A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier.
The mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries. In the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of Lagrange multipliers. In this paper, this Lagrange multiplier space is replaced by a dual space without losing the optimality of the method. The advantage of this new approach is that the matching condition is much easier to realize. In particular, all the basis functions of the new method are supported in a few elements. The mortar map can be represented by a diagonal matrix; in the standard mortar method a linear system of equations must be solved. The problem is considered in a positive definite nonconforming variational as well as an equivalent saddle-point formulation.
Introduction
. Discretization methods based on domain decomposition techniques
are powerful tools for the numerical approximation of partial differential equa-
tions. The coupling of different discretization schemes or of nonmatching triangulations
along interior interfaces can be analyzed within the framework of the mortar
methods [6, 7]. In particular, for time dependent problems, diffusion coefficients with
jumps, problems with local anisotropies as well as corner singularities, these domain
decomposition techniques provide a more flexible approach than standard conforming
formulations. One main characteristic of such methods is that the condition of
pointwise continuity across the interfaces is replaced by a weaker one. In a standard
primal approach, an adequate weak continuity condition can be expressed by appropriate
orthogonality relations of the jumps of the traces across the interfaces of the
decomposition of the domain [6, 7]. If a saddle point formulation arising from a mixed
finite element discretization is used, the jumps of the normal components of the fluxes
are relevant [29]. To obtain optimal results, the consistency error should be at least of
the same order as the best approximation error. Most importantly, the quality of the
a priori error bounds depends strongly on the choice of weak continuity conditions at
the interfaces.
Section 2 contains a short overview of the mortar finite element method restricted
to the coupling of P 1 -Lagrangian finite elements and a geometrically conforming sub-division
of the given region. We briefly review the definition of the discrete Lagrange
multiplier space and the weak continuity condition imposed on the product space as
it is given in the literature. In Section 3, we introduce local dual basis functions,
which span the modified Lagrange multiplier space. We also give an explicit formula
of projection-like operators and establish stability estimates as well as approximation
properties. Section 4 is devoted to the proof of the optimality of the modified
nonconforming variational problem. It is shown that we can define a nodal basis
function satisfying the constraints at the interface and which at the same time has
local support. This is a great advantage of this modified method compared with the
Math. Institut, Universitat Augsburg, Universitatsstr. 14, D-86 159 Augsburg, Germany.
Email: wohlmuth@math.uni-augsburg.de, http://wwwhoppe.math.uni-augsburg.de/~wohlmuth
standard mortar methods. Central results such as uniform ellipticity, approximation
properties and consistency error are given in separate lemmas. A saddle point formu-
lation, which is equivalent to these nonconforming variational problems is considered
in Section 5. Here, the weak continuity condition at the interface enters explicitly
in the variational formulation. As in the standard mortar case, we obtain a priori
estimates for the discretization error for the Lagrange multiplier. Here, we analyze
the error in the dual norm of H 1=2
00 , as well as in a mesh dependent L 2 -norm. Finally,
in Section 6, numerical results indicate that the discretization errors are comparable
with the errors obtained when using the original mortar method.
2. Problem setting. We consider the following model problem
where\Omega is a bounded, polygonal domain in IR 2 , and that f 2 L
Furthermore,
we assume a 2 L 1
(\Omega\Gamma to be an uniformly positive function and 0  b 2 L
We will consider a non-overlapping decomposition
of\Omega into polyhedral subdo-
Each
subdomain\Omega k is associated with a family of shape regular simplicial triangulations
is the maximum of the diameters of the elements in
T hk . The sets of vertices and edges of the
subdomains\Omega k and
of\Omega are denoted by P hk ,
respectively. We use P 1 -conforming finite elements
individual subdomains and enforce the homogeneous Dirichlet boundary conditions
on
@\Omega k .
We restrict ourselves to the geometrical conforming situation where the intersection
between the boundary of any two different subdomains
l, is either
empty, a vertex, or a common edge. We only call it an interface in the latter case. The
mortar method is characterized by introducing Lagrange multiplier spaces given on
the interfaces. A suitable triangulation on the interface is necessary for the definition
of a discrete Lagrange multiplier space. Each interface
@\Omega k is associated with
a one dimensional triangulation, inherited either from T hk or from T h l . In general,
these triangulations do not coincide. The interface in question will be denoted by
its triangulation is given by that
and\Omega l , respectively. We call
the inherited one dimensional triangulation on \Gamma kl and \Gamma lk , \Sigma kl and \Sigma lk , respectively
with the elements of \Sigma kl and \Sigma lk being edges of T hk and T h l , respectively. We remark
that geometrically \Gamma lk and \Gamma kl are the same.
Thus, each
@\Omega k can be decomposed, without overlap, into
where
