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


Xun Jiang
Department of Mathematics
University of California at Davis
Davis, CA 95616, USA



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