Nitsche type mortaring for some elliptic problem with corner singularities.
The paper deals with Nitsche type mortaring as a finite element method (FEM) for treating non-matching meshes of triangles at the interface of some domain decomposition. The approach is applied to the Poisson equation with Dirichlet boundary conditions (as a model problem) under the aspect that the interface passes re-entrant corners of the domain. For such problems and non-matching meshes with and without local refinement near the re-entrant corner, some properties of the finite element scheme and error estimates are proved. They show that appropriate mesh grading yields convergence rates as known for the classical FEM in presence of regular solutions. Finally, a numerical example illustrates the approach and the theoretical results.
Introduction
For the e-cient numerical treatment of boundary value problems (BVPs), domain decomposition
methods are widely used. They allow to work in parallel: generating the mesh in
subdomains, calculating the corresponding parts of the stiness matrix and of the right-hand
side, and solving the system of nite element equations.
There is a particular interest in triangulations which do not match at the interface of
the subdomains. Such non-matching meshes arise, for example, if the meshes in dierent
subdomains are generated independently from each other, or if a local mesh with some
structure is to be coupled with a global unstructured mesh, or if an adaptive remeshing
in some subdomain is of primary interest. This is often caused by extremely dierent
data (material properties or right-hand sides) of the BVP in dierent subdomains or by a
complicated geometry of the domain, which have their response in a solution with singular
or anisotropic behaviour. Moreover, non-matching meshes are also applied if dierent
discretization approaches are used in dierent subdomains.
There are several approaches to work with non-matching meshes. The task to satisfy some
continuity requirements on the interface (e.g. of the solution and its conormale derivative)
can be done by iterative procedures (e.g. Schwarz's method) or by direct methods like the
Lagrange multiplier technique.
There are many papers on the Lagrange multiplier mortar technique, see e.g. [5, 6, 9, 25]
and the literature quoted in these papers. There, one has new unknowns (the Lagrange
multipliers) and the stability of the problem has to be ensured by satisfying some inf-sup
condition (for the actual mortar method) or by stabilization techniques.
Another approach which is of particular interest here is related to the classical Nitsche
method [16] of treating essential boundary conditions. This approach has been worked out
more generally in [23, 20] and transferred to interior continuity conditions by Stenberg [21]
(Nitsche type mortaring), cf. also [1]. As shown in [4] and [10], the Nitsche type mortaring
can be interpreted as a stabilized variant of the mortar method based on a saddle point
problem.
Compared with the classical mortar method, the Nitsche type mortaring has several ad-
vantages. Thus, the saddle point problem, the inf{sup{condition as well as the calculation
of additional variables (the Lagrange multipliers) are circumvented. The method employs
only a single variational equation which is, compared with the usual equations (without any
mortaring), slightly modied by an interface term. This allows to apply existing software
tools by slight modications. Moreover, the Nitsche type method yields symmetric and
positive denite discretization matrices in correspondence to symmetry and ellipticity of
the operator of the BVP. Although the approach involves a stabilizing parameter
, it is
not a penalty method since it is consistent with the solution of the BVP. The parameter
can be estimated easily (see below). The mortar subdivision of the chosen interface
can be done in a more general way than known for the classical mortar method. This
can be advantageous for solving the system of nite element equations by iterative domain
decomposition methods.
Basic aspects of the Nitsche type mortaring and error estimates for regular solutions u 2
2) on quasi-uniform meshes are published in [21, 4]. Compared with these
papers, we extend the application of the Nitsche type mortaring to problems with non-regular
solutions and to meshes being locally rened and not quasi-uniform.
We consider the model problem of the Poisson equation with Dirichlet data in the presence
of re-entrant corners and admit that the interface with non-matching meshes passes the
vertex of such corners. For the appropriate treatment of corner singularities we employ
local mesh renement around the corner by mesh grading in correspondence with the
degree of the singularity. Therefore, the Nitsche type mortaring is to be analyzed on more
general triangulations. For meshes with and without grading, basic inequalities, stability
and boundedness of the bilinear form as well as error estimates in a discrete H 1 -norm are
proved. The rate of convergence in L 2 is twice of that in the H 1 -norm. For an appropriate
choice of some mesh grading parameter, the rate of convergence is proved to be the same
as for regular solutions on quasi-uniform meshes. Finally, some numerical experiments are
given which conrm the rates of convergence derived.
Analytical preliminaries
In the following, H s (X), s real (X some domain, H denotes the usual Sobolev
spaces, with the corresponding norms and the abbreviation Constants
C or c occuring in inequalities are generic constants.
For simplicity we consider the Poisson equation with homogeneous Dirichlet boundary
conditions as a model problem:
in
Here,
is a bounded polygonal domain in R 2 , with Lipschitz-boundary
@
consisting of
straight line segments. Suppose further that f 2 L
holds. The variational equation of
(2.1) is given as follows. Find
@
such that
with a(u; v) :=
Z
Z
We now decompose the
domain
into non-overlapping subdomains. For simplicity of
notation we consider two
subdomainsand
2 with interface , where
holds closure of the set X). We assume that the boundaries
@
of
are
also Lipschitz-continuous and formed by open straight line segments j such that
We distinguish two important types of interfaces :
case I1: the intersection
\@
consists of two points being the endpoints
of , and at least one point is the vertex of a re-entrant corner, like in Figure 1,
case I2: \
@
does not touch the boundary
@
, like in Figure 2.1

