Error control and adaptivity for a phase relaxation model

The phase relaxation model is a diffuse interface model with small parameter epsilon which consists of a parabolic PDE for temperature theta and an ODE with double obstacles for phase variable chi. To decouple the system a semi-explicit Euler method with variable step-size tau is used for time discretization, which requires the stability constraint tau less than or equal to epsilon. Conforming piecewise linear finite elements over highly graded simplicial meshes with parameter h are further employed for space discretization. A posteriori error estimates are derived for both unknowns theta and chi, which exhibit the correct asymptotic order in terms of epsilon, h and tau. This result circumvents the use of duality, which does not even apply in this context. Several numerical experiments illustrate the reliability of the estimators and document the excellent performance of the ensuing adaptive method.