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A multigrid method for nonconforming FE-discretisations with application to non-matching grids

20 Jul 1999-Computing (Springer Verlag)-Vol. 63, Iss: 1, pp 1-25
TL;DR: In this paper, a general scheme is proposed, which guarantees the approximation property of nonconforming finite element discretisations, and is applied to the discretisation by non-matching grids (mortar elements).
Abstract: Nonconforming finite element discretisations require special care in the construction of the prolongation and restriction in the multigrid process. In this paper, a general scheme is proposed, which guarantees the approximation property. As an example, the technique is applied to the discretisation by non-matching grids (mortar elements).

Summary (2 min read)

1. Introduction

  • Recently, domain decomposition methods have been applied to situations where subdomain meshes may be separately constructed and are non-matching along the interfaces.
  • When multigrid methods are designed, there is now the problem that the finite element spaces are not nested.
  • Therefore, the authors have to construct appropriate prolongation operators.
  • The approximation property for the convergence proof will be derived from an auxiliary problem.
  • After completing the paper, the authors learnt about the paper [11] investigating the mortar finite element method by other theoretical means.

3. Construction of the Prolongation

  • For the sake of simplicity, the authors omit these indices.
  • V` plays an important role when Stevenson [14] considers an axiomatic framework for the Cascadic multigrid algorithms suitable for nonconforming elements.

3.2. Conditions on π and σ

  • The space 6 will be equipped with the norm and the scalar product of H0, and S is assumed to become a Hilbert space by a norm ‖·‖S and a scalar product (·, ·)S whose specification may depend on the specific finite element space.
  • The introduction of the intermediate space S gives us more freedom in the construction of the mappings.
  • The authors emphasise that only boundedness in H0 is required for ι, while the concept of Brenner [8, 15] also refers to conditions with respect to the energy norm.

4. Coarse-Grid Correction and Approximation Property

  • The coarse-grid correction e`−1 ∈ V`−1 approximates the finite element function e` ∈ V`.
  • Note that it is required for converting the function w`−1 from V`−1 into an element of V`.

4.2. An Auxiliary Problem

  • To this end the authors introduce two Riesz representations of the residue.
  • After dividing by ‖g‖0 and inserting the estimate of r̄` above, the authors obtain the required inequality.

4.3. Approximation Property

  • The error enew` = φ` enew` from (4.5) after the coarse-grid correction will be estimated in the following proposition.
  • After inserting this estimate into (4.11) the proof is complete.

5. Smoothing Property and Multigrid Convergence

  • 1. Smoothing Property Specifically, the authors assume that the matrix.
  • The cases of A` not being positive definite or of other smoothing iterations are described in [12], too.

5.2. Multigrid Convergence

  • The iteration matrix of the two-grid iteration (with pre-smoothing, only) equals MTGM` (ν) = (A−1` − pA−1`−1 r)(A` Sν` ), where ν is the number of smoothing iterations described by the iteration matrix S`.
  • Replacing the exact solution of the coarse-grid problem by two iterations of the multigrid method, the authors obtain the W-cycle.
  • Since Lemma 5.1 implies (5.6), the multigrid convergence is proved.

6. Application to Non-Matching Grids

  • The authors describe and analyse a multigrid method for solving systems of algebraic equations arising from a finite element method based on non-matching triangulations.
  • The discretisation is done by the mortar technique, see [2, 3].
  • A multigrid method is presented and analysed which makes use of the general scheme presented in the previous sections.

6.1. Discrete Problem

  • In the mortar method nomenclature, this case is called geometrically conforming.
  • In particular, at a cross point there are several nodal points (xp, j(p)) with identical position xp but different j(p).
  • To define suitable spaces for discretisation of (6.1) the authors need to impose some constraints on the jumps of functions from X`( ) on 0ij which are called mortar 1 I maps the vertex into the index of the related subdomain.
  • Similarly, one considers the mesh T`,j ∩ δ̄m on δm.

6.3. Choice of 6, S, σ, and π

  • In the diagram below, V`−1 and V` are the mortar spaces introduced in Section 6.1.1.
  • After a slight modification at nodal points neighboured to the slave nodes, one obtains zero coefficients for all p ∈ Because of the Lagrange basis property, φ`v` recovers the function v` ∈ V`.
  • Hence, if the underlying second order boundary value problem satisfies the solvability and regularity conditions, all necessary requirements for the two- and multigrid convergence are satisfied.

7. Geometrically Nonconforming Case

  • This case is called the geometrically nonconforming case in the mortar method, and a typical situation is depicted in Fig 1.
  • The authors first formulate a discrete problem for (6.1) and then extend their treatment of the geometrically conforming case to this one.

7.1. Discrete Problem

  • To formulate the discrete problem, the authors will adapt some of their previous notations and introduce some new ones.
  • The edges 0j,k which are not masters are called slaves and denoted by δm, m ∈M.
  • The estimates of the consistency errors are local in nature and carry over without changes.
  • The authors goal is to design and analyse the multigrid method for solving (7.3) using the previous scheme.

