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

Paolini, Maurizio

doi:10.1090/S0025-5718-97-00771-0

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