Xh as follows. For each ;kl , let vkl 2 Skl0 be the function constructed in Lemma 7.3 (with v and
in (7.26) replaced by vkl and kl ) satisfying (7.25). Recall that by our notational convention k denotes the
non-mortar side of ;kl . On any k we dene a function vk 2 S1 (Tk ), rst at the nodal points x 2 @ k by
( v (x) if x 2 ; for some l 2 f1 : : : k g
kl
kl max
S max ; vk (x) := 0
if x 2 @ k n kl=1
kl
;
and then, on k , by #0 vk j@ k , where #0 is the extension operator as in Notation 5.3. The global function
v 2 Xh is now dened by vjk := vk , 1 k kmax. This function v satises v]j;kl = vkl and, in view of
(7.25),
b (v ) =
=
X
;kl S
X
;kl S
(v] )0;kl &
X
kvkl k21=2h;kl + kkl k2;1=2h;kl
;kl S
2
kvkl k1=2h;kl + kk2;1=2h:
To estimate the rst sum in the right-hand side above we use Theorem 4.1 and Lemma 5.2 to obtain
2
kvk2H 1 (k ) . h;1 v2L2 (k ) = h;1 #0 (vk )2L2 (k ) h;1+1=2 vk L2 (@ )
=
X ;1=2 2
X
kvkl k21=2h;kl :
h vkl L2(;kl) =
l:;kl @ k
l:;kl @ k
Combining this estimate with the denition of the kk1h -norm leads to
kvk21h =
Thus, we end up with
kX
max
k=1
kvk2H 1 (k ) +
X
;kl S
kv]k21=2h;kl .
X
;kl S
kvkl k21=2h;kl :
b (v ) & kvk21h + kk2;1=2h & kvk1h kk;1=2h:
22
k
We conclude this section with a few remarks on error estimates. The standard starting point is Strang's
second lemma, see e.g. 3], pp. 102-104, 5, Proof of Theorem 4.1], which says that
ku ; uhk1h c vhinf
ku ; vh k1h + sup
2Vh
vh 2Vh
R a @u v ]ds !
S @n h
kvh k1h
:
(7.28)
The rst term is referred to as the approximation error and the second one as the consistency error. Let us for
simplicity further assume that the solution u is in H 2 (k ) for each k. Since vh 2 Vh and hence is orthogonal
to Mh in the sense of (7.15) we may subtract an arbitrary element h 2 Mh from the conormal derivative of
u in the consistency error, to obtain
X
(a @u vh])0;kl X ka @u ; hk;1=2h;kl kvh]k1=2h;kl
;klS @n
;klS @n
s X @u
2
;kl S
ka @n ; h k;1=2h;kl kvh k1h :
(7.29)
For the last estimate, we have applied a Cauchy Schwarz' inequality and the denition of the kk1h -norm (cf.
(7.12)). The rst factor in (7.29) can be estimated by the same arguments as those used in the proof of (7.22)
by employing a@u=@n 2 H 1=2 (;), choosing h := Qkl (a@u=@n) and a trace inequality for a@u=@n at the end
@u 2
@u ; k2
@u
ka @n
. h% 2k kuk2H 2 (k ) : (7.30)
h ;1=2h;kl = kh1=2 (id ; Qkl ) a @n k2L2 (;kl ) h% 2k a @n H 1=2 (;kl )
Here, h% k denotes the maximal mesh size in Tk . Combining (7.29) and (7.30) results in the estimate of the
consistency term
kmax
!1=2
Z @u
a vh ]ds =kvhk1h . X h% 2k kuk2H2(k ) :
S @n
k=1
Furthermore, it is well known (cf. 3, Remark III, 4.6]) that, due to the validity of the LBB condition,
the approximation error on the right hand side of (7.28) can be bounded by the best approximation in the
unconstrained space Xh , i.e., the approximation vh can be chosen independently on each subdomain. Recall
that u 2 H 2 (k ) and dene vk := vh jk 2 S1 (Tk ) as the nodal interpolant of u with respect to the grid Tk .