7.2. Matrix Form

  • The basis functions b`,p are defined again by the Lagrange basis property similar to those in Section 6.
  • A` is symmetric and positive definite and its eigenvalues satisfy also the inequalities (6.10).

7.3. Two-Grid and Multigrid Method

  • This is sufficient to analyse the multigrid method.
  • Since ` is a variable level number, the space V`−1 and the mortar projection5`−1,m are defined as well.
  • Using this lemma, the authors check that Lemmata 6.2, 6.4 and 6.5 from Section 6 are also valid for the discussed case.

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Computing 63, 1–25 (1999)
c
Springer-Verlag 1999
Printed in Austria
A Multigrid Method for Nonconforming FE-Discretisations
with Application to Non-Matching Grids
D. Braess, Bochum, M. Dryja, Warsaw, and W. Hackbusch, Leipzig
Received October 15, 1998
Abstract
Nonconformingfiniteelementdiscretisationsrequire special care inthe constructionoftheprolongation
andrestrictioninthemultigridprocess. Inthispaper,ageneralschemeisproposed,whichguaranteesthe
approximationproperty. As an example, the technique is appliedto the discretisationby non-matching
grids (mortar elements).
AMS Subject Classifications: 65F10, 65N55, 65N30.
Key Words: Nonconforming finite element method, mortar, multigrid method.
1. Introduction
Recently, domain decomposition methods have been applied to situations where
subdomain meshes may be separately constructed and are non-matching along
the interfaces. The method was called mortar element method in [3]. When this
scheme is employed with finite elements, it may be considered as a nonconforming
method or as a mixed method.
In this paper, we will treat the mortar elements in the framework of nonconforming
methods, and we assume that the Lagrange multipliers have been eliminated as in
the setting of the second author [9]. When multigrid methods are designed, there
is now the problem that the finite element spaces are not nested.
Therefore, we have to construct appropriate prolongation operators. In the multi-
grid scheme for other elements, as, e.g., for the Crouzeix–Raviart elements in [5,
8] the L
2
-projectors could be chosen for the prolongations. We will abandon this
restriction and describe a more general framework which admits a lot of freedom
in the construction. In particular, a prolongation that is natural for the mortar el-
ements fits into our framework. The approximation property for the convergence
proof will be derived from an auxiliary problem. In essence, we will only assume
that an L
2
error estimate is known for the finite elements under consideration.

2 D.Braessetal.
In Section 2 we recall some notation for nonconforming finite elements. Section
3 is concerned with an extension of the prolongation operators which admits in
Section 4 to derive the central approximation property from an L
2
error estimate.
In Section 5 the associated smoothing property and the convergence is discussed.
Section 6 provides the application to mortar elements in the geometric conforming
case. We conclude with a generalisation to geometric nonconforming meshes.
After completing the paper, we learnt about the paper [11] investigating the mortar
finite element method by other theoretical means.
2. Multigrid Transfer
2.1. Variational Problem
Weconsider a variational problem of the followingform. Let
H
1
bea Hilbert space.
Given a bilinear form a(·,·) on
H
1
× H
1
and a functional f H
1
:= (H
1
)
0
,we
look for a solution u
H
1
of
a(u, w) = f(w) for all w
H
1
. (2.1)
Let
H
1
H
0
H
1
be the Gelfand triple, e.g., H
1
:= H
1
0
(), H
0
:= L
2
(), and H
1
:= H
1
().
In addition, we need a space
H
2
H
1
(e.g., H
2
= H
2
() H
1
0
()). The norms
of
H
k
are denoted by
k
·
k
k
. The scalar product in H
0
is written as (·, ·)
0
.
We assume:
(i) Solvability: For all f
H
1
, (2.1) has a unique solution u H
1
with
k
u
k
1
C
k
f
k
1
.
(ii) Regularity: If f
H
0
, (2.1) has a solution u H
2
with
k
u
k
2
C
k
f
k
0
.
2.2. Nonconforming Discretisation
Let V
H
0
for = 0, 1,... be a sequence of (nonconforming) finite element
spaces, i.e., we do not assume that the spaces are nested. Instead of the bilinear
forma(·, ·) a mesh-dependent bilinear form a
(·, ·) on V
×V
is used. For f H
0
,
(2.1) is discretised by
u
V
with a
(u
,w
)=f(w
) for all w
V
. (2.2)
We assume that also (2.2) is solvable and that the error estimate
k
u u
k
0
C
e
h
2m
k
u
k
2
(2.3)

