\documentstyle{amsart}
\begin{document}
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.\ 1997(1997), No.\ 10, pp. 1--4.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp) 147.26.103.110 or 129.120.3.113}
\thanks{\copyright 1997 Southwest Texas State University and
University of North Texas.}
\vspace{1.5cm}
\title[\hfilneg EJDE--1997/10\hfil Singularities for Ginzburg-Landau]{
Numerical Calculation of Singularities for \\
Ginzburg-Landau Functionals}
\author[J.W. Neuberger \& R.J. Renka\hfil EJDE--1997/02\hfilneg]
{J.W. Neuberger \& R.J. Renka}
\address{J.W. Neuberger\\ Dept. of Mathematics, University of North Texas,
Denton, TX 76203 USA}
\email{jwn@@unt.edu}
\address{R.J. Renka\\ Dept. of Computer Science, University of North Texas,
Denton, TX 76203 USA}
\email{renka@@cs.unt.edu}
\date{}
\thanks{Submitted May 1, 1997. Published June 18, 1997.}
\subjclass{35Q80, 65N06}
\keywords{Ginzburg-Landau, Singularities, Sobolev gradient}
\begin{abstract}
We give results of numerical calculations of asymptotic behavior of
critical points of a Ginzburg-Landau functional. We use both continuous
and discrete steepest descent in connection with Sobolev gradients in
order to study configurations of singularities.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\section{Location of Singularities} \label{S:intro}
Suppose $\epsilon > 0$ and $d$ is a positive integer.
Consider the problem of determining
critical points of the functional $\phi_{\epsilon}$:
\begin{equation} \label{functional}
\phi_{\epsilon}(u) =
\int_{\Omega} (\|\nabla u \|^2/2 + (|u|^2 -1)^2/(4 \epsilon^2)),
\; u \in H^{1,2}(\Omega,C),\; u(z) = z^d,\; z \in \partial \Omega,
\end{equation}
where $\Omega$ is the unit closed disk in $C$, the complex numbers. For
each such $\epsilon > 0$, denote by $u_{\epsilon,d}$
a minimizer of \eqref{functional}.
\medskip
In \cite{BBH} it is indicated that for various sequences
$\{\epsilon_n\}^{\infty}_{n=1}$
of positive numbers converging to $0$,
precisely $d$ singularities develop for $u_{\epsilon_n,d}$ as
$n \rightarrow \infty$. The open problem is raised
(Problem $12$, page $139$ of \cite{BBH}) concerning
possible orientation of such singularities. Our calculations suggest
that for a given $d$ there are (at least) two resulting families of
singularity configurations. Each configuration is formed by vertices
of a regular $d-$gon centered at the origin of $C$, with each
corresponding member of
one configuration being
about $.6$ times as large as a member of the other.
A family of which we speak is obtained by rotating a configuration through
some angle $\alpha$. That this results in another possible configuration
follows from the fact (page 88 of \cite{BBH}) that if
\begin{equation*}
v_{\epsilon,d} (z) = e^{-id\alpha} u_{\epsilon,d}(e^{i \alpha}z),
z\in \Omega,
\end{equation*}
then $\phi_{\epsilon}(v_{\epsilon,d}) = \phi_{\epsilon}(u_{\epsilon,d})$
and $v_{\epsilon,d}(z) = z^d, z \in \partial \Omega$.
\medskip
That there should be singularity patterns formed by vertices of
regular $d-$gons has certainly been anticipated although it seems that no
proof has been put forward. What we offer here is some
numerical support for this proposition. What surprised us in this work
is the indication of {\it two}
families for each positive integer $d$.
\medskip
We explain how these two families were
encountered. Our calculations use steepest descent with
numerical Sobolev gradients (\cite{rngl}, \cite{n}).
