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 ⊆ Ω P Ω (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.