Document [original]

TECHNISCHE UNIVERSIT¨

AT BERLIN

Moving Dirichlet Boundary Conditions

Robert Altmann

Preprint 2013/12

Preprint-Reihe des Instituts f¨ur Mathematik

Technische Universit¨at Berlin

http://www.math.tu-berlin.de/preprints

Preprint 2013/12 September 2013

MOVING DIRICHLET BOUNDARY CONDITIONS

R. ALTMANN†

Abstract. This paper develops a framework to include Dirichlet boundary conditions

on a subset of the boundary which depends on time. In this model, the boundary

conditions are weakly enforced with the help of a Lagrange multiplier method. In order

to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz

transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced.

An inf-sup condition as well as existence results are presented for a class of second order

initial-boundary value problems.

For the semi-discretization in space, a finite element scheme is presented which satisfies

a discrete stability condition. Because of the saddle point structure of the underlying

PDE, the resulting system is a DAE of index 3.

Key words. Dirichlet boundary conditions, operator DAE, inf-sup condition, wave equation

AMS subject classifications. 65J10,65M60,65M20

1. Introduction

Consider an initial-boundary value problem as it arises in many applications in which

a dynamic behavior is modeled. No matter which particular problem is analyzed, initial

and boundary values are needed in order to obtain a well-posed problem. Although the

boundary conditions may depend on time, the part of the boundary on which they are

specified is usually fixed. In this paper, we analyse Dirichlet boundary conditions on a

time-dependent boundary part ΓD(t). A simple example is shown in Figure 1.1. Therein,

an elastic body Ω is coupled through Dirichlet boundary conditions with a spring damper

system, which moves to the right with a given speed v0.

Ω

ΓD(t)

Figure 1.1. Example of a flexible body Ω coupled with a spring damper

system along ΓD(t)⊆∂Ω. The spring damper system moves to the right

with speed v0. In addition, a force Facts upwards on the spring damper.

Date: September 24, 2013.

The author’s work was supported by the ERC Advanced Grant ”Modeling, Simulation and Control of

Multi-Physics Systems” MODSIMCONMP and the Berlin Mathematical School BMS.

This is the pre-peer reviewed version of the following article: R. Altmann, Moving Dirichlet Boundary Con-

ditions, ESAIM Math. Model. Numer. Anal. (M2AN) 48 (2014), no. 6, 1859–1876. http://www.esaim-

m2an.org/articles/m2an/abs/2014/06/m2an140022/m2an140022.html.

In a more general framework, the here presented model can be used to couple problems

from different physics or to model flexible multibody systems [GC01, Sha05]. Consider

for example the pantograph and catenary dynamics [PEM+97] analyzed in [AS00]. This

one-dimensional benchmark problem contains a coupling of partial differential equations

(PDE) and differential-algebraic equations (DAE). The critical part of this model is the

contact between pantograph and catenary to achieve the transmission of electrical energy.

In the mentioned model [AS98, AS00], the contact is modeled by unilateral constraints

and thus, actually given by inequalities. This can be treated by slack variables or barrier

functions [CCFR07]. Then, the contact constraint is given in the form

u(xp(t), t) = g(t).

Therein, xp(t) is the position of pantograph, g(t) its height, and uthe deformation of the

wire. Because of well-known embedding theorems [Eva98, Ch. 5], point constraints are only

well-defined for one-dimensional problems. But even in this case they lead to numerical

instabilities such that the contact constraint is typically modeled via a regularized point

constraint including a regularized delta distribution.

Assume that we model the wire of the catenary in a more detailed way, for example

by a two-dimensional model. In this case, the pantograph has to be in contact with

the boundary of the wire. Since the train is moving with a certain speed, the coupling

constraint has the form of a moving Dirichlet boundary condition. Note that the example

in Figure 1.1 is a strongly simplified model of the pantograph and catenary system. The

spring damper system represents the pantograph which acts with a force Fupwards to

stay in contact with the wire.

In the here presented model, the moving Dirichlet boundary conditions are incorporated

in form of a weak constraint via the Lagrange multiplier method [Bab73, BG03, Bra81].

Since the boundary conditions are intended to model coupling constraints, they should not

be included in the ansatz space of the deformation, as suggested in many PDE text books,

e.g. [BS08, Ch. 5.4]. This already accounts for the coupling of flexible bodies through

fixed Dirichlet boundaries since the deformation along the boundary may depend on the

motion of adjacent bodies [Sim06, Sim13]. This modeling procedure leads to a dynamic,

also called transient, saddle point problem. The structure is then similar to the ansatz

used for mortar methods [BB99].

In this paper, we aim to formulate a framework to incorporate Dirichlet conditions on

moving boundary parts. Since we enforce the boundary conditions as a weak constraint,

we require a suitable ansatz space for the Lagrange multiplier. In order to avoid a time-

dependent ansatz space, the model is based on a bi-Lipschitz transformation of the moving

Dirichlet boundary. With this transformation, we can introduce a constraint operator

which satisfies the usual stability condition. This allows to formulate existence results of

solutions for the resulting constrained operator system.

Because of the saddle point structure, capable finite element spaces for the discretization

in space lead to DAEs of (differentiation) index 3. For a definition and a review of the

various index concepts of DAEs, we refer to [KM06, Ch. 1.2].

The paper is organized as follows. In Section 2 a time-dependent bi-Lipschitz trans-

formation is introduced, which maps a fixed interval onto the moving Dirichlet boundary

part. With this transformation we can formulate the constrained operator equations of

motion. The section ends with a discussion on the existence of solutions for second order

initial-boundary value problems and in particular for the linear wave equation.

The spatial discretized equations are subject of Section 3. We apply piecewise linear and

globally continuous finite elements combined with edge-bubble functions. Together with

a piecewise constant discretization of the Lagrange multiplier, this yields under certain

conditions a stable discretization scheme in the sense of a discrete inf-sup condition. In

Section 4 we close with some concluding remarks.

Throughout this paper, we write v|∂Ωfor the image of the trace operator applied to

v∈H1(Ω), which has a well-defined trace [Ste08, Ch. 2.5]. Furthermore, we write A.B

if there exists a generic constant c > 0 with A≤cB. This constant is independent of the

mesh-size and time. Finally, for an edge Ewe denote its length by |E|.