denotes the subset of f1; Kg such that
@\Omega k is an interface
for l 2
M(k): The union of all interfaces S can be decomposed uniquely in
A Mortar Method Using Dual Spaces 3
Here,
M(k) such that for each set fk; lg, 1  k  K, l 2
l 2 M(k) or k 2 M(l) but not both. The elements of f\Gamma kl j 1  k  K; l 2 M(k)g are
called the mortars and those of f\Gamma lk j 1  k  K; l 2 M(k)g the non-mortars. The
choice of mortars and non-mortars is arbitrary but fixed. We note that the discrete
Lagrange multiplier space will be associated with the non-mortars. To simplify the
analysis, we will assume that the coefficients a and b are constant in each subdomain,
with a k := a
It is well known that the unconstrained product space
Y
is not suitable as a discretization of (2.1). We also note that in case of non-matching
meshes at the interfaces, it is in general not possible to construct a global continuous
space with optimal approximation properties. It is shown [6, 7] that weak constraints
across the interface are sufficient to guarantee an approximation and consistency error
of order h if the weak solution u is smooth enough. The nonconforming formulation
of the mortar method is given by:
Find ~ u h 2 e
h such that
where a(v; w) :=
R\Omega
arv
1(\Omega
Here, the global space e
V h is defined by
e
where the bilinear form b(\Delta; \Delta) is given by the duality pairing on S
Y
1(\Omega
Y
Y
and [v] := v
j\Omega l
. Here, (H 1
denotes the dual space of H 1
Of crucial importance is the suitable choice of f
M h in (2.2)
f
Y
Y
f
where in general the local space f
chosen as a modified trace space of finite
element functions in S
can be found satisfying
is the trace space of S
f
subspace of W h (\Gamma lk ) of codimension 2 and given by
f
contains an endpoint of \Gamma lk g
and N lk := dim f
denotes the number of elements in \Sigma lk .
Here, we assume that N e  2. The nodal basis functions fOE i g N lk
of f
are
associated with the interior vertices given by
The space f
its nodal basis functions fOE i g N lk
are illustrated in Figure 2.1;
for a detailed analysis of f
Fig. 2.1. Lagrange multiplier space
Let us remark that continuity was imposed at the vertices of the decomposition
in the first papers about mortar methods. However, this condition can be removed
without loss of stability. Both these settings guarantee uniform ellipticity of the
bilinear form a(\Delta; \Delta) on e
as well as a best approximation error and a consistency
error of O(h) [6, 7]. Combining the Lemmas of Lax Milgram and Strang, it can be
shown that a unique solution of (2.2) exists and that the discretization error is of
order h if the solution of (2.1) is smooth; see [6, 7].
In a second, equivalent approach the space f
explicitly plays the role of a
Lagrange multiplier space. This approach is studied in [4] and used further in [11, 25,
26]. The resulting variational formulation gives rise to a saddle-point problem:
Find
M h such that
In particular, it can be easily seen that the first component of the solution of (2.3)
is the unique solution of (2.2). Observing that ~
h is an approximation of the normal
derivative of u on the interface, it makes sense to consider a priori estimates for
suitable norms. Here n lk is the outer unit normal
restricted to
lk . This issue was first addressed in [4] where a priori estimates in the (H 1=2
were established. Similar bounds are given in [26] for a weighted L 2 -norm. As in the
general saddle-point approach [13], the essential point is to establish adequate inf-sup
conditions; such bounds have been established with constants independent of h for
both these norms; see [4, 26].
In the following, all constants are generic depending on the local
ratio between the coefficients b and a, the aspect ratio of the elements and subdomains
but not on the mesh size and not on a. We use standard Sobolev notations and
Y
stand for the broken H 1 -norm and semi-norm. The dual space of a Hilbert space X
is denoted by X 0 and the associated dual norm is defined by
(2.
A Mortar Method Using Dual Spaces 5
3. Dual basis functions. The crucial point for the unique solvability of (2.2)
and (2.3) is the definition of the discrete space f
M h . As we have seen, the discrete
space of Lagrange multipliers is closely related to the trace space in the earlier work on
mortar methods; these spaces are only modified in the neighborhood of the interface
boundaries where the degree of the elements of the test space is lower. We note that
it has been shown only recently, see [23], that for Pn -conforming finite elements the
finite dimensional space of piecewise polynomials of only degree  can be used
instead of degree n in the definition of the Lagrange multiplier space without losing
the optimality of the discretization error . However, in none of these studies has
duality been used to construct an adequate finite element space for the approximation
of the Lagrange multiplier. We recall that the Lagrange multiplier in the continuous
setting represents the flux on the interfaces. Even if the weak solution of (2.1) is
does not have to be continuous on the interfaces. This observation has
motivated us to introduce a new type of discrete Lagrange multiplier space. We note
that local dual basis functions have been used in [22] to define global projection-like
operators which satisfy stability and approximation properties; in this paper we use
the same dual basis functions to define the discrete Lagrange multiplier space.
Let oe be an edge and e
be a polynomial space satisfying P 0 (oe) ae e
2g, be a basis satisfying
R
oe OE oe;i ds 6= 0. We can then
define a dual basis f/ oe;i g N
by the following relation
Z
oe
Z
oe
The definition (3.5) guarantees that f/ oe;i g N
is well defined. Each / oe;i can be written