Figure

1:1

Figure

2:
For the presentation of the method and error estimates we need the degree of regularity
of the solution u. Clearly, the functionals a(: ; :) and f(:) satisfy the standard assumptions
of the Lax-Milgram theorem and we have the existence of a solution u
of problem
(2.2) as well as the a priori estimate kuk
C kfk
.
Furthermore, the regularity theory of (2.2) yields
and kuk
C kfk
if
is
convex. If
@
has re-entrant corners with angles
can be represented by
I
with a regular remainder w 2 H
. Here, (r denote the local polar coordinates of a
point Pwith respect to the vertex
@
r 0j is the radius of some circle neighborhood with center at P j . Moreover, we have
2 <  j < 1), a j is some constant, and  j is a locally acting (smooth) cut-o function
around the vertex P j , with
The solution u
satises the relations
I
C kfk
and, owing to (2.3), also
for any ": 0 su-ciently small. For these
results, see e.g. [13, 7].
In the context of
dividing
into
subdomains;
2 , we introduce the restrictions v i :=
of some function v
on
i as well as the vectorized form of v by
i.e. we have
It should be noted that we shall use here the same symbol v
for denoting the function
on
as well as the vector
. This will not lead to confusion,
since the meaning will be clear from the context. The one-to-one correspondence between
the \eld function" v and the \vector function" v is given
2 . Moreover, vj is
dened by the trace. We shall keep the notation also in cases, where the traces v 1
on the interface are dierent (e.g. for interpolants
on
Using this notation, it is obvious that the solution of the BVP (2.1) is equivalent to the
solution of the following interface problem: Find
such that
in
@
@
are satised, where n i denotes the outward normal to
@
Introducing the
spaces given by
case
@
\@
for
@
@
case
@
@
and the space V := the BVP (2.5) can be formulated in a weak form (see e.g.
[2]). Clearly, we have u
as well as
. The
continuity of the solution u and of its normal derivative @u i
@n
on
to be required in the sense of H2
() and H2
() (the dual space of H2
()), respectively.
(@
by the range of V i by the trace operator and to be provided with
the quotient norm, see e.g. [9, 13]. So we use in case I1: H2
(@
@
@
case I2: H2
(@
@
@
means that
we identify the corresponding spaces. By
@
we shall denote the duality pairing of
(@
(@
3 Non-matching mesh nite element discretization
We
cover
by a triangulation T i
consisting of triangles. The triangulations
h and T 2
h are independent of each other. Moreover, compatibility of the nodes of T 1
and T 2
@
@
2 is not required, i.e., non-matching meshes on are admitted.
Let h denote the mesh parameter of these triangulations, with 0 < h  h 0 and su-ciently
small h 0 . Take e.g.
h denote the triangulations of dened by the traces
of T 1
h and T 2
h on , respectively.
Assumption 3.1
(i) For
holds
(ii) Two arbitrary triangles
are either disjoint or have a
common vertex, or a common edge.
(iii) The mesh
in
is shape regular, i.e., for the diameter h T of T and the
diameter % T of the largest inscribed sphere of T , we have
C for any T 2 T i
where C is independent of T and h.
Clearly, relation (3.2) implies that the angle  at any vertex and the length hF of any side
F of the triangle T satisfy the inequalities
with constants  0 and " 1 being independent of h and T . Owing to (3.2), the triangulations
do not have to be quasi-uniform in general.
For and according to V i from (2.6) introduce nite element spaces V i
h of functions
on
i by
@
\@
denotes the set of all polynomials on T with degree  k. We do not employ
dierent polynomial degrees
2 , which could also be done. The nite element space
V h of vectorized functions v h with components v i
on
i is given by
In general, v h 2 V h is not continuous across .
Consider further some triangulation E h of by intervals
where hE denotes the diameter of E. Furthermore, let
be some positive constant (to be
specied subsequently) and  1 ;  2 real parameters with
Following [21] we now introduce the bilinear form B h (: ; :) on V h  V h and the linear form
F h (:) on V h as follows:
for
(Note that in [4] a similar bilinear form with
2 and employed.) The
nite element approximation u h of u on the non-matching triangulation T
h is
now dened by u
h satisfying the equation
denotes the L
the H2
()-duality pairing
and product. Owing to u 2 H 3+"(
, the trace theorem yields
holds also for v
This will be
used subsequently for evaluating by the L 2 ()-scalar product. A natural choice for
the triangulation E h of is
h
cf.