A Multigrid Method for Nonconforming FE-Discretisations and Non-Matching Grids 3
holds, cf. Braess [4, p. 102], Hackbusch [13, (8.4.15b)]. Here, 2m is the order of
the differential operator, i.e.,
H
1
is a subspace of H
m
(). As usual, h
is the size
of the finite element mesh of V
.
Together with the regularity assumption
k
u
k
2
C
k
f
k
0
from above, we obtain
k
u u
k
0
C
0
h
2m
k
f
k
0
. (2.4)
2.3. Matrix Representation
Let
b
,i
: i I
be a basis of V
, where I
is the corresponding index set (e.g.,
the set of nodal points). The coefficient vector space
R
I
is denoted by U
.The
vectors in
U
are u
= (u
,i
)
iI
, and U
will be equipped with the usual Euclidean
norm k·k
U
(scaled by a suitable factor to ensure (2.6) below), so that the adjoint
mappings are given by the transposed matrices (maybe up to a fixed factor).
The isomorphism between
U
and V
is denoted by φ
:
φ
: U
V
with u
= φ
u
:=
X
iI
u
,i
b
,i
. (2.5)
The finite element matrix A
corresponding to a
(·, ·) has the coefficients a
,ij
=
a
(b
,j
,b
,i
). The variational problem (2.2) is equivalent to
A
u
= f
with
f
= φ
f, i.e., f
,i
= f(b
,i
) = (f, b
,i
)
0
.
As mentioned above, after a suitable scaling we require the equivalence of the
Euclidean norm k·k
U
and the H
0
-norm:
1
C
φ
1
kv
k
U
≤kv
k
0
C
φ
kv
k
U
for all v
= φ
v
. (2.6)
2.4. General Concept for the Multigrid Prolongation
Main ingredients of the multigrid algorithm are the prolongation
p :
U
1
U
and the restriction r = p
: U
U
1
.

4 D.Braessetal.
In the case of a conforming finite element discretisation with a finite element
hierarchyV
0
... V
1
V
,oneobtainsthefollowingcommutativediagram:
V
1
inclusion
ι
V
φ
1
x
x
φ
U
1
−→
p
U
.
In this case, the canonical prolongation is given by p = φ
1
φ
1
.
In the following we will admit V
1
6⊂ V
, and the inclusion is to be replaced by
a suitable mapping
ι : V
1
V
. (2.7)
Once ι has been given, we are able to define the prolongation and restriction by
p = φ
1
ι φ
1
and r = p
= φ
1
ι
)
1
. (2.8)
In the next section, we will propose a general construction of ι leading to the
approximation property
kA
1
pA
1
1
rk
U
U
C
A
h
2m
, (2.9)
which is an essential sufficient condition for the multigrid convergence (cf. Hack-
busch [12, §6.1.3]).
3. Construction of the Prolongation
3.1. Spaces and S
Although the algorithm needs only the mapping ι : V
1
V
(cf. (2.7)),
the theoretical consideration will lead to a variational problem (4.7) on the sum
V
1
+ V
and require ι to be defined and bounded on V
1
+ V
(or on a larger
space). Since ι : V
V
has to be the identity, we must construct a bounded
mapping ι : V
1
V
such that its restriction to V
1
V
is the identity.
In order to make the metric structure of the sum more transparent we will refer to
a (possibly larger) space with
V
1
+ V
H
0
.

A Multigrid Method for Nonconforming FE-Discretisations and Non-Matching Grids 5
The space and the space S defined below belong to the index pair ( 1,), and
1,
and S
1,
would be a more precise notation. For the sake of simplicity,
we omit these indices.
Here we also note that the sum V
1
+ V
plays an important role when Steven-
son [14] considers an axiomatic framework for the Cascadic multigrid algorithms
suitable for nonconforming elements.
Next, we need an auxiliary space S, which is connected with and V
via the
mappings σ and π , as shown in the following commutative diagram:
σ
−→ S
inclusion
x
-
y
π
V
1
−→
ι
V
φ
1
x
x
φ
U
1
−→
p
U
.
The desired mapping ι (more precisely, its extension to ) is the product
ι = π σ : V
. (3.1)
Before we will discuss the characteristic requirements concerning π, σ, and ι in
the next subsection, for elucidating the formalism, we specify the spaces and map-
pings for the Crouzeix-Raviart element, i.e., for the simplest nonconforming finite
elements (cf. Braess-Verf
¨
urth [5]).
Example 1. Let
T
1
be the coarse triangulation of the domain , while T
is
obtained by regular halving of all triangle sides. V
is the space of all piecewise
linear functions which are continuous at the midpoints of edges in
T
. Define the
nodal point set
N
by all midpoints of edges in T
(except boundary points in the
case of Dirichlet conditions). For all α
N
, basis functions b
,α
V
are defined
byb
,α
) = δ
αβ
(α, β N
) with theKronecker symbol δ. Then, U
:=
2
(N
) is
the coefficient space which is mapped by φ
: (c
,α
)
αN
7→ u
=
P
αN
c
,α
b
,α
onto V
. Similarly, V
1
, U
1
, and the isomorphism φ
1
are defined.
An appropriate space is the space of piecewise linear elements with respect to the
fine triangulation
T
that may be discontinuous at the edges of this triangulation.
Obviously, V
1
+ V
.
We set S :=
U
, π := φ
, and define σ as follows: Every nodal point α N
is the

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Frequently Asked Questions (1)
Q1. What are the contributions in "A multigrid method for nonconforming fe-discretisations with application to non-matching grids" ?

In this paper, a general scheme is proposed, which guarantees the approximation property.