Error Control and Adaptivity for a Phase Relaxation Model

Z. Chen
Institute of Mathematics, Academia Sinica, Beijing 100080, PR China

R.H. Nochetto
Department of Mathematics, University of Maryland, College Park, MD 20742, USA

A. Schmidt
Institut für Angewandte Mathematik, Hermann-Herder-Str. 10, 79104 Freiburg, Germany

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.


Phase variable isolines for eps=0.1 (top) and eps=0.01 (bottom)


Adaptive meshes for eps=0.1 (top) and eps=0.01 (bottom)

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