Asymptotic Anal., 10 (1995), 173-198.
Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA
Maurizio Paolini
Dipartimento di Matematica
Universita di Milano
Milano, 20133, Italy
Claudio Verdi
Dipartimento di Matematica
Universita di Milano
Milano, 20133, Italy
A singularly perturbed double obstacle problem is examined as a variational tool for the approximation of the geometric motion of fronts. The relaxation parameter is space-time dependent, thereby allowing the control of transition layer thickness and related interface pointwise accuracy. Optimal order interface error estimates of local character, namely depending on the relaxation parameter local magnitude, are derived for smooth evolutions. The proof is based on constructing suitable sub and supersolutions, which incorporate a number of shape corrections to the basic standing wave profile, and using a modified distance function to the front. Numerical simulations illustrate how such a local property can be exploited in dealing with large curvatures and ultimately in resolving singularities.