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

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

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

## 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 $h\approx\tau\leq\ep$.