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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem
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by Maurizio Paolini PDF
Math. Comp. 66 (1997), 45-67 Request permission

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
  • Email: paolini@dimi.uniud.it
  • 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.
  • © Copyright 1997 American Mathematical Society
  • Journal: Math. Comp. 66 (1997), 45-67
  • MSC (1991): Primary 35B25, 35J60, 65N30; Secondary 35A35, 49Q05
  • DOI: https://doi.org/10.1090/S0025-5718-97-00771-0
  • MathSciNet review: 1361810