One family appears using discrete steepest
descent and the other appears when continuous steepest descent is
closely tracked numerically. We can offer no explanation for this
phenomenon but simply report it. For a given $d$, the family of
singularities obtained with discrete steepest descent
is closer to the origin (by about a factor of $.6$) than the
corresponding family for continuous steepest descent. In either case,
the singularities are closer to the boundary of $\Omega$ for larger
$d$. It is emphasized that more than the usual caveats concerning
the deduction of analytical facts from numerical calculations
certainly apply here. We are using a descent method to calculate
critical points of a highly singular object
(for small $\epsilon$, a graph of $|u_{\epsilon,d}|^2$ would appear
as a plate of height one
above $\Omega$ with $d$ slim tornadoes coming down to zero). Moreover
for each $d$ as indicated above, one expects a continuum of
critical points (one obtained from another by rotation) from which
to `choose'. A feature of continuous steepest descent method
with Sobolev gradients is that
it tends to pick out the nearest (in perhaps some non-Euclidean sense)
critical point to the starting estimate. Such methods are suited to
problems in which there are many critical points.
In addition, the topography of $\phi_{\epsilon}$
over all competing functions
seems rather severe with critical points having rather small support.
This all makes for a fairly difficult calculation which calls for
more computing power than is available
to us at the moment. This is particularly true if one seeks evidence
that as $d \rightarrow \infty$, then points of developing singularities
approach $\partial \Omega$. In our opinion, owing to the importance of this
problem, independent and more extensive calculations should be made.
\medskip
\noindent{\bf Some questions.} Are there more than two (even infinitely many)
families of
singularities for each $d$? Does some other descent method (or some
other method entirely) lead one to new configurations? Are there in
fact configurations which are not symmetric about the origin?
\section{Description of Method} \label{method}
We indicate how to construct a Sobolev gradient for \eqref{functional}.
We use an equivalent real formulation of \eqref{functional} and
regard $\phi_{\epsilon}$ as a function from $H=H^{1,2}(\Omega,R)^2$ to $R$.
For $u \in H$, $\phi^{\prime}_{\epsilon}(u)$
is a continuous linear functional
on $H$. Thus there is a unique member of $H$, called
$(\nabla \phi_{\epsilon})(u)$, such that
\begin{equation} \label{sobrep}
\phi^{\prime}_{\epsilon}(u)h =
\langle h,(\nabla \phi_{\epsilon})(u) \rangle_H,\; u \in H,\; h \in H_0,
\end{equation}
where $H_0$ is the subspace of $H$ consisting of those members of $H$
which are zero on $\partial \Omega$. The reader might refer to \cite{rngl}
or \cite{n} for more details on an explicit construction of this
gradient.
\medskip
Once a gradient function for $\phi_{\epsilon}$ is available there
are two corresponding steepest descent processes, continuous steepest
descent and discrete steepest descent.
\medskip
Continuous steepest descent consists of picking $x \in H$ and
determining $z: [0,\infty) \rightarrow H$ such that
\begin{equation} \label{csd}
z(0) = x,\; z'(t) = - (\nabla \phi_{\epsilon})(z(t)),\; t \ge 0.
\end{equation}
For each $\epsilon > 0$, critical points $u_{\epsilon,d} \in H$
are sought so that
\begin{equation*}
u_{\epsilon,d} = \lim_{t \rightarrow \infty} z(t),
\end{equation*}
or at least so that
\begin{equation*}
u_{\epsilon,d} = \lim_{n \rightarrow \infty} z(t_n)
\end{equation*}
for some unbounded increasing sequence $\{t_n\}^{\infty}_{n=1}$ of
positive numbers.
\medskip
For fixed $\epsilon > 0$ discrete steepest descent, on the other hand,
consists of picking
$x \in H$ and determining $\{z_n\}^{\infty}_{n=1}$ so that
\begin{equation} \label{dsd}
z_{n+1} = z_n - \delta_n (\nabla \phi_{\epsilon})(z_n),\; n=1,2,\dots,
\end{equation}
where $\delta_n$ is chosen to be the smallest positive local minimum
$\delta$ of
\begin{equation*}
\phi_{\epsilon}(z_n - \delta (\nabla \phi_{\epsilon})(z_n)),\; \delta \ge 0.