2. Continuous Model

Thinking primarily about problems from elastodynamics, we discuss second order initial-

boundary value problems. Nevertheless, the presented method can be applied to first order

systems as well. Consider the second order initial-boundary value problem in operator form

M¨u(t) + D˙u(t) + Ku(t) = F(t)

for t∈(t0, T] with initial conditions for u(t0),˙u(t0) and Dirichlet boundary conditions of

the form

u(t) = uD(t) on ΓD(t)⊂∂Ω.(2.1)

In the dynamics of elastic media, the operator Mincludes the density of the investigated

material. The operator Kincorporates the stiffness, i.e., a possibly nonlinear material law,

and Da viscous damping term. This framework includes the wave equation, vibrating

membranes [LS65] as well as examples from nonconvex elastodynamics modeling shape

memory alloys [Eˇ

S12].

As mentioned in the introduction, we include the Dirichlet boundary condition (2.1)

on the time-dependent boundary part ΓD(t) in form of a constraint since uDmight be

unknown a priori [Sim06]. In operator form, the Dirichlet boundary condition reads

B(t)u(t) = G(t) with the linear operator B(t) defined in Section 2.4 below. Adding this

constraint to the system, we have to introduce a Lagrange multiplier [Bab73]. The deriva-

tion of a suitable ansatz space is subject of the following subsection.

In the sequel, we denote the Sobolev space on a domain Dof order αby Hα(D), see

[AF03] for an introduction. The corresponding norm is denoted by k·kα,D. This includes

the L2-norm (which equals H0) as well as negative norms,

k·kα,Ω:= k·kHα(Ω),k·k0,Ω:= k·kL2(Ω),k·k−1/2,Γ:= k·k[H1/2(Γ)]∗.

2.1. Preliminaries. Let Ω ⊂Rndenote an open, bounded, and connected domain with

Lipschitz boundary ∂Ω [Bra07, Ch. I]. We assume that the time-dependent part of the

boundary on which we have Dirichlet boundary conditions has positive measure and is

denoted by ΓD(t)⊂∂Ω. Furthermore, we assume that ΓD(t) changes continuously in

time which may include a change of length.

Depending on the underlying initial-boundary value problem, the solution is a time-

dependent mapping from Ω to Rd. Considering the wave equation, we seek for the velocity

u(t): Ω →R, i.e., d= 1. In the case of elastodynamics, the unknown is the deformation in

every space direction and thus, d=n. As search space for the deformation (respectively

velocity) we introduce the space of square integrable functions in dcomponents, which

also have a square integrable weak derivative,

V:= H1(Ω)d.

By V∗we denote its dual space. In order to shorten notation, we introduce the space

H:= L2(Ω)d.

Note that V,H,V∗form a Gelfand triple [Zei90, Ch. 23.4]. As a consequence, v∈ V is

embedded in V∗such that for all w∈ V,

hv, wiV∗,V= (v, w)H.

Therein, h·,·iV∗,Vdenotes the duality pairing of V∗,Vand (·,·)Hthe inner product in H.

Furthermore, we have the continuous embedding [Zei90, Ch. 23.6],

u∈L2(t0, T;V)|˙u∈L2(t0, T;V∗),→C([t0, T]; H).

It remains to find a suitable ansatz space of the Lagrange multiplier. If the Dirichlet

boundary is independent of time, the natural ansatz space for the Lagrange multiplier is

the dual space of the broken Sobolev space H1/2(ΓD), see [Sim06, Alt13a]. For a definition

of broken Sobolev spaces, we refer to [AF03]. However, the right choice for the dynamic

case is not clear since a direct adoption would lead to the dual space of H1/2(ΓD(t)) which

is time-dependent. Such a space would cause difficulties in the modeling process and also

within the discretization procedure.

A solution to this problem is the introduction of a bi-Lipschitz transformation which

maps a fixed, i.e., time-independent, (n−1)-dimensional domain Iwith positive Lebesgue

measure onto the Dirichlet boundary ΓD(t). More details on needed assumptions are given

in the following subsection. We then define the Lagrange multiplier on I. For this, we

define a Hilbert space Qvia its dual space,

Q∗:= H1/2(I)d.

Remark 2.1.Since Q∗is densely embedded in [L2(I)]d, the three spaces Q∗,[L2(I)]d,Q

form a Gelfand triple. Thus, the duality pairing of Q∗,Qis densely defined by the L2

inner product on I.

2.2. Bi-Lipschitz Transformation. This section is devoted to the transformation which

maps the time-independent (n−1)-dimensional domain Ionto ΓD(t). Clearly, the trans-

formation has to be time-dependent. We introduce

Φ: [t0, T]×Rn→Rn

(2.2)

and require the following properties. For every t∈[t0, T ] we assume Φ(t): Rn→Rnto

be a bi-Lipschitz transformation, i.e., the function is bijective and Φ(t) as well as Φ−1(t)

are Lipschitz continuous. Thus, by Rademacher’s theorem [Eva98, Ch. 5.8], Φ(t) and its

inverse are differentiable a.e. in Rn. We denote this derivative by D Φ(t) and assume

that it is (w.r.t. a n-dimensional measure) a.e. uniformly bounded in t, i.e., there exist

constants 0 < cΦ< CΦ<∞with

cΦ≤ |det D Φ(t)| ≤ CΦ.(2.3)

Clearly, also |det D Φ−1(t)|is a.e. bounded by the constants C−1

Φand c−1

Φ. In particular,

we assume (2.3) to hold (w.r.t. a (n−1)-dimensional measure) a.e. on I. Since the

Dirichlet boundary ΓD(t) moves continuously with respect to time, we assume that Φ is

continuous in tas well. Recall that we introduce a time-dependent transformation in order

to map Ionto ΓD(t). Therefore, we assume that

Φ(t)|I:I→ΓD(t)

is onto. The last requirement for Φ concerns the inverse image of Ω. Since Φ(t) is bijective,

we are able to define Σ(t) as the domain satisfying

Φt, Σ(t)= Ω.

For fixed t∈[t0, T ], we may also define the Sobolev space on Σ(t), [H1(Σ(t))]d. Since Iis

mapped onto ΓD(t)⊂∂Ω and thus I⊂∂Σ(t), the inverse trace theorem [Ste08, Th. 2.22]

gives a continuous map of the form

Q∗→H1Σ(t)d.

The involved continuity constant depends on the domain Σ(t) and therefore on time.

Assumption 2.1 (Uniform inverse trace constant).Let CinvTr(Σ(t)) denote the continuity

constant given by the inverse trace theorem with respect to Q∗and [H1(Σ(t))]d. We

assume that these constants are uniformly bounded in tby a constant CinvTr, i.e., for all

t∈[t0, T] we have that

CinvTr(Σ(t)) ≤CinvTr .(2.4)

Remark 2.2.Assumption 2.1 is certainly fulfilled if Σ(t) is independent of t, i.e., the inverse

image of Ω under Φ(t) is a fixed domain. The same is true in the case where Σ(s) and

Σ(t) only differ by a translation for all s, t ∈[t0, T].