Then, the estimate of the rst summand in the denition of the kk1h ;norm (cf. (7.12))
inf ku ; vh k1 C
vh 2Vh
kX
max
k=1
h% 2k kuk2H 2 (k )
!1=2
follows by well-known approximation results in Sobolev spaces.
It remains to discuss the second summands ku ; vh ]k1=2h;kl . Here, we have to assume a weak matching
condition for the mesh sizes of adjacent subdomains
h(lk) . h(kl) on ;kl :
(7.31)
Then employing (7.31), well-known approximation results, and the trace theorem at the end, one obtains
ku ; vh ]k1=2h;kl = k(h(kl) );1=2 u ; vh ]kL2 (;kl )
(kl) ;1=2
(lk) ;1=2
<
k(h ) (u ; vk )kL2(;kl ) + k(h ) (u ; vl )kL2(;kl )
(kl)
(lk)
<
h% kukH 3=2 (;kl ) + h% kukH 3=2 (;kl )
<
h% k kukH 2 (k ) + h% l kukH 2 (l ) :
Thus, in summary, one obtains an error estimate for the discrete solution uh of (7.10) of the familiar type
ku ; uh k21h <
kX
max
k=1
23
h% 2k kuk2H 2 (k ) (7.32)
where the constant, however, depends on the bound in (7.31).
Assuming lower Sobolev regularity one obtains analogous bounds with a correspondingly lower power of
h% k .
Remark 7.4 The matching condition (7.31) allows meshes with possibly very dierent mesh sizes by choosing
the Mortar sides in an appropriate way. The stronger condition
h(kl) (h(lk) );1 1 on ;kl
ensures convergence without any restrictions on the choice of the Mortar sides.
For highly non-uniform meshes, error estimates in terms of the number of degrees of freedom (cf. Theorem
6.1 and Appendix A.2) are preferable compared to estimates in terms of the maximal step size. Here, our
main focus was to prove the unconditional stability of the dual basis Mortar method for rather general
meshes (without the matching condition (7.31)) and to derive usual convergence results under additional weak
assumptions on the meshes.
A Appendix
A.1 Besov norms and proof of Theorems 4.1 and 4.2.
There are various equivalent versions of Besov norms. To introduce one which is suitable for the present
purposes, for y 2 Rd and k 2 N , let 'ky v denote the kth order forward dierence of v in the direction y and
let yk := fx 2 : x + ly 2 l 2 N 0 l kg. Then, for t 0, we dene the kth order Lp modulus of
smoothness of v by
!k (v t )p := sup k'ky vkLp(y k ) jyjt
with the usual interpretation for p = 1. Let s > 0. Then, for any k > s, the quantity
kvkqBqs(Lp()) := kvkqLp() +
1
X
j =0
2sqj !k (v 2;j )qp
(A.1)
denes a norm for the Besov space of smoothness s in Lp (). Norms for dierent k > s are equivalent. Here
p q only need to satisfy 0 < p q 1 where as usual for p q = 1 sums are replaced by sup, see e.g. 14]. Of
prime importance for us is the fact that when p = q 1 and s 62 N , one has the norm equivalence:
kvkW s p() kvkBps(Lp ()) :
(A.2)
When p < 1 the classical denition of Sobolev spaces has to be modied. In view of (A.2) it is natural to use
the expression (A.1) with p = q as a denition in this case. Therefore we will prove now the following more
general statement which covers the assertions of Theorem 4.1 as a special case.
Theorem A.1 Suppose that 0 < p0 p 1 and that 0 s < i + 1=p. Then the estimate
;s
khukBps(Lp ()) <
kh ukLp()
holds uniformly in u 2 Si (T ), 2 ], p0 p 1 and i 2 f0 1g.
(A.3)
Proof. Let k be the smallest integer greater than or equal to i + 1=p to obtain the required estimate for the
last term on the right-hand side of (A.1).