\end{equation*}
Critical points $u_{\epsilon,d}$ of $\phi_{\epsilon}$ are sought
as a limit of a subsequence of $\{z_n\}^{\infty}_{n=1}$.
\medskip
In the case of either continuous steepest descent or discrete
steepest descent we are interested in asymptotic behavior of
$u_{\epsilon,d}$ as $\epsilon \rightarrow 0$. For computations we seek
a critical point of $\phi_{\epsilon,d}$ for
`small' $\epsilon$ as indicated in the
following section. As explained in \cite{BBH}, there is not a limit
in $H^{1,2}(\Omega)$ of $u_{\epsilon,d}$ as $\epsilon \rightarrow 0$
but that $u_{\epsilon,d}$ becomes more nearly singular as
$\epsilon \rightarrow 0$.
\medskip
For calculations we use a finite dimensional version of the above.
The region $\Omega$ is broken into pieces using some number ($180$ to $400$,
depending on $d$) of evenly spaced radii together with $40$ to
$80$ concentric circles. References \cite{n},\cite{rngl} contain details
of Sobolev gradient construction in these finite dimensional settings.
We mention that in our results, $u = \lim_{t \rightarrow \infty}z(t)$ appears
to exist in the case of continuous steepest descent and
$u = \lim_{n \rightarrow \infty} z_n$ appears to exist in the case of
discrete steepest descent. Thus no `taking of subsequences' seems
to be necessary.
\section{Results} \label{results}
For continuous steepest descent, using $d=2,\dots,10$ we started
each steepest descent iteration with a finite dimensional version of
$u_{\epsilon,d}(z) = z^d$. To emulate continuous steepest descent, we
used discrete steepest descent with small step size (on the order of
.0001) instead of the optimal step size of \eqref{dsd}. In all runs
reported on here we used $\epsilon = 1/40$
except for the discrete steepest descent run with $d=2$.
In that case $\epsilon = 1/100$ was used (for $\epsilon = 1/40$ convergence
seemed not to be forthcoming in the single precision code used
- the value $.063$ given is likely smaller
than a successful run with $\epsilon = 1/40$ would give).
Runs with somewhat larger
$\epsilon$ yielded a similar pattern except the corresponding
singularities were a little farther from the origin. In all cases we
found $d$ singularities arranged on a regular $d$-gon centered at the
origin.
\medskip
Results for continuous steepest descent are indicated by the following pairs:
\begin{equation*}
(2,.15),(3,.25),(4,.4),(5,.56),(6,.63),(7,.65),(8,.7),(9,.75),(10,.775)
\end{equation*}
where a pair $(d,r)$ above indicates that a (near) singularity
of $u_{\epsilon,d}$ was found at a
distance $r$ from the origin with $\epsilon = 1/40$.
In each case the other $d-1$ singularities are located by rotating
the first one through an angle that is an integral multiple of $2 \pi/d$.
\medskip
Results for discrete steepest descent are indicated by the following pairs:
\begin{equation*}
(2,.063),(3,.13),(4,.18),(5,.29),(6,.34),(7,.39),(8,.44),(9,.48),(10,.5)
\end{equation*}
using the same conventions as for continuous steepest descent.
Computations with a finer mesh would surely yield more precise results.
\medskip
\noindent{\bf Acknowledgment.} We thank Pentru Mironescu for his careful description of this problem
to JWN in December 1996 at the Technion in Haifa.
\begin{thebibliography}{9}
\bibitem{BBH}
F. B\'ethuel, H. Brezis, F. H\'elein, {\em Ginzburg-Landau Vortices},
Birkhauser (1994).
\medskip
\bibitem{n}
J.W. Neuberger, {\em Sobolev Gradients and
Differential Equations}, Springer-Verlag Lecture Notes (to appear).
\medskip
\bibitem{rngl}
{J.W. Neuberger and R.J. Renka}, {Sobolev Gradients and
the Ginzburg-Landau Functional}, SIAM J. Sci. Comp. (to appear).
\medskip
\end{thebibliography}
\end{document}