Solutions of the so-called prescribed curvature problem $\min_{A\subseteq\Omega} P_\Omega (A) - \int_A g(x)$, 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 $O(\epsilon^2|\log\epsilon|^2)$ 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.

Paolini, M., A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem, <<MATHEMATICS OF COMPUTATION>>, 1997; (66): 45-67 [http://hdl.handle.net/10807/21235]

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

#### Abstract

Solutions of the so-called prescribed curvature problem $\min_{A\subseteq\Omega} P_\Omega (A) - \int_A g(x)$, 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 $O(\epsilon^2|\log\epsilon|^2)$ 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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Paolini, M., A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem, <<MATHEMATICS OF COMPUTATION>>, 1997; (66): 45-67 [http://hdl.handle.net/10807/21235]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/21235
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