Note rst that, since we are considering meshes T 2 MK" , there exists a 2 (0 1) such that, for each 2 T
and each x 2 , the ball B (x kah ), centred on x with radius kah satises
B (x kah ) ^ := f 0 : 0 \ 6= g :
For each j 2 N 0 := N f0g, let us dene
Tj := f 2 T : 2;j < ah g j := f 2 T : 62 Tj g:
24
(A.4)
With these preliminaries we can write
1
X
j =0
2sjp !k (h u 2;j )pp +
1
X
j =0
1
X
j =0
2sjp sup;j k'ky (h u)kpLp(j \y k )
jyj2
X
2sjp
2Tj
sup;j k'ky (h u)kpLp( \y k )
jyj2
=: A1 + A2 :
First we estimate A1 . Note that, for each j , there exists y = y (j ) such that jy j 2;j and
k k X
k
sup k'y (h u)kLp(j \y k ) kh ukLp(j +ly(j)\)
l
jyj2;j
l=0
Now with the convention + z = fx + z : x 2 g, dene
(
Cj := 2 T : \
k
l=0
!
:
)
(j + ly(j )) \ 6= :
For each 2 T nTj , we have, by denition, ah 2;j . Moreover, since T 2 MK" , we know that the in Cj must satisfy bh 2;j , again with b depending only on the mesh regularity parameters. Hence with
:= maxfa;1 b;1 g we can write
A1 C p
1
X
j =0
X
2spj
2T
h 2;j
kh ukpLp( )
for some constant C depending only on k. If we now dene for each 2 T an integer j by requiring
2;j ;1 h 2;j then we have
A1 C p
j
XX
2T j=0
C p sp
X
2T
2spj khukpLp(
)
. Cp
X
2T
(A.5)
2spj kh ukpLp(
)
h;sp khukpLp( ) . C p sp kh;s ukpLp() the pth root of which is in the appropriate form, since s is bounded above and below. (Throughout the rest of
the appendix we use the convenient notation Ap . B p when we actually mean that A CB with a constant
C independent of p.)
To estimate the quantity A2 , consider y 2 Rd with jyj 2;j . Then, for 2 Tj we have, by denition,
jyj 2;j < ah . Consider the regions y := fx 2 : dist(x @ ) kjyjg and set 0 := n y so that
k'ky (h u)kpLp( ) = k'ky (h u)kpLp( 0 ) + k'ky (h u)kpLp(y ) :
(A.6)
On 0 the function h u is smooth and we can estimate kth order dierences by kth order derivatives of h u
on times kth powers of the step size jyj 2;j . Taking the linearity of h and u on 0 into account, recalling
the boundedness of rh and estimating the gradient of u (which is constant) by kukL1( )h;1 , one obtains
;kjp ;kp d p
k'ky (h u)kpLp( 0) <
2 h h kh ukL1( ):
Now note that
2;kjp h;kp hd = 2;jp(i+ p ) hd;ip;1 (2j h )1+ip;kp :
1
25
Our choice of k ensures that ip + 1 ; kp 0. Thus, making again use of the fact that for 2 Tj one has
h;1 a2j , the factor in parentheses can be estimated by apk;1;ip , whence we conclude
k'ky (h u)kpLp( 0 ) < 2;pj(i+1=p) hd;ip;1 kh ukpL1( ) i 2 f0 1g:
(A.7)
To estimate the second term on the right-hand side of (A.6) note the volume of y is of the order of
2;j hd;1 . When i = 0 we estimate the kth order dierence on y by a constant times absolute values which
yields
;j d;1 p
k'ky (h u)kpLp(y ) <
2 h kh ukL1(^) S
where ^ = f
y + tky : 0 t 1g.
