A P1-P1 finite element method for a phase relaxation model I:: Quasi-uniform mesh

We study a simple model of phase relaxation which consists of a parabolic PDE for temperature theta and an ODE with a small parameter epsilon and double obstacles for phase variable chi. The model replaces sharp interfaces by diffuse ones 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 less than or equal to epsilon is enforced. Asymptotic behavior of the fully discrete model is examined as epsilon, tau, h down arrow 0 independently, which leads to a rate of convergence of order O((tau + h)epsilon(-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 approximate to tau less than or equal to epsilon.