as a linear combination of the OE oe;i , 1  i  N and the coefficients are obtained by
solving a N \Theta N mass matrix system. Furthermore (3.5) yields
Z
oe
ds
and thus
1. The f/ oe;i g N
also form a linearly independent set. To see
this, let us assume that
R
oe
ds
R
oe
0:
As a consequence, we obtain e
Ng.
Let us consider the case that f oe;i g 2
are the nodal basis functions of
Then, the dual basis is given by
2:
Based on these observations, we introduce a global space M h (\Gamma lk ) on each non-
6 BARBARA I. WOHLMUTH
be the nodal basis function of f
introduced in Section 2. Then,
each OE i can be written as the sum of its local contributions
=: OE oe;i
where the local contributions are linearly independent. We set e
g. In particular by construction, it is guaranteed that P 0 (oe) ae
e
Using the local dual
basis functions on each oe, the global basis functions of M h (\Gamma lk ) are defined as
The support of / i is the same as that of OE i and the f/ i g N lk
form a linear independent
system. Figure 3.2 depicts the two different types of dual basis functions.
G lk
G lk
Fig. 3.2. Dual basis functions in the neighborhood of the boundary of \Gamma lk (left) and in the
interior (right)
Remark 3.1. The following global orthogonality relation holds
Z
Z
oe
Z
oe
Z
We note the similarity with (3.5).
The central point in the analysis of the consistency and approximation error will
be the construction of adequate projection-like operators. We refer to [6, 7] for the
standard mortar approach. Here, we use different operators onto M h (\Gamma lk ), f
and W h (\Gamma lk
defined by
R
ds
R
ds
A dual operator
M h (\Gamma lk ), is now given by
R
ds
R
ds
A detailed discussion of this type of operator can be found in [22]. It is easy to see
that P lk and Q lk restricted to M h (\Gamma lk ) and f
respectively, is the identity
A Mortar Method Using Dual Spaces 7
In addition, using (3.6), (3.7) and (3.8), we find that for v; w 2 L 2 (\Gamma lk ),
and it therefore makes sense to call Q lk a dual operator to P lk . Furthermore, the
operators are L 2 -stable. We have
0;oe
R
ds
R
ds
R
where the domain D oe is defined by
Here, we have used the fact that D oe contains at most three elements and that
independently of the length of the edges. The same
type of estimate holds true for Q lk
0;oe
R
ds
R
ds
R
0;D oe
Thus, P lk and Q lk are L 2 -projection-like operators which preserve the constants.
Using (3.9) and (3.10), it is easy to establish an approximation result.
Lemma 3.2. There exist constants such that for
C
Proof. The proof of (3.12) follows by applying the Bramble-Hilbert Lemma and
using the stability (3.10) and the identity (3.9); it is important that the constants are
contained in the space M h (\Gamma lk ). For each v we define a constant c v in the following
way
Z
where jD oe j is the length of D oe . We remark that the constant c v depends only on the
values of v restricted on D oe . Now, by means of P lk c
oe jvj s;D oe :
The global estimate (3.12) is obtained by summing over all local contributions and
observing that each oe 0 is only contained in a fixed number of D oe .
Although dim f
we get the same type of estimate as (3.14)
for Q lk instead of P lk by using (3.11).
For the estimate (3.13) in the dual norm, we use the definition (2.4)
R
ds
R
ds
In a next step, we consider the last integral in more detail. Using (3.14) for Q lk
instead of P lk and setting
0;oe  CjOEj 2
(D oe )
Summing over all oe 2 \Sigma lk and using that the sum over jOEj 2
is bounded by
Combining this upper bound with (3.14) gives (3.13).
4. Nonconforming formulation. Replacing the space f
M h in the definition of
e
we get a new nonconforming space
The original nonconforming variational problem (2.2) is then replaced by:
Find h such that
In what follows, we analyze the structure of an element v h 2 V h . Each v 2 X h
restricted to a non-mortar side \Gamma lk can be written as
are defined in Section 2 and OE 0 and OE N lk +1 are the nodal basis
functions of W h (\Gamma lk ) associated with the two endpoints of \Gamma lk . The following lemma
characterizes the elements of V h .
Lemma 4.1. Let v 2 X h restricted on \Gamma lk be given as in (4.17). Then,
and only if for each non-mortar \Gamma lk
R
(v
ds
R
ds
The proof follows easily from (4.17) and the global orthogonality relation (3.6).
As in case of e
h the values of a function v 2 V h at the nodal Lagrange interpolation
points in the interior, p i , 1  i  N lk , of any non-mortar \Gamma lk are uniquely determined
by its values on the corresponding mortar side \Gamma kl and the values at the endpoints of
A Mortar Method Using Dual Spaces 9
. The nodal values in the interior of the non-mortars \Gamma lk are obtained by combining
(4.18) with a basis transformation. In particular, these values can be directly obtained
by the simple formula
j\Omega l
R
ds
R
ds
For the two interior nodal points p 1 and pN lk next to the endpoints of \Gamma lk , we get
j\Omega l
R
(v
ds
R
ds
j\Omega l
(p N lk
R
(v
j\Omega l (pN lk +1 )OE N lk +1 ) ds
R
ds
Here, we have used that v
j\Omega l
j\Omega l
(p N lk +1 and that
are identically 1 on the edges next to the endpoints of \Gamma lk . We note that
by definition of the basis functions there exist
Z
Z
Z
Z
OE N lk ds:
If we have a closer look at the nodal basis functions of e
we realize that
there is a main difference in the structure of the basis functions. Figures 4.3 and
4.4 illustrate this difference for the special situation that we have a uniform but
nonmatching triangulation on the mortar and the non-mortar side.
nodal
basis
functions
mortar side
nodal
basis
functions
non-mortar side
nodal
basis
functions
non-mortar side
Fig. 4.3. Nodal basis function on a mortar side (left) and on the non-mortar side in e
and in V h (right)
In