Figure

3.

Figure

3:
We require the asymptotic behaviour of the triangulations T 1
h and of E h to be consistent
on in the sense of the following assumption.
Assumption 3.2 For
there are positive
constants C 1 and C 2 independent of h T , hE and h (0 < h  h 0 ) such that the condition
is satised.
Relation (3.9) guarantees that the diameter h T of the triangle T touching the interface
at E is asymptotically equivalent to the diameter hE of the segment E, i.e., the equivalence
of h T ; hE is required only locally.
4 Properties of the discretization
First we show that the solution u of the BVP (2.1) satises the variational equation (3.7),
i.e., u is consistent with the approach (3.7).
Theorem 4.1 Let u be the solution of the BVP (2.1). Then
solves (3.7), i.e.,
we have
Proof. Insert the solution u into B h (:; v h ). Owing to the properties of u, B h (u; v h ) is well
dened and, since u 1
and @u 1
hold, cf. (2.5), we get
Taking into account (2.4) and using Green's formula on the
domains
i , the relations
are derived for any v h 2 V h . This proves the assertion.
Note that due to (4.1) and (3.7) we also have the B h -orthogonality of the error u u h on
For further results on stability and convergence of the method, the following \weighted
discrete trace theorem" will be useful, which describes also an inverse inequality.
Lemma 4.2 Let Assumption 3.1 and 3.2 be satised. Then, for any v h 2 V h the inequality

C IX

holds, where F
h is the face of a triangle TF 2 T i
h touching by F (TF
constant C I does not depend on h; h T ; hE .
Note that extending the norms on the right-hand side of (4.3) to the whole
of
implies

C IX

For inequalities on quasi-uniform meshes related with (4.4) see [23, 21, 4].
Proof. For
h yields @v i
@xs

2X

holds. Let hF denote the length of side F belonging to triangle Since the shape
regularity of T is given, the quantities hF and h T are asymptotically equivalent. Owing to


and to inequality


which is derived by means of the trace theorem on TF and of the inverse inequality, we get


where TF
i has the edge
h . The constants c i do not depend on h; c 2 is
also uniform in T . Inequality (4.5) combined with the previous inequalities yields (4.3).
The constant C I in the inequalities (4.3) and (4.4) can be estimated easily if special assumptions
on E h and on the polynomial degree k are made. For example, let us choose
h from (3.8),
Then, on the triangle T the derivatives
are constants which can be calculated explicitely, together with their
2 -norms on E and on . Thus, we get


