A P1-P1 finite element method for a phase relaxation model II:: Adaptively refined meshes

We examine the effect of adaptively generated refined meshes on the P-1 - P-1 finite element method with semi-explicit time stepping of part I, which applies to a phase relaxation model with small parameter epsilon > 0. A typical mesh is highly graded in the so-called refined region, which exhibits a local meshsize proportional to the time step tau, and is coarse in the remaining parabolic region where the meshsize is of order root tau. Three admissibility tests guarantee mesh quality and, upon failure, lead to remeshing and so to incompatible consecutive meshes. The most severe test checks whether the transition region, where phase changes take place, belongs to the refined region. The other two tests monitor equidistribution of pointwise interpolation errors. The resulting adaptive scheme is shown to be stable in various Sobolev norms and to converge with a rate of order O (tau/root epsilon) in the natural energy spaces. Several numerical experiments illustrate the scheme's efficiency and enhanced performance as compared with those of part I.