Figure

4.3, the mortar side is associated with the finer triangulation whereas
in

Figure

4.4 it is associated with the coarser one.
As in the standard finite element context, nodal basis functions can be defined for
contained in a circle of diameter Ch: This is in general not possible
for e
h . In the latter case, the support of a nodal basis function associated with a
nodal point on the mortar side is a strip of length j\Gamma lk j and width h, see Figure 4.5,
and the locality of the basis functions is lost.
We conclude this section, by establishing a priori bounds for the discretization
error. As in [6, 7] a mortar projection will be a basic tool in the analysis of the
best approximation error. We now use the new Lagrange multiplier space M h in
nodal
basis
functions
mortar side0.20.61
nodal
basis
functions
non-mortar side0.20.61
nodal
basis
functions
non-mortar side
Fig. 4.4. Nodal basis function on a mortar side (left) and on the non-mortar side in e
and in V h (right)
lk
G
lk
Fig. 4.5. Support of a nodal basis function in e
the definition of suitable projection-like operators. For each non-mortar side the new
mortar projection is given by \Pi
Z
By using (4.19) and (4.20), it can be easily seen that the operator \Pi lk is well defined.
To analyze the approximation error, it is sufficient to show that the mortar projection
is uniformly stable in suitable norms. The stability in the L 2 - and H 1 -norms is given
in the following lemma.
Lemma 4.2. The mortar projection \Pi lk is L 2 - and H 1 -stable
Proof. Using the explicit representation (4.19) and (4.20) where v
has to be
replaced by v and v
j\Omega l
j\Omega l
(p N lk +1 ) have to be set to zero, (4.22) is obtained.
It can be easily seen that even the local estimate
holds true. By means of an inverse inequality, we find for each p 2 W h (\Gamma lk
satisfying
const. and
R
ds
R
ds
We remark that if @D oe " @ \Gamma lk 6= ;, then p was set to zero. However, due to the
boundary conditions of v we obtain kvk 0;D oe
Ch oe jvj 1;D oe in this case.
A Mortar Method Using Dual Spaces 11
The mortar projection can be extended to the space H 1=2
in the following
way:
Z
Then, an interpolation argument together with Lemma 4.2 gives the H 1=2
-stability
Ckvk
It is of interest to compute the stability constant in (4.22) in the special case of
a uniform triangulation of \Gamma lk with h := joej. Then,
are the two endpoints of oe. Using the mortar definition and summing
over all elements in \Sigma lk , we get
In general, the constants in a priori estimates depend on the coefficients. Here,
we will give a priori estimates which depends explicitly on the coefficient a. For each
ff k in the following way
sup
min
min
a j
~
sup
a j
We note that ff k and ~
ff k are bounded by 2 if the non-mortar side is chosen as that
with the smaller value of a.
The uniform ellipticity of the bilinear form a(\Delta; \Delta) on V h \Theta V h is important for
the a priori estimates. For the standard mortar space, it is well known, see [6, 7, 8].
Moreover in [8], it is shown that the bilinear form a(\Delta; \Delta) is uniform elliptic on Y \Theta Y ,
where
Y
1(\Omega
Z
The starting point of the proof is a suitable Poincar'e-Friedrichs type inequality. For
general considerations on Poincar'e-Friedrichs type inequalities in the mortar situation,
we refer to [24]. In [17, Theorem IV.1], it is shown that the ellipticity constant does
not depend on the number of subdomains. A similar estimate is given for the three
field formulation in [14]. We refer to [17] for a detailed analysis of the constants
in the a priori estimates in terms of the number of subdomains and their diameter.
Observing that V h is a subspace of Y , it is obvious that that the bilinear form a(\Delta; \Delta)
on V h \Theta V h is uniform elliptic.
4.1. Approximation property. To establish an approximation property for
h , we follow [6, 7]. One central point in the analysis is an extension theorem. In
[9], a discrete extension is used such that the H 1 -norm of the extension
on\Omega k is
bounded by a constant times the H 1=2 -norm on the boundary
@\Omega k . The support of
such an extension is in
general\Omega k and it is assumed that the triangulation is quasi-
uniform. However, it can be generalized to the locally quasi-uniform case. Combining
the approximation property of
using the mortar projection \Pi lk ,
we obtain the following lemma.
Lemma 4.3. Under the assumption that u 2
the best
approximation error is of order h s ,
vh 2Vh
k a k kuk 2
where the ff k are defined in (4.25).
Proof. The proof follows exactly the same lines as for e
h and the Laplace operator;
we therefore omit the details and refer to [6, 7]. For each
subdomain\Omega k , we use the
Lagrange interpolation operator I k . Then, we define w
I k v. We
note that w h is not in general, contained in V h . To obtain an element in V h , we
have to add appropriate corrections. For each interface \Gamma lk , we consider the jump
apply the mortar projection. The result is extended as a discrete
harmonic function into the interior
of\Omega l . Finally, we define
H l (\Pi lk [w h ])
where H l denotes the discrete harmonic extension operator
kH l
vk1;\Omega l  Ckvk H 1(@\Omega l )
see [9, Lemma 5.1]. Here, \Pi lk [w h ] is extended by zero onto
vanishes
outside\Omega l . By construction, we have
Z
Z
and thus v h 2 V h .
A coloring argument yields,
a l k\Pi lk [w h ]k 2
C
a k h 2s
a l k\Pi lk [w h ]k 2
C
a k h 2s
