# A $P^1-P^1$ finite element method for a phase relaxation model. Part II: Adaptively refined meshes

SIAM J. Numer. Anal., 36 (1999), 974-999.

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

xun@galois.ucdavis.edu

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA

rhn@math.umd.edu

Claudio Verdi
Dipartimento di Matematica
Universita di Milano
Milano, 20133, Italy

verdi@paola.mat.unimi.it

## Abstract

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 $\ep>0$. 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 $\sqrt\tau$. 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 $\Or(\tau/\sqrt\ep)$ in the natural energy spaces. Several numerical experiments illustrate the scheme's efficiency and enhanced performance as compared with those of Part I.