# A $P^1-P^1$ finite element method for a phase relaxation model.
Part II: Adaptively refined meshes

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SIAM J. Numer. Anal., 36 (1999), 974-999.
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Xun Jiang

Department of Mathematics

University of California at Davis

Davis, CA 95616, USA

xun@galois.ucdavis.edu
Ricardo H. Nochetto

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

rhn@math.umd.edu
Claudio Verdi

Dipartimento di Matematica

Universita di Milano

Milano, 20133, Italy

verdi@paola.mat.unimi.it
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Abstract

We examine the effect of adaptively generated
refined meshes on the $P^1-P^1$
finite element method with semi-explicit time stepping of Part I,
which applies to a phase relaxation model with
small parameter $\ep>0$. A typical mesh is highly graded in
the so-called refined region, which
exhibits a local meshsize proportional to the time step $\tau$, and
is coarse in the remaining parabolic region where the meshsize is of
order $\sqrt\tau$. Three admissibility tests guarantee mesh quality
and, upon failure, lead to remeshing and so to incompatible consecutive
meshes. The most severe test checks whether the transition region,
where phase changes take place, belongs to the refined region.
The other two tests monitor equidistribution of
pointwise interpolation errors.
The resulting adaptive scheme is shown to be stable in
various Sobolev norms and to converge with a rate of order
$\Or(\tau/\sqrt\ep)$ in the natural energy spaces. Several
numerical experiments illustrate the scheme's efficiency and
enhanced performance as compared with those of Part I.