a l
C
a k h 2s
a l
l kuk 2
C
ff k a k h 2s
A Mortar Method Using Dual Spaces 13
Here, we have used the stability of the harmonic extension; see [9], the stability of the
mortar projection (4.24) and the approximation property of the Lagrange interpolant.
4.2. Consistency error. The space V h is in general not a subspace of H 1
Therefore, we are in a nonconforming setting and in addition to uniform ellipticity and
the approximation property we need to consider the consistency error [10] to obtain
a stable and convergent finite element discretization. In Strang's second Lemma, the
discretization error is bounded by the best approximation error and the consistency
error [10].
Lemma 4.4. The consistency error for
[arun lk is of order h s
sup
R
a @u
@n lk
ds
k a k kuk 2
1where ff k is defined in (4.25).
Proof. The proof generalizes that given for e
V h in [6, 7]. Here, the Lagrange
multiplier space M h is used and we also consider the effect of discontinuous coefficients.
By the definition of V h , we have
Z
and thus
Z
a
@n lk
Z
(a
@n lk
@n lk
ds:
where P lk is defined in (3.7). Using a duality argument and the continuity of the
trace, we get
Z
a
@n lk
ds
where  := a @u
@n lk
. To replace, in the last inequality, the H 1 -norm by the H 1 -semi-
norm, we take into account that
j\Omega l
where \Pi lk denotes the L 2 -projection onto piecewise constant functions on \Gamma lk . In the
duality argument, the H 1=2 -norm can therefore be replaced by the H 1=2 -semi-norm
kw
hj\Omega l
j\Omega l
C
j\Omega l
j\Omega l
Finally, Lemma 3.2, which states the approximation property of P lk  in the (H 1=2
norm, yields
l jj H
l min(a l juj
14 BARBARA I. WOHLMUTH
Remark 4.5. In case that the coefficient a is smaller on the non-mortar side
then ff k is bounded by 2 independently of the jumps in a. Otherwise the upper bound
for ff k depends on the jumps in a. A possibly better bound might depend on the ratio of
the mesh size across the interface; see (4.25). However, numerical results have shown
that in case of adaptive mesh refinement controlled by an a posteriori error estimator,
remains bounded independently of the jump in the coefficients; see [26, 27].
Using Lemmas 4.3 and 4.4, we obtain a standard a priori estimate for the modified
mortar approach (4.16). Under the assumptions that [arun lk
ff k a k h 2(s\Gamma1)
Remark 4.6. The a priori estimates in the literature [6, 7] are often given in
the following form
This is weaker than the estimate (4.26), since for generally we only have
K
4.3. A priori estimates in the L 2 -norm. Finite element discretizations pro-
vide, in general, better a priori estimates in the L 2 -norm than in the energy norm.
In particular, if we assume H 2 -regularity, we have the following a priori estimate for
in the L 2 -norm
The proof can be found in [11] and is based on the Aubin-Nitsche trick. In addition,
the nonconformity of the discrete space has to be taken into account. An essential
role in the proof of the a priori bound is the following lemma. It shows a relation
between the jumps of an element across the interfaces and its nonconformity.
The same type of result for v 2 e
V h can be found in [26].
Lemma 4.7. The weighted L 2 -norm of the jumps of an element v 2 V h is bounded
by its nonconformity
a l
0;oe  inf
~
Proof. The proof follows the same ideas as in case for v h 2 e
use
the orthogonality of the jump and the definition (3.8) to obtain
j\Omega l
Now, it is sufficient to consider an interface \Gamma lk at a time, and we find
a l
a l
j\Omega l
j\Omega l
(v
A Mortar Method Using Dual Spaces 15
Using (3.15) and the continuity of the trace operator, we get for each w
a l
a l jv
j\Omega l
a l jv
Summing over the subdomains and using the definition for ~
ff k give the assertion.
Using the dual problems:
Find h such that
gives
a @w
@n lk
ds
Z
a @u
@n lk
ds
Then, the H 2 -regularity, Lemma 3.2, Lemma 4.7, and observing that the jump of an
element in V h is orthogonal on M h yield
a l
0;oe
a l
0;oe
a l
0;oe
a l
0;oe
:= a @w
@n lk
. Using the a priori estimate for the energy norm (4.26), we
obtain an a priori estimate for the L 2 -norm. The following lemma gives the a priori
estimate for the modified mortar approach.
Lemma 4.8. Assuming H 2 -regularity, the discretization error in the L 2 -
norm is of order h 2 .
5. Saddle point formulation. A saddle point formulation for mortar methods
was introduced in [4]. In particular, a priori estimates involving the (H 1=2
for the Lagrange multiplier were established in that paper whereas estimates in a
weighted L 2 -norm were given in [26]. Here, we analyze the error in the Lagrange
multiplier for both norms and obtain a priori estimates of the same quality as for the
standard mortar approach.
The norm for the Lagrange multiplier is defined by
l2M(k)a l
Y
Y
The weight a \Gamma1
l is related to the fact that we use the energy norm for in the a
priori estimates.
Working within the saddle point framework, the approximation property on V h ,
which is given in Lemma 4.3, is a consequence of the approximation property on X h ,
the continuity of the bilinear form b(\Delta; \Delta), and an inf-sup condition [13]. A discrete
inf-sup condition is necessary to obtain a priori estimates for the Lagrange multiplier.
The saddle point problem associated with the new nonconforming formulation
(4.16) involves the space (X
We get a new saddle point
problem, with exactly the same structure as (2.3):
Find