denotes the height of over the side E, hE the length of E. Taking the sum
h for all inequalities (4.6), we obtain the value of C I to be
Thus, for equilateral triangles and isosceles rectangular triangles (see the mesh on the left-hand
sides of Figures 6, 7) near , we get C I = 4=
3 and C I = 2, respectively.
For deriving the V h -ellipticity and V h -boundedness of the discrete bilinear form B h (: ; :) from
(3.6), we introduce the following discrete norm


cf. [21] and [9, 4] (uniform weights). Then we can prove the following theorem.
Theorem 4.3 Let Assumptions 3.1 and 3.2 for T i
the constant
in (3.6) independently of h and such that
> C I holds, C I from (4.3). Then,
holds, with a constant  1 > 0 independent of h.
Proof. For from (3.6) we have the identity


Using Cauchy's inequality and Young's inequality (2ab  a 2

i"



Utilizing inequality (4.3) yields (4.8), with
"g > 0, if " is chosen according
to C I < " <
Beside of the V h -ellipticity of B h (: ; :) we also prove the V h -boundedness.
Theorem 4.4 Let Assumption 3.1 and 3.2 be satised. Then there is a constant  2 > 0
such that the following relations holds,
Proof. We apply Cauchy's inequality several times (also with distributed weights hE , h 1
inequality (4.3) and get relation (4.9) with a constant
g.
estimates and convergence
Let u be the solution of (2.1) and u h from (3.7) its nite element approximation. We
shall study the error u u h in the norm k : k 1;h given in (4.7). For functions v satisfying
introduce the mesh-dependent norm
by



First we bound ku u h k 1;h by the norm jj : jj
of the interpolation error u I h u, where
I h u := (I h
h , and I h u i denotes the usual Lagrange interpolant of u i in the
space
Lemma 5.1 Let Assumption 3.1 and 3.2 be satised. For u; u h from (2.1), (3.7), respec-
tively, and
> C I , the following estimate holds,
Proof. Obviously, I h holds, and the triangle inequality yields
Owing to I h u and to the V h -ellipticity of B h (: ; :), we have
In relation (5.4) we utilize (4.2) and get
For abbreviation we use here w := I h u u and v h := I h u u h . Clearly
yields
2 L 2 (). Because of I h u; u
denoting
the interpolant of u i in V i
h and u i
belong only to H 3"(
Unfortunately, w 62 V h holds,
We now apply the same inequalities as used for the proof of Theorem 4.4, with the modication
that inequality (4.3) is only employed with respect to the function v h . This leads
to the estimate
which gives together with (5.5) the inequality
This inequality combined with (5.3) and with the obvious estimate kI h u uk 1;h  kI h u uk
conrms assertion (5.2). The positive constant c 1 depends on
and C I .
An estimate of the error jju u h jj 1;h for regular solutions u is given in [20] and in [4] by
citation of results contained in [23]. Nevertheless, since we consider a more general case,
and since we need a great part of the proof for regular solutions also for singular solutions,
the following theorem is proved.
Theorem 5.2 Let u
(l  2) be the solution of (2.1) and u its nite element
approximation according to (3.7), with
> C I . Furthermore, let the mesh from Assumptions
3.1, 3.2 be quasi-uniform, i.e. max T2T h h T
C. Then the following error estimate holds,
with k  1 being the polynomial degree in V i
Proof. We start from inequality (5.2) which bounds ku u h k 1;h by the interpolation error
and, in the following, take into account tacitly the assumptions on the mesh.
Note that the traces on of the interpolants I h u i of u i in V i
do not coincide,
in general. First we observe that the weighted squared norms
0;E can be rewritten
such that interpolation estimates involve the edge F of the triangle T
h , for 2:

I h u i u i
F

I h u i u i


r I h u i u i
Moreover, we apply the rened trace theorem
for
which is proved in [24], cf. also [23]. Replace v by I h u i u i and @(I h u i
@xs
using (5.9) and some simple estimates, we get

I h u i u i

I h u i u i

I h u i u i

1;T

r I h u i u i
1;T
2;T
Taking the well-known interpolation error estimate on triangles T ,

I h u i u i

ch l j


see e.g. [8, 11], we derive from the inequalities (5.10) and (5.11) the estimates