Theorem 2.1 (Bi-Lipschitz equivalence).Consider two domains Σ,Ωand a bi-Lipschitz

transformation Φwith Φ(Σ) = Ω and Φ(ΓΣ)=ΓΩfor boundary parts ΓΣ⊆∂Σand

ΓΩ⊆∂Ω. Then, the operators

(a) A1:H1(Ω) →H1(Σ)

u7→ u◦Φ,

(b) A1/2:H1/2(ΓΩ)→H1/2(ΓΣ)

q7→ q◦Φ,

and their continuous extension

γ7→ γ◦Φ

are bounded and have bounded inverses.

Proof. (a) The proof of the first claim is given in [Neˇc67, Ch. 2, Lem. 3.2], see also

[GGKR02]. We denote the operator norms by kA1kand kA−1

1k. The corresponding

transformation formula is stated in [EG92, Ch. 3.3, Th. 2].

(b) We show the boundedness of A1/2. Note that we can write the operator in terms of

A1and trace operators,

H1/2(ΓΩ)inverse

−−−−→

trace H1(Ω) A1

−−→ H1(Σ) trace

−−−→ H1/2(ΓΣ).

Then, the boundedness of A1, the trace operator [Ste08, Th. 2.21], and the inverse trace

operator [Ste08, Th. 2.22] imply

kA1/2k ≤ Ctr(Ω)kA1kCinvTr(Σ).

remains true for f∈H1/2(ΓΩ), i.e.,

ˆΓΣ

fΦ(x)dx=ˆΓΩ=Φ(ΓΣ)

f(y)|det D Φ−1(y)|dy.(2.6)

As an extension of A1/2, the operator A−1/2is defined for γ∈[H1/2(ΓΩ)]∗as the limit

A−1/2γ= lim

j→∞ A1/2γj

for a sequence {γj} ⊂ H1/2(ΓΩ) with γj→γin [H1/2(ΓΩ)]∗. By the transformation

formula (2.6) and part (b) of this theorem, we obtain

kA−1/2γjk−1/2,ΓΣ=kγj◦Φk−1/2,ΓΣ

= sup

p∈H1/2(ΓΣ)´ΓΣ(γj◦Φ) ·pdx

kpk1/2,ΓΣ

≤ kA−1

1/2ksup

p∈H1/2(ΓΣ)´ΓΣγj·(p◦Φ−1)◦Φ dx

kp◦Φ−1k1/2,ΓΩ

=kA−1

1/2ksup

q∈H1/2(ΓΩ)´ΓΣγj·q◦Φ dx

kqk1/2,ΓΩ

=kA−1

1/2ksup

q∈H1/2(ΓΩ)´ΓΩγjq|det D Φ−1|dy

kqk1/2,ΓΩ

=kA−1

1/2k kγj|det D Φ−1|k−1/2,ΓΩ

≤ kA−1

1/2kc−1

Φkγjk−1/2,ΓΩ.

Thus, the operator A−1/2is bounded with constant kA−1

1/2kc−1

Φ. The boundedness of the

inverse operator A−1

−1/2follows by the same arguments. 

The shown bi-Lipschitz equivalence from Theorem 2.1 is one of the main properties to

proof the stability of the boundary constraint. The definition of the constraint operator

and the proof of the inf-sup stability is subject of the remaining two subsections.

2.3. Continuous Inf-Sup Condition. In order to include the boundary conditions as

a weak constraint, we need a bilinear form which is defined on the moving boundary part

ΓD(t). For this, we introduce for t∈[t0, T ],

b(·,·;t): V ×Q → R.

With v∈ V, the bilinear form bis densely defined, i.e., for q∈[L2(I)]d, by

b(v, q;t) := ˆΓD(t)

v·q◦Φ−1(t)dx.(2.7)

Note that bis well-defined because of part (c) of Theorem 2.1. In the case of a fixed

Dirichlet boundary, i.e., Φ(t, x) = Φ(x) = xand ΓD(t) = ΓD, the bilinear form bis

independent of time and equals the bilinear form used in that setting [Sim06, Alt13a].

Remark 2.3.We could equivalently define bwith the transformation term |det D Φ−1(t)|.

This corresponds to a scaling of the Lagrange multiplier and is used in [Alt13b] to model

flexible multibody systems.

One important property of the bilinear form bis the so-called inf-sup, LBB, or stability

condition [Bra07, Ch. III.4]. In the fixed boundary case, the inf-sup condition is easy to

show with the help of the inverse trace theorem [Ste08, Th. 2.22]. The proof uses the fact

that the dual of the ansatz space for the Lagrange multiplier equals the space of traces

of V. The situation in the time-dependent case is slightly changed since Q∗contains the

traces of the transformed functions of V. However, the proof of the stability condition of

bfollows the same ideas [Ste08, Lem. 4.7].

Lemma 2.2 (Inf-sup condition).Let Φbe the time-dependent bi-Lipschitz transformation

from (2.2) satisfying (2.3) and Assumption 2.1. Then, the bilinear form bfrom (2.7)

satisfies an inf-sup condition, i.e., there exists a positive constant βsuch that for all

t∈[t0, T],

inf

q∈Q

sup

v∈V

b(v, q;t)

kvkVkqkQ≥β > 0.

Proof. Let t∈[t0, T ] be arbitrary but fixed and let Σ(t) be the inverse image of Ω under

Φ(t). Consider an arbitrary element q∈ Q. Since the determinant of D Φ(t) is bounded, it

follows that (q|det D Φ(t)|)∈ Q. According to the Riesz representation theorem [Ste08,

Th. 3.3], there exists an element w(t)∈ Q∗such that for all v∈ Q∗,

w(t), vQ∗=q|det D Φ(t)|, vQ,Q∗.

Therein, (·,·)Q∗denotes the inner product in Q∗and h·,·iQ,Q∗the duality pairing given by

the Gelfand triple from Remark 2.1. In addition, it holds that kw(t)kQ∗=kq|det D Φ(t)|kQ.

By the inverse trace theorem, there exists an extension of w(t) on the domain Σ(t). This

extension, namely v(t)∈[H1(Σ(t))]d, satisfies

v(t)|I=w(t)

and, because of the uniform bound in tby (2.4),

kv(t)k1,Σ(t)≤CinvTr(Σ(t)) kw(t)kQ∗≤CinvTr kw(t)kQ∗.