The inf-sup condition, established in [4] for the pairing (X
Lemma 5.1. There exists a constant independent of h such that
sup
c:
Proof. Using the definition of the dual norm (2.4), we get
OE2H2
OE2H2
C sup
OE2H2
The maximizing element in W h (\Gamma lk
called
and a v lk 2 X h is defined in the following way
j\Omega n\Omega l
\Omega l
where OE h is extended by zero on
We then find
and a(v lk
21;\Omega
l
. Finally, we set
a l
and observe that a(v h ; v h only if  coloring argument gives
Summing over all interfaces yields
C
l2M(k)a l
By construction, we have found for each
C
A Mortar Method Using Dual Spaces 17
The proof of the inf-sup condition (5.28) together with the approximation Lemma
3.2 and the first equation of the saddle point problem gives an a priori estimate similar
to (4.26) for the Lagrange multiplier.
Lemma 5.2. Under the assumptions u 2 Q K
[arun lk the following a priori estimate for the Lagrange multiplier holds true
C
ff k a k h 2(s\Gamma1)
Proof. Following [4] and using the first equation of the saddle point problem, we
get
Taking (5.29) into account, we find that the inf-sup condition even holds if the supremum
over X h is replaced by the supremum over a suitable subspace of X h . For the
proof of (5.30), we start with (5.29) and not with the inf-sup condition (5.28)
constructed as in the proof of Lemma 5.1. We recall
that w is defined as a linear combination of discrete harmonic functions
a l
where w
1. A coloring argument shows that
the energy norm of w is bounded by the H 1=2
00 -dual norm of  h \Gamma  h , moreover we
find
a l
a l
K
l2M(k)a l
combining (5.31) and (5.32), we obtain
kwk
Applying the triangle inequality, choosing  h
using
we find that (3.13) yields, for
C
a l
C
ff k a k h 2
Here, we have used that  restricted on \Gamma lk is arun lk and a trace theorem.
We note that in spite of Lemma 3.2 we cannot obtain a priori estimates of order h
for the norm of the dual of H 1=2 (S). This is due to the fact that the inf-sup condition
(5.28) cannot be established for that norm.
Remark 5.3. The a priori estimate (5.30) also holds if we replace the (H 1=2
norm by the weighted L 2 -norm
a l
Using (3.12) and the techniques of the proof of Lemma 5.2, it is sufficient to have
a discrete inf-sup condition similar to (5.28) for the weighted L 2 -norm, i.e.
sup
c:
The only difference in the proof is the definition of v lk . Instead of using a discrete
harmonic extension
onto\Omega l , we use a trivial extension by zero, i.e. we set all nodal
values on
@\Omega l n\Gamma lk and
on\Omega l to zero. Then, v lk is non zero only on a strip of length j\Gamma lk j
and width h l and a(v lk ; v lk ) is bounded form below and above by
a l
0;oe .
6. Numerical results. We get a priori estimates of the same quality for the
error in the weak solution and the Lagrange multiplier as in the standard mortar case
[4, 6, 7]. In contrast to e
, we can define nodal basis functions for V h which have local
supports. Efficient iterative solvers for linear equation systems arising from mortar
finite element discretization are very often based on the saddle point formulation or
work with the product space X h instead of the nonconforming mortar space. Different
types of efficient iterative solvers are developed in [1, 2, 3, 11, 15, 16, 19, 20, 18, 25].
However, most of these techniques require that each iterate satisfies the constraints
exactly. In most studies of multigrid methods, these constraints have to be satisfied
even in each smoothing step [11, 12, 18, 25]. If we replace e
the constraints
are much easier to satisfy, since instead of solving a mass matrix system, the nodal
values on the non-mortar side can be given explicitly.
Fig. 6.6. Decomposition and initial triangulation (left) and solution (right) (Example 1)
Here, we will present some numerical results illustrating the discretization errors
for the standard and the new mortar methods in the case of P 1 Lagrangian finite
elements. We recall that in the standard mortar approach the Lagrange multipliers
belong to f
M h whereas we use M h in the new method. We have used a multigrid
method which satisfies the constraints in each smoothing step; see [11, 25] for a
A Mortar Method Using Dual Spaces 19
discussion of the standard mortar case. This multigrid method can be also applied
without any modifications to our modified mortar setting. It does not take advantage
of the diagonal mass matrix on the non-mortar side of the new formulation. To obtain
a speedup in the numerical computations, special iterative solvers for the new mortar
setting have to be designed. We will address this issue in a forthcoming paper [28].
We start with an initial triangulation T 0 , and obtain the triangulation T l on level l by
uniform refinement of T l\Gamma1 .
Both discretization techniques have been applied to the following test example:
where the right hand side f and the Dirichlet boundary conditions
are chosen so that the exact solution is
yy. The solution and the initial triangulation
are given in Figure 6.6. The domain is decomposed into nine subdomains defined by
and the triangulations do not
match at the interfaces. We observe two different situations at the interface, e.g. the
isolines of the solution are almost parallel at
@\Omega 12 whereas at
@\Omega 21 the
angle between the isolines and the interface is bounded away from zero. In case that
the isolines are orthogonal on the interface the exact Lagrange multiplier will be zero.