(A.8)
When i = 1 the function h u is Lipschitz continuous on the set ^, with Lipschitz constant of order
l k,
all belong to ^, when x 2 . Hence, estimating this time the kth order dierence by a sum of rst order
dierences, we have the estimate
h;1 khukL1(^) (cf. proof of Lemma 2.5 (e)) and (by denition (A.4)), and the points x + ly, 0
k'ky (h u)kpLp(y ) . (2;j hd;1)h;p jyjp kh ukpL1(^) 2;j;jp hd;1;p khukpL1(^) (A.9)
Note that (A.8) and (A.9) can be combined as
;jp(i+ p1 ) hd;1;ip kh ukp 1 i 2 f0 1g u 2 Si (T )
(A.10)
k'ky (h u)kpLp(y ) <
L (^)
2
which is valid for all 2 Tj and which is exactly of the form (A.7). Combining (A.7) and (A.10) in turn
implies
A2 1 X
X
j =0 2Tj
2;pj(i+1=p;s) hd;1;ip khukp 1
L (^)
X
C p hd;1;ip
2T
1
X
j =j
2;pj(i+1=p;s) kh ukpL1(^) (A.11)
for some constant C , where j is dened by (A.5), but with replaced by a;1 .
The sum on the right-hand side of (A.11) is convergent whenever s < i + 1=p, allowing us to write
A2 . C p
X
2T
hd;1;ip 2;j p(i+1=p;s) khukpL1( ) C p
X
= C p hd;ps kh ukpL1(^)
2T
:
Recalling Proposition 3.9, we obtain
A2 . C p
X
2T
hd;ps kh;d=ph ukpLp(^) = C p
X
2T
hd;1;ip hp(i+1=p;s) kh ukpL1(^)
X
kh;s h ukpLp(^) <
C p kh;s ukpLp() 2T
where we have used that, again due to the K -mesh property the overlap of the domains ^ stays controlled.
This completes the proof.
A.2 Comments on Theorem 6.1
To see how the Besov scale enters the picture in connection with direct estimates we need rst good local
regularity-free error bounds for Lp approximation on K -meshes. Recall that now p 1. We will only treat
approximations from S1 (T ). The case S0 (T ) is analogous but simpler. To this end, let us denote by i i 2 N1
(the set of nodes in T ), the standard nite element hat functions normalized as before such that i (j ) = ij ,
i j 2 N1 . One can construct piecewise linear (discontinuous) polynomials i supported on supp i such that
(i j ) :=
Z
i j = ij ki kL1 () 1 i j 2 N1 (see 4] and the literature cited there. This requires shape-regularity). Now consider the projector
QT v :=
X
i2N1
26
(v i )i :
(A.12)
For every 2 T let
^ := fsupp i : supp i g:
Since QT is a projector one has QT P = P for every polynomial P of maximal degree one. Thus for every
2 T one has
p
p
p
kv ; QT vkpLp( ) <
kv ; P kLp( ) + kQT (v ; P )kLp( ) kv ; P kLp( )
0
1p
X
+@
kv ; P kLp(supp i ) ki kLp~(supp i ) ki kLp( )A i: supp i
where p1 + p1~ = 1. By the fact that T 2 MK" , it follows from (A.12) that ki kLp~(supp i ) ki kLp( ) 1. Hence
p
kv ; QT vkpLp( ) <
(A.13)
P :deginfP 1 kv ; P kLp(^) :
Thus the local error can be estimated by best linear polynomial approximation on a somewhat larger domain.
We wish to estimate the local polynomial approximation in a possibly weak smoothness scale and consider
integrability indices r0 satisfying now
1 s+1
(A.14)
r0
d p
and recall that for equality in (A.14) one0 is on the critical line. Recall from the discussion in section 6 that for
strict inequality it is known that Bqs (Lr ()) is compactly embedded in Lp(). For convenience let us denote
the discrepancy
:= s ; rd0 + dp :
(A.15)
As in (6.4) we will write r instead of r0 when equality holds in (A.14).
Let us assume now that v 2 Lp () (p 1) also belongs to the Besov space Bqs (Lr0 ()) where at this point
q is arbitrary when > 0 and q = r when = 0. From Lemma 3.10 in 12] we know that for (A.14) (under
the assumption of shape regularity of the triangles)
inf kv ; P kLp(^) <
(A.16)
h kvkBqs(Lr0 (^)) :
P :deg P 1
Moreover, when p <0 1 one can replace r0 by r from (6.4) in which case = 0. 0Clearly the smaller the
larger the space Brs0 (Lr ()). In particular, as mentioned earlier in section 6 Brs0 (Lr ()) is then signicantly
larger than W sp ().
