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 minA⊆Ω PΩ(A)− ∫ A g(x), g being the curvature field, are approximated via a singularly perturbed elliptic PDE of bistable type. For nondegenerate relative minimizers A ⊂⊂ Ω we prove an O( 2| log |2) error estimate (where 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.

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

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Cite this paper

@article{Paolini1997AQE, title={A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem}, author={Maurizio Paolini}, journal={Math. Comput.}, year={1997}, volume={66}, pages={45-67} }