Table
Discretization errors (Example 1)
standard approach modified approach
Lagrange multiplier f
level # elem. L 2 -err. energy err. L 2 -err. energy err.
In

Table

6.1, the discretization errors are given in the energy norm as well as in the
for the two different mortar methods. We observe that the energy error is
of order h whereas the error in the L 2 -norm is of order h 2 . There is no significant
difference in the accuracy between the two mortar algorithm. The discretization errors
in the energy norm as well as in the L 2 -norm are almost the same.
Fig. 6.7. Decomposition and initial triangulation (left) and solution (right) (Example 2)
In our second example, we consider the union square with a slit decomposed
into four subdomains, see Figure 6.7. Here, the right hand side f and the Dirichlet
boundary conditions of are chosen so that the exact solution is given by
sin OE. The solution
has a singularity in the center of the domain. We do not have H 2 -regularity, and we
therefore cannot expect an O(h) behavior for the discretization error in the energy
norm.

Table
Discretization errors (Example 2), Energy error in 1e \Gamma 01
standard approach modified approach
Lagrange multiplier f
level # elem. L 2 -err. energy err. L 2 -err. energy err.
2:069586
The discretization errors are compared in Table 6.2. In this case, we observe a difference
in the performance of the different mortar methods. The L 2 -error of the modified
mortar method is asymptotically better than that of the standard method. The situation
is different for the energy error; the standard mortar approach gives slightly
better results. A non-trivial difference can only be observed in this example where
there is no H 2 -regularity. In that case, the modified mortar method gives better
results in the L 2 -norm.
Our last example illustrates the influence of discontinuous coefficients. We consider
the diffusion equation \Gammadiv the coefficient a is dis-
continuous. The unit
square\Omega is decomposed into four
as in Figure 6.8.
Fig. 6.8. Decomposition and initial triangulation (left) and solution (right) (Example
The coefficients on the subdomains are given by a
The right hand side f and the Dirichlet boundary conditions are chosen to match
a given exact solution,
solution is continuous with vanishing [arun] on the interfaces. Because of the discontinuity
of the coefficients, we use a highly non-matching triangulation at the interface,
see

Figure

6.8.
A Mortar Method Using Dual Spaces 21
The discretization errors in the energy norm as well as in the L 2 -norm are given
for the two different mortar algorithms in Table 6.3. We observe that the energy error
is of order h. As in Example 1, there is only a minimal difference in the performance
of the two mortar approaches.

