# A $P^1-P^1$ finite element method for a phase relaxation model.
Part I: Quasi-uniform mesh

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SIAM J. Numer. Anal., 35 (1998), 1176-1190.
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Ricardo H. Nochetto

Department of Mathematics

University of Maryland, College Park

College Park, MD 20742, USA

rhn@math.umd.edu
Xun Jiang

Department of Mathematics

University of California at Davis

Davis, CA 95616, USA

xun@galois.ucdavis.edu
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Abstract

We study a simple model of phase relaxation which consists of
a parabolic PDE for temperature $\theta$ and an ODE with a small
parameter $\ep$ and double obstacles for phase variable $\chi$.
The model replaces sharp by diffuse interfaces and
gives rise to superheating effects. A semi-explicit time
discretization with uniform time-step $\tau$ is combined with
continuous piecewise linear finite elements for both
$\theta$ and $\chi$, over a fixed quasi-uniform mesh of size $h$.
At each time step, an inexpensive
nodewise algebraic correction is performed to update $\chi$,
followed by the solution of a linear positive definite symmetric
system for $\theta$ by a preconditioned conjugate
gradient method. A priori estimates for both
$\theta$ and $\chi$ are derived in $L^2$-based
Sobolev spaces provided the stability constraint $\tau\leq\ep$
is enforced.
Asymptotic behavior of the fully discrete model is examined
as $\ep,\tau,h\da0$ independently, which leads to a
rate of convergence of order $\O((\tau+h)\ep^{-1/2})$,
provided a natural compatibility condition on the
initial data is satisfied.
Numerical experiments illustrate the performance
of the proposed method for the natural choice
$h\approx\tau\leq\ep$.