A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem. (English) Zbl 0854.35008
Summary: 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.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35J60 | Nonlinear elliptic equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35A35 | Theoretical approximation in context of PDEs |
| 49Q05 | Minimal surfaces and optimization |
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