#
Error Control and Adaptivity for a Phase Relaxation Model

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Model. Math. Anal. Numer., 34 (2000), 775-797.
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Zhiming Chen

Institute of Mathematics

Academia Sinica

Beijing 100080, PR China

zmchen@lsec.cc.ac.cn
Ricardo H. Nochetto

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

rhn@math.umd.edu
Alfred Schmidt

Institut fuer Angewandte Mathematik

Universitaet Freiburg

79104 Freiburg

Schmidt@math.uni-bremen.de
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Abstract

The phase relaxation model is a diffuse interface model with
small parameter $\eps$ which
consists of a parabolic PDE for temperature
$\theta$ and an ODE with double obstacles
for phase variable $\chi$.
To decouple the system a semi-explicit Euler method with variable
step-size $\tau$ is used for time discretization, which requires
the stability constraint $\tau\leq\eps$. Conforming piecewise
linear finite elements over highly graded simplicial meshes
with parameter $h$ are further employed for space discretization.
A posteriori error
estimates are derived for both unknowns $\theta$ and $\chi$, which
exhibit the correct asymptotic order in terms of $\eps$, $h$ and
$\tau$. This result circumvents the use of duality, which does not
even apply in this context.
Several numerical experiments illustrate the reliability of the
estimators and document the excellent performance of the ensuing
adaptive method.