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A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem

Author: Maurizio Paolini
Journal: Math. Comp. 66 (1997), 45-67
MSC (1991): Primary 35B25, 35J60, 65N30; Secondary 35A35, 49Q05
MathSciNet review: 1361810
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Abstract: Solutions of the so-called prescribed curvature problem $\min _{A\subseteq \Omega } \mathcal {P}_ \Omega (A) - \int _A g(x)$, $g$ being the curvature field, are approximated via a singularly perturbed elliptic PDE of bistable type. For nondegenerate relative minimizers $A \subset \subset \Omega$ we prove an $\mathcal {O}( \epsilon ^2 |\log \epsilon |^2)$ error estimate (where $\epsilon$ stands for the perturbation parameter), and show that this estimate is quasi-optimal. The proof is based on the construction of accurate barriers suggested by formal asymptotics. This analysis is next extended to a finite element discretization of the PDE to prove the same error estimate for discrete minima.

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Additional Information

Maurizio Paolini
Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, 33100, Udine, Italy

Received by editor(s): September 28, 1994
Received by editor(s) in revised form: August 9, 1995
Additional Notes: This work was partially supported 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.
Article copyright: © Copyright 1997 American Mathematical Society