One expects to obtain a good mesh by equilibrating these local error bounds. This can be done by a simple
adaptive renement scheme, see 12, 15]. We describe this for simplicity for the case that is the unit d-cube
which will be successively subdivided into dyadic subcubes 2. Each subcube is subdivided in a canonical way
into d! simplices so that the translates of all subcubes of xed dyadic level induce a conforming (uniform)
triangulation of . For each dyadic cube 2 let P (2) denote its parent and let U (2) be the union of 2 and
all its neighbours on the dyadic level of 2. Here neighbour of a cube means any other (closed) cube that
intersects that cube. Now (A.16) suggests to associate with any dyadic cube 2 the error functional
E (2) = E (v 2) := (diam 2) kvkBqs(Lr0 (U (2)))
(A.17)
where we will always assume q = r when r0 = r as in (6.4).
Now x a tolerance . We create a collection G = G () of `good' dyadic cubes as follows. Set B := fg,
G := . If for 2 in the current set of `bad' cubes B one has E (2) remove 2 from B and add it0 to G . If
E (2) > subdivide 2 into its 2d children and replace 2 by these children in B. For any v 2 Bqs (Lr ()) this
process terminates after nitely many steps. The resulting collection G is characterized by the fact that for
each 2 2 G one has E (2) while E (P (2)) > . Therefore the global error satises
p (v) :=
X
22G
E (2)p
27
!1=p
(#G )1=p :
(A.18)
Since for P := fP (2) : 2 2 Gg one still has #P N := #G <
#P , the fact that E (P (2)) > for any
2 2 G yields
0
r ;1
r0 =p
r0
p (v)r0 <
N E (P (2)) N p
X
P (2)2P
E (P (2))r0 :
(A.19)
Now suppose for a moment that r0 = r satises (6.4) (i.e. = 0 in (A.17)). Then the above estimate provides
0
11=r
0
11=r
X
X
p (v) <
E (P (2))r A = N ;s=d @
E (P (2))r A :
Np r @
P (2)2P
P (2)2P
1;1
(A.20)
Thus the question is how to bound the sum on the right-hand side of (A.20). To this end, let us further assume
for a moment that only a uniformly (in ) bounded nite number of the U (P (2)) dened above overlap. Then
one can
P show by the arguments
1=r in 12] that the sum of local Besov norms can be bounded by the global one,
r
<
i.e., P (2)2P E (P (2))
kvkBrs(Lr ()) so that
;s=d
p (v) <
N kvkBrs(Lr ()) (A.21)
;s=d
p (v) <
N kvkBqs(Lr0 ()) (A.22)
;s=d
kv ; QT vkLp() <
p (v) <
N kvkBqs(Lr0 ()) :
(A.23)
which would give an optimal convergence rate for the above algorithm under weakest possible regularity s.
The problem is that the required nite overlap property, which was needed to bound the sum of the
local Besov norms by a global one, will in general not hold. It is shown, however, in 15, 12] that slightly
more regularity does allow one to recover the rate N ;s=d. More precisely, when r0 satises (A.14) with strict
inequality so that > 0 in (A.17) with given by (A.15), the above renement scheme produces for the
corresponding version of the error functional E (2) a global error satisfying
with a constant depending on the discrepancy > 0, where 0 < q 1. This holds in the case of piecewise
linear approximations for the full range s 2 even when the corresponding r0 given by (A.14) is smaller than
one.
It is now easy to construct a K -mesh providing the above convergence rate. First one can show that since
U (2) smears the error the dyadic renements produced by the above scheme are automatically graded 12].
That means that neighbouring cubes dier at most by one generation. Therefore the still nonconforming
triangulation induced by the collection G can easily be closed to a conforming one T satisfying the K -mesh
property. Moreover the above estimates (A.13), (A.16) ensure that still
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