I h u i u i
ch 2l 1


r I h u i u i
ch 2l 3

Using these estimates and (5.7), (5.8), we realize that

I h u i u i

0;EA  ch 2l 2

holds. For the interpolation error I h u i u i
on
i , the estimate

r I h u i u i
ch 2l 2
obviously follows from (5.12). Clearly, (5.13) and (5.14) lead via (5.2) to (5.6).
6 Treatment of corner singularities
We now study the nite element approximation with non-matching meshes for the case
that has endpoints at vertices of re-entrant corners (case I1). Since the in
uence region
of corner singularities is a local one (around the vertex P 0 ), it su-ces to consider one
corner. For basic approaches of treating corner singularities by nite element methods see
e.g. [3, 7, 13, 17, 19, 22]. For simplicity, we study solutions u 62 H
in correspondence
with continuous piecewise linear elements, i.e.
h from (3.3). We shall consider the
error u u h on quasi-uniform meshes as well as on meshes with appropriate local renement
at the corner.
Let be the coordinates of the vertex P 0 of the corner, (r; ') the local polar coordinates
with center at P 0 , i.e. x x 4.
y
'r
P(x;y)
r

Figure

4:
some circular sector G around P 0 , with the radius r 0 > 0 and the angle ' 0 (here:
G :=

@G boundary of G. For dening a mesh with grading, we employ the real grading parameter
the grading function R i real constant b > 0, and the
step size h i for the mesh associated with layers [R
Here n := n(h) denotes an integer of the order h 1 , n :=
for some real  >
integer part). We shall choose the numbers ; b > 0 such that 2
holds, i.e., the
mesh grading is located within G from (6.1).
Lemma 6.1 For h; the following relations hold
We skip the proof of Lemma 6.1 since it is comparatively simple.
Using the step size h i in the neighbourhood of the vertex P 0 of
the corner a mesh with grading, and for the remaining domain we employ a mesh which
is quasi-uniform. The triangulation T
h is now characterized by the mesh size h and the
grading parameter , with 0 < h  h 0 and 0 <   1. We summarize the properties of T
in the following assumption.
Assumption 6.2 The triangulation T
h satises Assumption 3.1, Assumption 3.2 and is
provided with a grading around the vertex P 0 of the corner such that h T := diam T depends
on the distance R T of T from P 0 , R T := dist in the following way:
with some constants % some real R g , 0 < R g < R g < R
are xed and independent of h.
Here, R g is the radius of the sector with mesh grading and we can assume R (w.l.o.g.
Outside this sector the mesh is quasi-uniform. The value
in the whole
region
min T2T
C holds. In [3, 17, 19] related types of mesh
grading are described. In [15] a mesh generator is given which automatically generates a
mesh of type (6.4).
For the error analysis we introduce several subsets of the triangulation T
h near the vertex
of the re-entrant corner, viz.
with Rn from (6.2). The set C 0h is now decomposed into layers (of triangles) D jh ,
holds:
According to 2
are located in G, G from (6.1). Owing
to Assumption 6.2 (cf. also Lemma 6.1), the asymptotic behaviour of h T is determined by
the relations (given for the case of one corner)
with well as n taken from (6.2). Note that the number of
all triangles T 2 T
h (0 <   1) and nodes of the triangulation is of the order O(h 2 ). The
number n j of all triangles T 2 D jh is bounded by C  j (j
independent of h, cf. [14].
First we investigate the interpolation error of a singularity function s from (2.3) in the class
of polynomials with degree 1. Employ the restrictions s i :=
and take always into
account that
Lemma 6.3 Let
be the singularity function with
respect to the corner at vertex P 0 . Further, let T
h be the triangulation
of
with mesh
grading within G according to Assumption 6.2 (cf. (6.2){(6.5)). Then, the interpolation
error s i I h s i in the seminorm
can be bounded as follows:
where (h; ) is given by
for  <   1
Proof. According to the mesh layers D jh (j n), the norms of the global interpolation
error s i I h s i are represented by the local interpolation error s
for local P 1 {Lagrange interpolation operator) as follows
0h
jh
with
case T 2 D i
First, we consider triangles
0h and employ the estimate
1;T
1;T
Using the explicit representation of s i and I T s i , we calculate the norms on the right-hand
side of (6.8) and get the following bound:
case T 2 D i
We now consider triangles
jh which do not touch the vertex P 0 (center of singularity),
i.e.
0h . In this case, s 2 H 2 (T ) holds owing to R T > 0. Hence, the well-known
interpolation error estimate   s i I T s i
1;T  ch T
2;T (6.10)
can be applied, where c is independent of the triangle T . The norm
2;T is estimated
easily by
a r  sin(')
r
for
Taking into account the relations between h, h T , R T , j and  from Assumption 6.2, cf. also
(6.5), a we nd easily bounds of the right-hand side in (6.11). This
leads together with (6.10) to the estimates
2. Since the number of triangles in the layer D i
jh grows not faster than with j,
we get by summation of the error contributions of the triangles T 2 C 0h n D i
0h the estimate
jh
2: (6.12)
Using monotonicity arguments and the estimation of sums by related integrals, it is not
hard to derive the following set of inequalities,
Some simple estimates of the right-hand side of (6.12) allow to apply (6.13) and n  ch 1
for getting the inequality
jh
with (h; ) given at (6.7) and for 2.
Finally, combining the estimates (6.8), (6.9) from case (i) and (6.14) from case (ii), we
easily conrm (6.6).
We now study the interpolation error s i I h s i and its rst order derivatives in the trace
norms.
Lemma 6.4 Under the assumption of Lemma 6.3 and with (h; ) from (6.7), the following
interpolation error estimates hold for the singularity function 2:


Proof. Clearly, due to the assumption on E h we have for the inequal-
ities

F



Consider now faces F of triangles touching and the local interpolate I T s i .
case T 2 D i
Here we use a similar approach like at (6.8) and get by direct evaluation of the norms the
following estimates:
F

F

F

I T s i


rs i

case T 2 D i
For the remaining faces F and adjacent triangles T which do not touch the vertex P 0 of the
corner, Therefore, inequalities (5.10), (5.11) can be applied. We insert
the well-known estimates
l;T  ch 2 l
2;T
for any triangle with face F  :
F

ch 2

ch 2
Calculating and estimating
and summation over all triangles T 2 C 0h n D 0h touching
near the singularity yields by analogy to (6.14) the estimate
2: (6.20)
Finally, we combine the inequalities (6.16){(6.20) and get (6.15).
Lemma 6.5 Assume that there is one re-entrant corner and that the triangulation T
h is
provided with mesh grading according to the Assumption 6.2. Then the following estimate
holds for the error u I h u of the Lagrange interpolant I h with u from (2.3) and
(h; ) from (6.7):
I h uk
Proof. According to (2.3), the solution u of the BVP (2.1) can be represented by
a r  sin(')
denotes the regular part of the solution, and s is the
singular part. Apply the triangle inequality jju I h ujj
jjs I h sjj
.
holds, the norm jjw I h wjj
has been already estimated in the
proof of Theorem 5.2. Thus, using the estimates (5.13) and (5.14) for
with (2.4), we get
kw I h wk
ch kwk
ch kfk
Bounds of the norm ks I h sk
can be derived from Lemma 6.3 and Lemma 6.4. The
combination of (6.6), (6.15) and (2.4) yields the inequalities
ks I h sk
with (h; ) from (6.7). Estimate (6.21) is obvious by (6.22) and (6.23).
The nal error estimate is given in the next theorem.
Theorem 6.6 Let u and u h be the solutions of the BVP (2.1) with one re-entrant corner
and of the nite element equation (3.7), respectively. Further, for T
h let Assumption 6.2
be satised. Then the error u u h in the norm k : k 1;h (4.7) is bounded by
with
for  <   1
Proof. The combination of Lemma 5.1 with Lemma 6.5 immediately yields the assertion.
Remark 6.7 Estimate (6.24) holds also for more than one re-entrant corner, with a slightly
modied function (h; ). For example, if the mortar interface touches the vertices
of two re-entrant corners with angles '
holds. According to  1 ;  2 , we employ meshes with grading
parameters holds now with
h jln hj2 for
Remark 6.8 Under the assumption of Theorem 6.6 and for the error in the L 2 {norm, the
estimate
holds. In particular, we have the O(h 2 ) convergence rate for meshes with appropriate
grading. Estimate (6.25) is proved by the Nitsche trick with additional ingredients, e.g.
include again some interpolant (cf. the proof of Lemma 5.1). For the proof in the conforming
case see e.g. [14].
7 Numerical experiments
We shall give some illustration of the Nitsche type mortaring in presence of some corner
singularity. In particular we investigate the rate of convergence when local mesh renement
is applied. Consider the BVP
in
@
where
is the L-shaped domain of Figure 5. The right-hand side f is chosen such that the
exact solution u is of the form
a
a
y
x

