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\begin{center}
{\bf Estimates and Computations for Melting and Solidification Problems}

\vspace{.25in} 

J. M. Greenberg
\vspace{.15in} 

Carnegie Mellon University\\
Department of Mathematical Sciences\\
Pittsburgh, PA  15213
\end{center} 

\begin{abstract}
In this paper we focus on melting and solidification processes
described by phase-field models and obtain rigorous estimates for such
processes.  These estimates are derived in Section 2 and guarantee the
convergence of solutions to non-constant equilibrium patterns.  The
most basic results conclude with the inequality (2.31).  The
estimates in the remainder of Section 2 illustrate what obtains if the
initial data is progressively more regular and may be omitted on first
reading.  We also present some interesting numerical simulations which
demonstrate the equilibrium structures and the approach of the system
to non-constant equilibrium patterns.  The novel feature of these
calculations is the linking of the small parameter in the system,
$\delta$, to the grid spacing, thereby producing solutions with
approximate sharp interfaces.  Similar ideas have been used by
Caginalp and Sokolovsky [1].  A movie of these simulations may be
found at {\it http:www.math.cmu.edu/math/people/greenberg.html}.
\end{abstract}

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