In this paper we consider the so called prescribed curvature problem approximated by a singularly perturbed double obstacle variational inequality. We extend [Pao97] with the introduction of the same nonregular potential used for the evolution problem [NPV94c] and we prove an optimal O(e2) error estimate for nondegenerate minimizers (where e stands for the perturbation parameter). Following [Pao97] the result relies on the construction of precise barriers suggested by formal asymptotics combined with the use of the maximum principle. Key ingredients are the construction of a sub(super)solution containing appropriate shape corrections and the use of a modified distance function based on the principal eigenfunction of the second variation of the prescribed curvature functional. This analysis is next extended to a piecewise linear finite element discretization of the elliptic PDE of bistable type to prove the same error extimate for discrete minima using the Rannacher-Scott L\infty-estimates and under appropriated restrictions on the mesh size (h2=O(es) with s > 5/2).

Cecon, B., Paolini, M., Romeo, M., Optimal interface error estimates for a discrete double obstacle approximation to the prescribed curvature problem, <<MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES>>, 1999; (Agosto): 799-823. [doi:10.1142/S0218202599000385] [http://hdl.handle.net/10807/20929]

### Optimal interface error estimates for a discrete double obstacle approximation to the prescribed curvature problem

#### Abstract

In this paper we consider the so called prescribed curvature problem approximated by a singularly perturbed double obstacle variational inequality. We extend [Pao97] with the introduction of the same nonregular potential used for the evolution problem [NPV94c] and we prove an optimal O(e2) error estimate for nondegenerate minimizers (where e stands for the perturbation parameter). Following [Pao97] the result relies on the construction of precise barriers suggested by formal asymptotics combined with the use of the maximum principle. Key ingredients are the construction of a sub(super)solution containing appropriate shape corrections and the use of a modified distance function based on the principal eigenfunction of the second variation of the prescribed curvature functional. This analysis is next extended to a piecewise linear finite element discretization of the elliptic PDE of bistable type to prove the same error extimate for discrete minima using the Rannacher-Scott L\infty-estimates and under appropriated restrictions on the mesh size (h2=O(es) with s > 5/2).
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1999
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Cecon, B., Paolini, M., Romeo, M., Optimal interface error estimates for a discrete double obstacle approximation to the prescribed curvature problem, <<MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES>>, 1999; (Agosto): 799-823. [doi:10.1142/S0218202599000385] [http://hdl.handle.net/10807/20929]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/20929
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