Figure

5: The L-shaped
domain
. Clearly, uj
@
3 and, therefore,
is satised. We apply the Nitsche type mortaring method to this BVP and
use initial meshes shown in Figure 6 and 7. The approximate solution u h is visualized in

Figure

9.

Figure

Triangulations with mesh ratio renement
(right).

Figure

7: Triangulations with mesh ratio renement
(right).
The initial mesh is rened globally by dividing each triangle into four equal triangles such
that the mesh parameters form a sequence fh
:g. The ratio
of the number of mesh segments on the mortar interface is given by
Figure 7). Furthermore, the values  are chosen, i.e., the trace
of the triangulation T 1
of
1 on the interface forms the partition E h
(for

Figure

5). For the examples the choice
su-cient to ensure stability. (For numerical
experiments with
and also with regular solutions, cf. [18]). Moreover, we also apply local
renement by grading the mesh around the vertex P 0 of the corner, according to section 6.
The parameter is chosen by
Let u h denote the nite element approximation according to (3.7) of the exact solution u
from (7.1). Then the error estimate in the discrete norm jj : jj 1;h is given by (6.24). We
assume that h is su-ciently small such that
holds with some constant C which is approximately the same for two consecutive levels of
h, like h; h
2 . Then (observed value) is derived from (7.2) by  obs := log 2 q h , where

. The same is carried out for the L 2 {norm, where ku u h k
Ch
is supposed. The values of  and  are given in Table 1 and Table 2, respectively.
mesh

Table

1: Observed convergence rates   for dierent pairs (h i , h i+1 ) of h-levels, for
and for
3 ) in the norm
mesh
=0:7 2.0093 2.0835 2.2252 2.0863 2

Table

2: Observed convergence rates   for dierent pairs (h i , h i+1 ) of h-levels, for
and for
in the norm jj : jj
.
The numerical experiments show that the observed rates of convergence are approximately
equal to the expected values. Furthermore, it can be seen that local mesh grading is suited
to overcome the loss of accuracy (cf. Figure 9) and the diminishing of the rate of convergence
on non-matching meshes caused by corner singularities.
number of elements
error
in
different
norms
1,h-norm
mr: 2:3
mr: 2:5
number of elements
error
in
different
norms
1,h-norm
mr: 2:3
mr: 2:511

Figure

8: The error in dierent norms on quasi-uniform meshes (left) and on meshes with
grading (right).
-0.50.500.20.4x
y
approximate
solution
-0.50.500.20.4y
x
approximate
solution
-22
x
y
pointwise
error
-22
x
y
pointwise
error

Figure

9: The approximate solution u h in two dierent perspectives (top), the local point-wise
error on the quasi-uniform mesh (bottom left) and the local pointwise error on the
mesh with grading (bottom right).



--R

Discontinuous Galerkin methods for elliptic problems.
Approximation of elliptic boundary value problems.


The Mortar
A new nonconforming approch to domain decomposition: The mortar element method.


A Multigrid Algorithm for the Mortar Finite Element Method.
Stabilization techniques for domain decomposition methods with non-matching grids

Elliptic Partial Di
Elliptic Problems in Nonsmooth Domains.
The Fourier- nite-element method for Poisson's equation in axisymmetric domains with edges
On adaptive grids in multilevel methods.




On some techniques for approximating boundary conditions in the
Mortaring by a Method of Nitsche.
An analysis of the


A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier.
--TR
corner singularities;finite element method;nitsche type mortaring;non-matching meshes;domain decomposition;mesh grading