# Error Control and Adaptivity for a Phase Relaxation Model

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

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

## 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.