By the first part of Theorem 2.1, the transformation of v(t) satisfies ¯v(t) := v(t)◦Φ−1(t)∈

V. Thus, we can insert ¯v(t) into the bilinear form band obtain by a sequence {qj} ⊆

[L2(I)]dwith qj→qin Qand the transformation formula (2.6),

b¯v(t), q;t

k¯v(t)kV

= lim

j→∞

b¯v(t), qj;t

k¯v(t)kV

= lim

j→∞ qj|det D Φ(t)|, v(t)Q,Q∗

k¯v(t)kV

=q|det D Φ(t)|, v(t)Q,Q∗

k¯v(t)kV

=q|det D Φ(t)|, w(t)Q,Q∗

k¯v(t)kV

=kw(t)k2

Q∗

k¯v(t)kV

With the first part of Theorem 2.1 and the inverse trace theorem, the norm of ¯v(t) is

bounded by

k¯v(t)kV.kv(t)k1,Σ(t)≤CinvTrkw(t)kQ∗.

Furthermore, we can bound kw(t)kQ∗from below with (2.3) by

kw(t)kQ∗=kq|det D Φ(t)|kQ≥cΦkqkQ.

All together, we yield the time-independent estimate

sup

u∈V

b(u, q;t)

kukV≥b(¯v(t), q;t)

k¯v(t)kV

&kw(t)kQ∗

CinvTr ≥cΦ

CinvTr kqkQ.

2.4. Saddle Point Formulation. With the bilinear form bfrom (2.7) we are in the

position to enforce the Dirichlet boundary conditions in a weak form. The needed Lagrange

multiplier is defined on the time-independent domain I, as described in Section 2.1. The

weak formulation in operator form reads: find u∈L2(t0, T;V) with sufficiently smooth

time derivatives and λ∈L2(t0, T;Q) such that

M¨u(t) + D˙u(t) + Ku(t) + B∗(t)λ(t) = F(t) in V∗,(2.8a)

B(t)u(t) = G(t) in Q∗

(2.8b)

for a.e. t∈[t0, T] with initial conditions

u(t0) = g∈ V,(2.8c)

˙u(t0) = h∈ H.(2.8d)

Therein, Fincludes the applied forces and Gcontains the Dirichlet data uD∈H2(t0, T;V)

and therefore the boundary conditions,

G(t) := b(uD,·;t)∈ Q∗

The time-dependent operator B(t): V → Q∗and its dual B∗(t): Q→V∗are defined via

the bilinear form bfrom (2.7),

B(t)u:= b(u, ·;t)∈ Q∗,B∗(t)λ:= b(·, λ;t)∈ V∗.

Since (2.8) is an operator DAE, i.e., a DAE in an infinite dimensional setting, the initial

values have to satisfy a consistency condition of the form B(t0)g=G(t0). The precise

definitions of the operators M,D, and Kdepend on the given problem. Note that the

time derivatives in (2.8) should be understood in the generalized sense [Zei90, Ch. 23.5].

An advantage of the formulation (2.8) is the time-independence of the spaces in which

the equations are stated. It remains to show that the operator DAE (2.8) is well-posed in

the sense that it is solvable if the corresponding PDE, constrained by Dirichlet boundary

conditions on ΓD(t), is solvable.

Theorem 2.3 (Existence of the Lagrange multiplier).Consider operators M,D, and K

with right-hand side F ∈ L2(t0, T;V∗). Further, let u∈L2(t0, T;V)be a solution of

M¨u(t) + D˙u(t) + Ku(t) = F(t),

where the test functions from Vvanish along ΓD(t)at time t, for given initial conditions

g, h and the constraint u(t) = uD(t)along ΓD(t)for a.e. t∈[t0, T]. Then, there exists a

unique λ∈L2(t0, T;Q)such that (u, λ)is a solution of (2.8).

Proof. The claim follows directly from the inf-sup condition of Lemma 2.2 together with

[Bra07, Th. III.3.6], see also [Alt13a, Th. 4.11]. 

It remains the question of the existence of a solution u. Therefore, we give a particular

existence result for the linear wave equation with moving Dirichlet conditions.

Example 2.1 (Linear wave equation).Consider the wave equation ¨u−∆u=f, i.e., M=

id, D= 0, and Kcorresponds to the Laplacian. Here, we assume that the transformation

Φ has a time-independent preimage Σ, i.e., Φ(t, Σ) = Ω. Besides, we assume Φ(t) and its

inverse to be continuously differentiable in tand ∇(det D Φ(t)) to be uniformly bounded

from above. Then, there exists a unique solution of (2.8), see Appendix A for a proof.

For a numerical example, for which these assumptions are satisfied, we refer to [Alt13b].

3. Semi-Discretized Model

In this section, we analyse the saddle point formulation (2.8) after semi-discretization in

space. For this, we restrict ourselves to the two-dimensional case. For the discretization we

use finite elements and need to introduce triangulations of Ω ⊆R2as well as I. In the two-

dimensional case, we may assume that I⊂Ris an interval. The presented discretization

scheme is stable in the sense that it satisfies a discrete inf-sup condition, which is crucial

to ensure stable approximations of the Lagrange multiplier.

3.1. Finite Element Scheme. Let Tbe a regular triangulation of Ω ⊆R2in the sense

of [Cia78], i.e., we exclude hanging nodes. Furthermore, we assume Tto be shape regular

[Bra07, Ch. II.5]. By TIwe denote a partition of the interval I. The set of edges of a

triangulation or partition is denoted by E(·).

ΓD(t)

∂Ω

Figure 3.1. Illustration of the closure ΓD(t) with respect to the triangu-

lation T.

In the sequel, we also need the partition of the moving boundary part which arises from

the restriction of Ton ΓD(t). This partition contains all edges of Twhich have a non-zero

intersection with ΓD(t) in a one-dimensional measure,

TΓ(t) := {E∈ E(T)|int(E)∩ΓD(t)6=∅}.

With respect to this partition, we define the ’closure’ of ΓD(t) by

ΓD(t) := [

E∈E(TΓ(t))

An illustrative picture of the closure is given in Figure 3.1. Clearly, it holds that ΓD(t)⊆

ΓD(t) and they are equal if and only if the endpoints of ΓD(t) are nodes of the triangulation

For the discretization in space, we introduce several finite element spaces. The space of

piecewise polynomials of degree one which are globally continuous is denoted by

Sh:= P1(T)∩C(Ω)d= span{ϕ1, . . . , ϕn1}⊂V.

