We consider a finite element approximation for a system
consisting of the evolution of a closed planar curve by forced curve shortening flow
coupled to a reaction-diffusion equation on the evolving curve.
The scheme for the curve evolution is based on a parametric description
allowing for tangential motion,
whereas the discretisation for the PDE on the curve uses an idea from
. We prove optimal error bounds for the resulting fully
discrete approximation and present numerical experiments. These confirm our estimates
and also illustrate the advantage of the tangential motion of the mesh points in practice.
urface PDE, forced curve shortening flow, diffusion induced grain boundary motion,
parametric finite elements, tangential motion, error analysis
65M60, 65M15, 35K55, 53C44, 74N20
The aim of this paper is to analyse a fully discrete numerical scheme for approximating a solution
of the following system: find a family of planar, closed curves and a function
subject to the initial conditions
Here, and are the normal velocity and the curvature of corresponding to
the choice of a unit normal, while is the arclength parameter on . Furthermore,
denotes the material derivative of , i.e. .
, , the closed curve and are all given.
The system (1a,b) couples the evolution of the curves
by forced curve shortening flow to a parabolic PDE on the moving
curves. It occurs for example as a sharp–interface model for diffusion induced grain
boundary motion: in this setting represents a grain boundary separating the
crystals of a thin polycrystalline film of metal that is placed in a vapour containing
another metal; atoms from the vapour diffuse into the
film along the grain boundaries causing them to move. A thorough description of the
physical set-up can be found in
, while an existence and uniqueness result has been obtained
In what follows we shall describe the evolving curves
with the help of a parametrisation ,
where is the periodic unit interval. Then
are a unit tangent and unit normal to respectively,
where denotes counter-clockwise rotation by .
The normal and tangential velocities of are then
Note that (1a) only prescribes
and with it the shape of , so that there is a certain freedom
in choosing the tangential velocity. Since one may consider
Clearly, (1a) holds with
the additional property that the velocity vector points in the normal direction.
to the PDE satisfied by , Pozzi and Stinner have derived and analysed in
 a finite element scheme for (1a,b) (with )
based on continuous piecewise linears. They are able to
prove the following error bounds in the spatially discrete case:
A major difficulty in the analysis arises from the fact that (5) is only
weakly parabolic. Following ,
this problem is solved in  by deriving additional equations for the continuous
and discrete length elements and by splitting the error into
a tangent part and a length element part.
Apart from these analytical difficulties, the motion in purely the normal direction also
may lead to the accumulation of
mesh points in numerical simulations.
A natural way to handle the above mentioned difficulties is to introduce a tangential
part in the velocity, which can be seen as a reparametrisation, an approach that has recently been
explored in a systematic way by Elliott and Fritz
in . The underlying idea uses the DeTurck trick
in coupling the motion of the curve to the harmonic
map heat flow, which results in the following equation replacing (5) (cf. (3.1) in ):
In the above,
is a parameter so
that corresponds to the diffusion coefficient in the harmonic map heat flow.
Note that we
obtain (1a) by taking the scalar product of (6) with .
It turns out that gets
closer to a parametrisation proportional to arclength as gets small.
At the numerical level this means that mesh points along
the curve become more and more equidistributed. Setting formally one recovers an
approach introduced by Barrett, Garcke and Nürnberg, see , .
A nice feature of (6) is that
the problem now is strictly parabolic allowing for a more straightforward error analysis;
for the spatially discrete case with ,
see ,  for ,
and  for .
It remains to derive the equation for , which in contrast to 
will also involve the tangential velocity . It is easily seen that
Inserting this relation into (1b) and recalling (4), we obtain
The initial conditions for (6), (7) now are: ,
a parametrization of .
We shall derive in Section 2 a weak formulation for (6),
(7) which forms the basis for a discretisation
by continuous piecewise linear finite elements in space and
a backward Euler scheme in time. As the main result of our paper we shall
present an error analysis for the resulting fully discrete scheme.
The precise result will be formulated at the end of Section 2,
while the proof is carried out in detail in Section 3.
In Section 4 we present a series of test calculations that
confirm our theoretical results and demonstrate the above mentioned improvement of the mesh quality
for small values of .
Finally, we end this section with a few comments about notation.
We adopt the standard notation
for Sobolev spaces, denoting the norm of
and a bounded
interval in ) by
and the semi-norm by . For
, will be denoted by
with the associated norm and semi-norm written,
For ease of notation, in the common case when
the subscript “” will be dropped on the above norms and semi-norms.
The above are naturally extended to vector functions, and we will write
for a vector function with two components.
In addition, we adopt the standard notation
(, , an
interval in , a Banach space)
for time dependent spaces
with norm .
Once again, we write if .
Furthermore, denotes a generic constant independent of
the mesh parameter and the time step , see below.
2 Weak formulation and finite element approximation
We shall assume that the data and the solution of (6), (7) (writing again instead
of for ease of notation) are such that
|for some .|
Since our error analysis will also include the discretisation in time our regularity assumptions are
slightly stronger than those made in , see Assumption 2.2 there.
Let us fix .
For a test function
, we obtain on
multiplying (6) by that
In order to derive a weak formulation for (7)
we employ an idea from ,
, and calculate for with the help of integration by parts and (3)
where we have also used (4), (7) and the fact that
We now use (2), (3) in order to discretise our system and begin by
introducing the decomposition
. We set
where and assume that
and be the
standard Lagrange interpolation operator such that ,
. We require also the local interpolation operator ,
and recall for , ,
As well as the standard inner product ,
we introduce the discrete inner product defined by
where are piecewise continuous functions on the partition
We note for and for all that
The result (8b) follows immediately from (6a,b).
The inner products and are naturally extended to
In addition to the above spatial discretisation, let be a partitioning of
with time steps ,
, and .
we define our scheme we assign to an
element (the upper index referring to the time level )
a piecewise constant discrete unit tangent and normal by
Our discretisation of (2) now reads: given and ,
find such that
Here, and in what follows, we abbreviate .
We next use the solution of (10) in order to discretise (3). To do so,
we define approximations , of the normal and tangential velocities by
Then find such that
Let us formulate
the main result of this paper, which will be proved in Section 3.
Let and .
There exists such that for all and
the discrete problem (10), (12)
has a unique solution ,
and the following error bounds hold:
3 Error analysis
To begin, it follows from (1c,d) and (4) that for
where . We abbreviate for
Our proof of Theorem 2 will be based on induction.
We assume for
where the function is defined by
and is chosen so small that
Here, and are independent of and ,
and will be chosen a posteriori.
Clearly, (3) holds for in view of our choice of initial data
for and .
It follows from (4), (6c),
(3), (1) and (5) that
provided that is chosen small enough. Combining this inequality with
(1e) we infer that
If we use (7) and argue similarly as in (6) we further obtain
In the same way as (6),
we obtain from (8a), (7), (3)
and (5) that
which combined with (6a,c), (4) and (1) yields for
3.1 The curve equation
We assume throughout that . Since (10) forms a linear problem it is
easily seen that (7) implies the existence of
a unique solution to (10).
We deduce from (10) and (2) with that
Using (7) and (8a)
we find, with the help of an elementary calculation,
Let us analyse the term defined in (11) and note that
We now bound the terms defined in (13) on recalling
(7), (3) and (9).
It follows from (8b) and (6b) that
Next, we have from Sobolev embedding, (6b) and (1) that
We now turn our attention to the term in (11) and note that