Affiliations: Department of Mathematics, University of Maryland, and Institute for Physical Science and Technology, College Park, MD 20742, USA | Dipartimento di Matematica, Universitá di Milano, 20133 Milano, and Istituto di Analisi Numerica del CNR, 27100 Pavia, Italy
Note: [] Correspondence to: R.H. Nochetto, Department of Mathematics, University of Maryland, and Institute for Physical Science and Technology, College Park, MD 20742, USA.
Note: [] This work was partially supported by NSF Grant DMS-9008999, and by MURST (Progetto Nazionale “Equazioni di Evoluzione e Applicazioni Fisico-Matematiche” and “Analisi Numerica e Matematica Computazionale”) and CNR (IAN and Contracts 92.00833.01, 93.00564.01) of Italy.
Abstract: 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 are derived for smooth evolutions. The estimates have a local character for small time, namely they depend on the relaxation parameter local magnitude. 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 the variable transition layer thickness can be exploited in dealing with large curvatures and ultimately in resolving singularities.