Therein, ϕ1, . . . , ϕn1denote the standard hat-functions [Bra07, Ch. II] in dcomponents

and therefore a basis of Sh. Thus, the dimension of Shequals dtimes the number of

vertices in T, namely n1.

A second finite element space is given by edge-bubble functions as introduced in [Ver96,

Ch. 1]. Here, we only consider edge-bubble functions on the boundary and in particular

only edges which are part of ΓD(t) at some point in time. Let E1, . . . , Er∈ E(T) denote

these boundary edges, i.e.,

[

t∈[t0,T]

ΓD(t) =

[

j=1

Ej.

We define the space

Bh:= span{ψ1, . . . , ψn2}⊂V

where n2:= d·rand ψ1, . . . , ψn2denote the standard edge-bubble functions in dcompo-

nents for the rboundary edges. Note that the dimension n2of the space Bhis independent

ψE

Figure 3.2. Illustration of an edge-bubble function ψEcorresponding to

a boundary edge E. The support of ψEis given by the triangle T.

of time. We summarize some properties of edge-bubble functions, which are important for

later estimates. Recall that k·k0,T and k·k0,E denote the L2-norm on a triangle Tand

on an edge E, respectively.

Lemma 3.1 (Properties of edge-bubble functions).Let ψEdenote the edge-bubble function

for a boundary edge Eof length h=|E|and bordering triangle T, as shown in Figure 3.2.

Furthermore, let Ebe partitioned into two intervals E1, E2with α:= |E1|/h ≥1/2. Then,

(a) ´EψEdx= 2h/3,

(b) k∇ψEk0,T .h−1/2k1k0,E,

(d) ´E2ψEdx≤(1 −α)´EψEdx.

The involved constant in (b) only depends on interior angles of the triangle T.

Proof. The first two claims are taken from [Lip04, Lem. 2.3.1]. The third claim follows by

an easy calculation and the last claim follows directly from (c). 

As finite dimensional approximation of the space V, we use a combination of hat-

functions and edge-bubble functions on the boundary,

Vh:= Sh⊕Bh= span{ϕ1, . . . , ϕn1, ψ1, . . . , ψn2}.

The dimension of this space is given by n:= n1+n2. The ansatz space of the Lagrange

multiplier Qis approximated by the space of piecewise constant functions on the interval I.

For this, we introduce the functions χiwhich are constant along one edge of the partition

TIand vanish elsewhere. Since these ansatz functions are in [L2(I)]d, this provides a

discontinuous but still conforming discretization,

Qh:= P0(TI)d= span{χ1, . . . , χm} ⊂ Q.

The dimension of Qh, namely m, equals dtimes the number of edges in TI. At this point,

we assume m<n. As semi-discrete finite element approximations of uand λ, we define

uh(t, x) :=

j=1

qj(t)ϕj(x) +

j=1

qn1+j(t)ψj(x), λh(t, y) :=

j=1

µj(t)χj(y).

The introduced discretization scheme also determines the positive definite n-by-nmass

matrix M, the damping matrix D, and the stiffness matrix Kas discrete representations

of the operators M,D, and K, respectively [Gus08, Ch. 12]. For nonlinear operators, D

and Kmay be replaced by some nonlinear functions. With ϕn1+k:= ψkfor k= 1, . . . , n2,

the time-dependent m-by-ncoupling matrix B(t) is given by

B(t)ji := b(ϕi, χj;t) = ˆΓD(t)

ϕi·χj◦Φ−1(t)dx.

The described semi-discretization in space results in a DAE for the coefficient vectors

q= [qj] and µ= [µj],

M¨q(t) + D˙q(t) + Kq(t) + BT(t)µ(t) = f(t),

B(t)q(t) = g(t).

(3.1)

Because of the saddle point structure, the DAE (3.1) has index 3 if the matrix B(t) is

of full rank for all t∈[t0, T]. For a precise definition of the index of a DAE see [KM06,

Ch. 3.3]. Roughly speaking, the index gives the needed smoothness of the inhomogeneity

to guarantee a continuously differentiable solution.

In the following subsection, we present assumptions under which the discretization

scheme Vh−Qhfulfills a discrete inf-sup condition. The importance of this condition is

commented in [Bra07, Ch. III.4]. Such a condition also implies the full rank property of

B(t) for all t∈[0, T].

3.2. Discrete Inf-Sup Condition. In order to guarantee stability in the sense of a

discrete inf-sup condition, the triangulations TΓ(t) and TIhave to be compatible in the

sense that the partition of Iis not too fine. A more precise formulation is given in the

following assumption. Therein, hI∈L2(I) and hΓ∈L2(ΓD(t)) denote the piecewise

constant functions which involve the local mesh-sizes, i.e.,

hI|E(x) := |E|for E⊆I, hΓ|F(x) := |F|for F⊆ΓD(t).

Assumption 3.1 (Compatibility of TΓ(t) and TI).We assume that there exists a constant

0<ε<1/4, independent of t, such that

cΦhI◦Φ−1(t)≥(3/2 + ε)hΓ

(3.2)

is satisfied a.e. on ΓD(t) with constant cΦfrom (2.3). The condition ε < 1/4 is just

included in order to unify the computations below.

The assumption states that the mesh-size of TI, transformed to ΓD(t), should be larger

than the mesh-size of Talong the moving boundary. In addition TΓ(t) has to be quasi-

uniform in the following sense.

Assumption 3.2 (Quasi-uniformity of TΓ(t)).Let κdenote the largest ratio of two adjacent

edges in the partition TΓ(t). Then, we assume that κ≤2.

Remark 3.1.If the triangulation on the boundary TΓ(t) is uniform, i.e., κ= 1, then

Assumption 3.1 can be weakened to cΦ(hI◦Φ−1(t)) ≥(1 + ε)hΓ.

In preparation for the main result of this section, we need to construct for a given

function qh∈ Qha piecewise constant function γh∈[P0(TΓ(t))]dwhich is a good approxi-

mation of qh◦Φ−1(t). The construction of γhis only necessary for the analysis of the finite

element scheme and does not have to be computed in the actual simulation. To clarify

the notation, in the sequel we neglect the time dependence of Φ−1.

3.2.1. Construction of γh.Assume that t∈[t0, T] is arbitrary but fixed. Consider qh∈

P0(TI) and its transformed analogon qh◦Φ−1, which is also piecewise constant and hence

in L2(ΓD(t)). Without relabeling, we extend this function by zero such that qh◦Φ−1∈

L2(ΓD(t)). Note that qh◦Φ−1is piecewise constant but not necessarily with respect to

TΓ(t), i.e., qh◦Φ−16∈ P0(TΓ(t)).

