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We study a simple model of phase relaxation which consists of a parabolic PDE for temperature and an ODE with a small parameter " and double obstacles for phase variable. The model replaces sharp by di use interfaces and gives rise to superheating e ects. A semi-explicit time discretization with uniform time-step is combined with continuous piecewise linear nite elements for both and , over a xed quasi-uniform mesh of size h. At each time step, an inexpensive nodewise algebraic correction is performed to update , followed by the solution of a linear positive de nite symmetric system for by a preconditioned conjugate gradient method. A priori estimates for both and are derived in L 2-based Sobolev spaces provided the stability constraint " is enforced. Asymptotic behavior of the fully discrete model is examined as "; ; h # 0 independently, which leads to a rate of convergence of order O((+ h)" ?1=2), provided a natural compatibility condition on the initial data is satis ed. Numerical experiments illustrate the performance of the proposed method for the natural choice h ".
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