no. 2
A-posteriori error estimates for linear exterior problems via mixed-FEM and DtN mappings
ESAIM: Modélisation mathématique et analyse numérique, Volume 36 (2002) no. 2, pp. 241-272

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the usual Cea error estimate and the corresponding rate of convergence. In addition, we develop two different a-posteriori error analyses yielding explicit residual and implicit Bank-Weiser type reliable estimates, respectively. Several numerical results illustrate the suitability of these estimators for the adaptive computation of the discrete solutions.

DOI: 10.1051/m2an:2002011
Classification: 65N15, 65N30, 65N38, 65N50
Keywords: Dirichlet-to-Neumann mapping, mixed finite elements, Raviart-Thomas spaces, residual based estimates, Bank-Weiser approach 
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     author = {Barrientos, Mauricio A. and Gatica, Gabriel N. and Maischak, Matthias},
     title = {\protect\emph{A-posteriori} error estimates for linear exterior problems \protect\emph{via} {mixed-FEM} and {DtN} mappings},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {241--272},
     publisher = {EDP Sciences},
     volume = {36},
     number = {2},
     year = {2002},
     doi = {10.1051/m2an:2002011},
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     zbl = {1028.65114},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an:2002011/}
}
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Barrientos, Mauricio A.; Gatica, Gabriel N.; Maischak, Matthias. A-posteriori error estimates for linear exterior problems via mixed-FEM and DtN mappings. ESAIM: Modélisation mathématique et analyse numérique, Volume 36 (2002) no. 2, pp. 241-272. doi: 10.1051/m2an:2002011

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