Proposition 3.2. Under Assumption 3.1 or the weaker condition of Remark 3.1, the

piecewise constant function qh◦Φ−1can only take two different values on an edge E∈

E(TΓ(t)).

Proof. Suppose that qh◦Φ−1has more than two values on E. Then, there exists an

edge F∈ E(TI) with Φ(F)⊂Eand |Φ(F)|<|E|. Equation (2.3) then implies that for

x∈Φ(F),

cΦhI◦Φ−1(x) = cΦ|F|≤|Φ(F)|<|E|=hΓ(x)

which is a contradiction to Assumption 3.1 as well as Remark 3.1. 

We define the approximation of qh◦Φ−1in P0(TΓ(t)) edge-wise. For this, consider an

edge E∈ E(TΓ(t)) with a partition E=E1∪E2such that

qh◦Φ−1|E(x) = (αfor x∈E1,

βfor x∈E2.

(3.3)

Such a partition always exists because of Proposition 3.2. If qh◦Φ−1is constant on E,

then E2vanishes. With the help of this decomposition, we define γh∈ P0(TΓ(t)) by

γh|E:= (αif |E1| ≥ |E2|,

βotherwise.

(3.4)

Before we show that γhis a reasonable approximation of qh◦Φ−1, we define weighted

norms of L2(ΓD(t)) and L2(ΓD(t)),

k·k2

h:= X

E∈E(TΓ(t))

hEk·k2

L2(E∩ΓD(t)),k·k2

h,¯

Γ:= X

E∈E(TΓ(t))

hEk·k2

L2(E).

The approximation property of the constructed function γhin (3.4) is given in the following

lemma.

Lemma 3.3 (Approximation property of γh).Let ψEdenote the edge-bubble function

for an edge Eand consider qh∈ P0(TI)as well as Assumptions 3.1 and 3.2. Then, the

corresponding function γh∈ P0(TΓ(t)) defined in (3.4) satisfies

E∈E(TΓ(t))

hEˆE∩ΓD(t)qh◦Φ−1·γhψEdx≥ε

E∈E(TΓ(t))

hEˆE

γ2

hψEdx(3.5a)

and

kqh◦Φ−1kh≤√3kγhkh,¯

Γ.(3.5b)

Proof. Recall that 0 <ε<1/4 and that hE=|E|denotes the length of an edge E. With

the partition of Eas in (3.3), we distinguish two types of edges:

type 1: qh◦Φ−1is constant along E, i.e., E=E1,

type 2: qh◦Φ−1takes two different values, i.e., |E2| 6= 0 and |E1| ≥ hE/2.

Consider an arbitrary edge E∈ E(TΓ(t)) with E⊆ΓD(t). If Eis of first type, then

hEˆEqh◦Φ−1·γhψEdx=hEˆE

γ2

hψEdx.(3.6)

If Eis of second type, we obtain with parts (c) and (d) of Lemma 3.1 and Young’s

E1E2

D E F

Figure 3.3. An edge E, partitioned into E1and E2, with neighboring

edges Dand Fas in the proof of Lemma 3.3.

inequality 2ab ≤λa2+b2/λ for λ > 0 [Eva98, App. B] that

hEˆEqh◦Φ−1·γhψEdx=hEˆE1

α2ψEdx+hEˆE2

αβψEdx

≥hE

2ˆE

α2ψEdx−hE

2λˆE2

α2ψEdx−hEλ

2ˆE2

β2ψEdx

≥hE

42−1

λˆE

γ2

hψEdx−hFλκ2

4ˆF

γ2

hψFdx.

Thereby, Fdenotes the edge adjacent of E2as shown in Figure 3.3. With the choice

λ= 1/(κ−ε), we obtain an estimate of the form

hEˆEqh◦Φ−1·γhψEdx≥c1(ε)hEˆE

γ2

hψEdx−c2(ε)hFˆF

γ2

hψFdx.(3.7)

Because of Assumption 3.2, the constants c1and c2satisfy

c1(ε) = 1

42−κ+ε≥ε

4, c2(ε) = κ2

4(κ−ε).

For an edge Ewith E6⊆ ΓD(t), i.e., an edge with only one neighbor in ΓD(t) (recall

Figure 3.1), it holds that

ˆE∩ΓD(t)qh◦Φ−1·γhψEdx≥1

2ˆE

γ2

hψEdx.

We are now in the position to sum up all contributions which gives

E∈E(TΓ(t))

hEˆE∩ΓD(t)qh◦Φ−1·γhψEdx≥X

E∈E(TΓ(t))

cEhEˆE

γ2

hψEdx.

It remains to show cE≥ε/4 for all edges E∈ E(TΓ(t)). Because of (3.2), negative

contributions can only arise for edges of first type. Thus, it holds that cE≥c1(ε)≥ε/4

for edges Eof second type or E6⊆ ΓD(t). If Eis of first type, we distinguish two cases:

First, there is only one negative contribution coming from a neighboring edge in form of

(3.7). Then, with (3.6) we obtain the estimate

cE= 1 −c2(ε) = 1 −1

κ2

κ−ε≥1−1

2−ε>3

In the second case, we have two negative terms for the edge E, i.e., there are negative

contributions from both neighboring edges. We show that (3.2) then locally implies a

stricter bound on κ. Let Dand Fdenote the neighboring edges of Eas illustrated in

Figure 3.3 (here with E=E1) and D2,F2the adjacent parts, respectively. The restriction

of the mesh-size (3.2) implies |D2| ≤ |D|/2 = hD/2 and |F2| ≤ |F|/2 = hF/2. Locally,

the largest ratio of two adjacent edges is given by the maximum of the ratios hD/hE

and hF/hE. We assume w.l.o.g. that hD≥hFand thus, obtain the local edge ratio

κE:= hD/hE. Then, Assumption 3.1 implies

3/2 + εκEhE=3/2 + εhD< cΦ(hI◦Φ)|D2≤ |D2|+|E|+|F2| ≤ (1 + κE)hE.

Thus, κE<2/(1 + 2ε) which leads to

cE= 1 −2c2(ε)=1−1

κ2

κE−ε≥1−2

(1 + 2ε)(2 −ε−2ε2)≥ε

In total, this yields the stated estimate (3.5a).