Table
Discretization errors (Example 3), Energy error in 1e \Gamma 01
standard approach modified approach
Lagrange multiplier f
level # elem. L 2 -err. energy err. L 2 -err. energy err.
The following two figures illustrate the numbers given in Tables 6.1 - 6.3. In

Figure

6.9, the errors in the energy norm are visualized whereas in Figure 6.10 the
errors in the L 2 -norm are shown. In each figure a straight dashed line is drawn below
the obtained curves to indicate the asymptotic behavior of the discretization errors.1100 1000 10000 100000
in
the
energy
norm
Number of elements
Example 1
standard
modified
in
the
energy
norm
Number of elements
Example 2
standard
modified
in
the
energy
norm
Number of elements
Example 3
standard
modified
Fig. 6.9. Discretization errors in the energy norm versus number of elements0.0010.1100 1000 10000 100000
in
the
norm
Number of elements
Example 1
standard
modified
in
the
norm
Number of elements
Example 2
standard
modified
100 1000 10000 100000
in
the
norm
Number of elements
Example 3
standard
modified
Fig. 6.10. Discretization errors in the L 2 -norm versus number of elements
In Examples 1 and 2, almost from the beginning on the predicted order h for the energy
norm and the order h 2 for the L 2 -norm can be observed. In these two examples only
one plotted curve for the standard and the new mortar approach can be seen. The
numerical results are too close to see a difference in the pictures. In Example 2,
where we have no full H 2 -regularity, the asymptotic starts late. We observe for both
22 BARBARA I. WOHLMUTH
mortar methods an O(h 1=2 ) behavior for the discretization error in the energy norm.
During the first refinement steps the error decreases more rapidly. For the L 2 -norm
the asymptotic rate is given by O(h 3=2 ). Moreover, it seems to be the case that the
new mortar method performs asymptotically better than the standard one. However,
this cannot be observed for other examples without full regularity.

Acknowledgment

The author would like to thank Professor Olof B. Widlund
for his continuous help and for fruitful discussions as well as the referees for their
valuable comments.



--R

Substructuring preconditioners for finite element methods on nonmatching grids.
Substructuring preconditioners for the Q 1 mortar element method.
Iterative substructuring preconditioners for mortar element methods in two dimensions.
The mortar finite element method with Lagrange multipliers.
The mortar element method for three dimensional finite elements.
Domain decomposition by the mortar element method.
A new nonconforming approach to domain de- composition: The mortar element method
Raffinement de maillage en elements finis par la methode des joints.
Iterative methods for the solution of elliptic problems on regions partitioned into substructures.

A multigrid algorithm for the mortar finite element method.
Stability estimates of the mortar finite element method for 3- dimensional problems
Mixed and hybrid finite element methods.
estimates for the three-field formulation with bubble stabilization submitted to Math
A hierarchical preconditioner for the mortar finite element method.
Adaptive macro-hybrid finite element methods
On the mortar finite element method.
Multigrid for the mortar finite element method.
Analysis and parallel implementation of adaptive mortar finite element methods.
Domain decomposition with nonmatching grids: Augmented Lagrangian approach.

Finite element interpolation of nonsmooth functions satisfying boundary conditions.
Convergence results for non-conforming hp methods: The mortar finite element method
Poincar'e and Friedrichs inequalities for mortar finite element methods.
The coupling of mixed and conforming finite element dis- cretizations
Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers.
Mortar finite element methods for discontinuous coefficients.
Multigrid methods based on the unconstrained product space arising from mortar finite element discretizations.
A mixed finite element discretization on non-matching multiblock grids for a degenerate parabolic equation arising in porous media flow
--TR

--CTR
Q. Hu, Numerical integrations and unit resolution multipliers for domain decomposition methods with nonmatching grids, Computing, v.74 n.2, p.101-129, March 2005
Dan Stefanica, Parallel FETI algorithms for mortars, Applied Numerical Mathematics, v.54 n.2, p.266-279, July 2005
Ralf Unger , Matthias C. Haupt , Peter Horst, Application of Lagrange multipliers for coupled problems in fluid and structural interactions, Computers and Structures, v.85 n.11-14, p.796-809, June, 2007
S. Heber , M. Mair , B. I. Wohlmuth, A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems, Applied Numerical Mathematics, v.54 n.3-4, p.555-576, August 2005
Barbara I. Wohlmuth, An a Posteriori Error Estimator for Two-Body Contact Problems on Non-Matching Meshes, Journal of Scientific Computing, v.33 n.1, p.25-45, October   2007
a priori estimates;lagrange multiplier;nonmatching triangulations;dual norms;mortar finite elements