For the second claim (3.5b), consider an arbitrary edge E∈ E(TΓ(t)). If qh◦Φ−1is

constant along E, then kqh◦Φ−1kL2(E)=kγhkL2(E). Otherwise, we distinguish between

the cases E⊆ΓD(t) and E6⊆ ΓD(t). In the first case, we have (w.l.o.g. |E2|≤|E1|)

kqh◦Φ−1k2

L2(E)=ˆE1

α2dx+ˆE2

β2dx≤ˆE

α2dx+κ

2ˆF

β2dx≤ kγhk2

L2(E∪F).

Therein, Fdenotes the neighboring edge of Eon which γhtakes the value β. For a

boundary edge, γheither equals the value of qh◦Φ−1,

kqh◦Φ−1k2

L2(E∩ΓD(t)) =kγhk2

L2(E∩ΓD(t)) ≤ kγhk2

L2(E)

or vanishes along E. Then, again with neighboring edge F,

kqh◦Φ−1k2

L2(E∩ΓD(t)) ≤ kγhk2

L2(E∪F).

The summation over all edges finally proves the claim. 

3.2.2. Proof of the discrete inf-sup condition. With the approximation γhfrom (3.4) in

hand, we are able to proof the stability condition of the discretization scheme Vh− Qh

introduced in Section 3.1.

Theorem 3.4 (Discrete inf-sup condition).Under Assumptions 2.1, 3.1 and 3.2, the

bilinear form bfrom (2.7) satisfies a discrete inf-sup condition w.r.t. the discrete spaces

Vhand Qh, i.e., there exists a positive constant βdisc(ε), independent of the mesh-sizes

and time, with

inf

qh∈Qh

sup

vh∈Vh

b(vh, qh;t)

kvhkVkqhkQ≥βdisc(ε)>0.

Proof. The proof basically works as for a fixed Dirichlet boundary [Lip04, Th. 2.3.7].

Nevertheless, the involved transformation requires several adjustments such that we give

the details here.

Consider an arbitrary qh∈ Qhwith kqhk−1/2,I = 1. As in [Lip04], we show the existence

of constants c1, c2, c3, which may depend on εbut not on the mesh-size or time, such that

(i) sup

vh∈Vh

b(vh, qh;t)

kvhkV≥c1kqh◦Φ−1(t)kh,(ii) sup

vh∈Vh

b(vh, qh;t)

kvhkV≥c2−c3kqh◦Φ−1(t)kh.

Since k·kh≥0, the claim then follows from

sup

vh∈Vh

b(vh, qh;t)

kvhkV≥max c1kqh◦Φ−1(t)kh, c2−c3kqh◦Φ−1(t)kh≥c1c2

c1+c2

In the proof we use generic constants which are independent of the mesh-size and time.

Furthermore, we neglect the time-dependence of variables.

Proof of (i): As described at the beginning of this subsection, qh◦Φ−1can be extended

by zero to a piecewise constant function on ΓD(t). In addition, γh∈[P0(TΓ(t))]ddenotes

the function defined componentwise as in (3.4). We define uh∈ Bh⊂ Vhby

uh:= X

E∈E(TΓ(t))

hE·γh|E·ψE

with edge-bubble function ψE. Inserting uhinto the bilinear form b, by Lemma 3.3 we

obtain

b(uh, qh;t) = ˆΓD(t)

uh·qh◦Φ−1dx

E∈E(TΓ(t))

hEˆE∩ΓD(t)

γh·ψE·qh◦Φ−1(t)dx

≥ε

E∈E(TΓ(t))

hEˆE

γ2

h·ψEdx=ε

E∈E(TΓ(t))

hEˆE

γ2

hdx=ε

6kγhk2

h,¯

Γ.(3.8)

In the following, we use part (b) of Lemma 3.1. For a boundary edge Ewe denote the

adjacent triangle by TE. By a Poincar´e-Friedrichs inequality (e.g. [PW60] for convex

domains), the H1-norm of uhis bounded by

kuhk2

V.k∇uhk2

0,Ω=X

E∈E(TΓ(t))

Ek∇(γhψE)k2

0,TE.X

E∈E(TΓ(t))

hEkγhk2

0,E =kγhk2

h,¯

Γ.

(3.9)

Together with the estimates (3.8) and (3.9), Lemma 3.3 yields

sup

vh∈Vh

b(vh, qh;t)

kvhkV≥b(uh, qh;t)

kuhkV≥ε

6kγhk2

h,¯

kuhkV

&εkγhkh,¯

Γ&εkqh◦Φ−1kh.

Proof of (ii): Note that Theorem 2.1 implies qh◦Φ−1∈[H1/2(ΓD(t))∗]dand

1 = kqhk−1/2,I ≤ kA−1/2k kqh◦Φ−1k−1/2,ΓD(t).

Thus, the norm of qh◦Φ−1is bounded from below. By the definition of the dual norm,

there exists a ˆq∈[H1/2(ΓD(t))]dwith kˆqk1/2,ΓD(t)= 1 such that

2kA−1/2k≤1

2kqh◦Φ−1k−1/2,ΓD(t)≤ˆΓD(t)

ˆq·qh◦Φ−1dx.(3.10)

Just as in [Lip04], we introduce ˜u(t) as weak solution of the Poisson equation

−∆˜u= 0 in Ω,˜u= ˆqon ΓD(t).

By ˜uhwe denote its Cl´ement interpolation [Cl´e75] in Sh. Standard stability estimates and

properties of the Cl´ement interpolation give

k˜uhk1,Ω.k˜uk1,Ω.kˆqk1/2,ΓD(t)= 1,(3.11)

k˜u−˜uhk0,E .h1/2

Ek˜uk1,ωE.(3.12)

Therein, ωEdenotes the set of triangles which have at least one common point with the

edge E. Inserting ˜uhinto the bilinear form b, we obtain by (3.10)-(3.12) and the Cauchy-

Schwarz inequality,

b(˜uh, qh;t) = ˆΓD(t)

˜u·(qh◦Φ−1(t)) dx+ˆΓD(t)

(˜uh−˜u)·qh◦Φ−1(t)dx

=ˆΓD(t)

ˆq·(qh◦Φ−1(t)) dx+X

E∈E(TΓ(t)) ˆE∩ΓD(t)

(˜uh−˜u)·(qh◦Φ−1(t)) dx

≥1

2kA−1/2k−X

E∈E(TΓ(t))

h−1/2

Ek˜u−˜uhk0,E∩ΓD(t)h1/2

Ekqh◦Φ−1k0,E∩ΓD(t)

≥c2−c3 X

E∈E(TΓ(t)) k˜uk2

1,ωE!1/2 X

E∈E(TΓ(t))

hEkqh◦Φ−1k2

0,E∩ΓD(t)!1/2

≥c2−c3k˜uk1,Ωkqh◦Φ−1kh.

Note that the constant c3only depends on the minimal interior angle of the triangulation

and kA−1/2k. The latter is independent of tbecause of Assumption 2.1. Thus, with (3.11)

we obtain the estimate

sup

vh∈Vh

b(vh, qh;t)

kvhkV≥b(˜uh, qh;t)

k˜uhkV

&c2−c3kqh◦Φ−1kh.

4. Conclusion

We have introduced a theoretical and numerical applicable framework to include Dirich-

let boundary conditions on moving boundary parts. By initiating a time-dependent bi-

Lipschitz transformation, we were able to formulate the dynamical system as a saddle

point problem within time-independent ansatz spaces. Although the proofs work with the

transformation of the entire domain, for practical computations it suffices to transform

the Dirichlet boundary. Because of the saddle point structure, the key for the analysis of

the continuous as well as the semi-discrete model is the verified inf-sup condition.

We have presented a spatial discretization scheme which is stable under some com-

patibility and quasi-uniformity condition. The compatibility assumption is necessary to

ensure that the number of constraints along the boundary is not larger than the number

of degrees of freedom.

Possible fields of application include flexible multibody dynamics. In this context,

Dirichlet boundary conditions may be used as dynamic coupling conditions. In order to

stay within this framework, the coupling surfaces have to be known beforehand. This

gives the main difference between this model and nonlinear contact problems. However,

the framework is not restricted to model interconnections of flexible bodies. The presented

model also allows to couple different kinds of physics.

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Appendix A. Proof of Example 2.1

The idea of the proof is to rewrite (2.8) such that a transformation by Φ leads to a

fixed Dirichlet boundary problem. As a consequence, the transformed operator ¯

Kwill be

time-dependent. The stated result then follows from the general framework in [Wlo87,

Ch. V]. In this part, we often neglect the time-dependence of the variables.

As a first step, we reformulate (2.8) with the operator ˆ

B(t): V → Q∗, which is defined

hˆ

B(t)u, qiQ∗,Q:= ˆ

b(u, q;t) := ˆΓD(t)

u·q◦Φ−1(t)|det D Φ−1(t)|dx.

To obtain an equivalent system, we have to adjust the right-hand side Gand λ, respectively.

Therefore, we introduce hˆ

G(t), qi:= ˆ

b(uD, q;t) and define ˆ

λ∈L2(t0, T;Q) by the one-to-

one relation ˆ

λ◦Φ−1|det D Φ−1|=λ◦Φ−1.

Thus, system (2.8) has a solution (u, λ) if and only if there exists a solution of

¨u(t) + Ku(t) + ˆ

B∗(t)ˆ

λ(t) = F(t) in V∗,(A.1a)

B(t)u(t) = ˆ

G(t) in Q∗

(A.1b)

with the same initial values.

In the second step, we transform the equations by Φ. For this, we use the variational

formulation of (A.1). The transformation formula applied to (A.1b) then yields

ˆI

(u◦Φ)qdx=hˆ

G(t), qifor all q∈ Q.

Introducing ϕ(t) := |det D Φ(t)|, which is bounded away from zero by (2.3), we obtain the

equivalent equation

ˆI

(u◦Φϕ1/2)qϕ−1/2dx=hˆ

G(t), qϕ−1/2ϕ1/2i=: h¯

G(t), qϕ−1/2ifor all q∈ Q.(A.2)

Defining ¯q:= qϕ−1/2∈ Q, we can equivalently test equation (A.2) for all ¯q∈ Q instead of

q∈ Q. Recall that Σ denotes the preimage of Ω under Φ, i.e., Φ(t, Σ) = Ω. Then, setting

a(t) := u(t)◦Φ(t)ϕ1/2(t)∈ VΣ:= [H1(Σ)]d(by Theorem 2.1), we obtain the equation

ˆI

a(t) ¯qdx=h¯

G(t),¯qifor all ¯q∈ Q

or, equivalently, in operator form with the time-independent trace operator ¯

B:VΣ→ Q∗,

Ba(t) = ¯

G(t) in Q∗.(A.3)

The transformation formula applied to the variational formulation of (A.1a) yields

dt2u◦Φϕ1/2, v ◦Φϕ1/2Σ+hKu, vi+ˆI

v◦Φˆ

λdx=hF(t), vifor all v∈ V.

Because of Theorem 2.1 and (2.3), the functions w:= (v◦Φ) ϕ1/2satisfy w∈ VΣ. The

bi-Lipschitz equivalence allows to test the equation with functions w∈ VΣinstead of

v∈ V. Since we test with all functions in VΣ, we may assume that the test functions are

independent of t. By a rescaling of ˆ

λ, which we denote by ¯

λ, and the introduction of a

new right-hand side ¯

F, we obtain the equivalent equation

dt2a(t), wΣ+h¯

K(t)a(t), wi+h¯

Bw, ¯

λi=h¯

F(t), wifor all w∈ VΣ,

or, as operator equation,

¨a(t) + ¯

K(t)a(t) + ¯

B∗¯

λ(t) = ¯

F(t) in V∗

Σ.(A.4)

Therein, the symmetric and time-dependent operator ¯

K(t): VΣ→ V∗

Σis defined by

¯

K(t)a, wV∗

Σ,VΣ

:= K(aϕ−1/2)◦Φ−1,(wϕ−1/2)◦Φ−1V∗,V.

Since we have only used the transformation Φ and a rescaling by the bounded function ϕ,

system (2.8) has a solution (u, λ) if and only if there exists a solution (a, ¯

λ) of (A.3)-(A.4).

In the last step we show that system (A.3)-(A.4) has a unique solution. Since the trace

operator ¯

Bsatisfies an inf-sup condition, by [Bra07, Th. III.3.6] it is sufficient to show

that the operator equation

¨a(t) + ¯

Ka(t) = ¯

F(t) in (VΣ,I)∗

has a unique solution a∈L2(t0, T;VΣ,I). Thereby, VΣ,I denotes the subspace of VΣwith

vanishing boundary values along I⊂∂Σ. The in Example 2.1 assumed smoothness of

the transformation Φ implies that the time-dependent operator ¯

Kfits in the framework of

[Wlo87, Ch. V]. Thus, the existence of a unique solution follows by [Wlo87, Th. 29.1].

†Institut f¨

ur Mathematik MA4-5, Technische Universit¨

at Berlin, Straße des 17. Juni

136, D–10623 Berlin, Germany

E-mail address:[email protected]