A posteriori error estimates for linear exterior problems via mixed-FEM and DtN mappings

Mauricio A. Barrientos; Gabriel N. Gatica; Matthias Maischak

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 2, page 241-272
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Barrientos, Mauricio A., Gatica, Gabriel N., and Maischak, Matthias. "A posteriori error estimates for linear exterior problems via mixed-FEM and DtN mappings." ESAIM: Mathematical Modelling and Numerical Analysis 36.2 (2010): 241-272. <http://eudml.org/doc/194103>.

@article{Barrientos2010,
abstract = { 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. },
author = {Barrientos, Mauricio A., Gatica, Gabriel N., Maischak, Matthias},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Dirichlet-to-Neumann mapping; mixed finite elements; Raviart-Tho mas spaces; residual based estimates; Bank-Weiser approach.; Raviart-Thomas spaces; residual based estimates; Bank-Weiser approach; linear exterior transmission problems; second order elliptic equations; Laplace equation; unbounded region; error estimate; convergence; numerical estimate; performance},
language = {eng},
month = {3},
number = {2},
pages = {241-272},
publisher = {EDP Sciences},
title = {A posteriori error estimates for linear exterior problems via mixed-FEM and DtN mappings},
url = {http://eudml.org/doc/194103},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Barrientos, Mauricio A.
AU - Gatica, Gabriel N.
AU - Maischak, Matthias
TI - A posteriori error estimates for linear exterior problems via mixed-FEM and DtN mappings
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 2
SP - 241
EP - 272
AB - 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.
LA - eng
KW - Dirichlet-to-Neumann mapping; mixed finite elements; Raviart-Tho mas spaces; residual based estimates; Bank-Weiser approach.; Raviart-Thomas spaces; residual based estimates; Bank-Weiser approach; linear exterior transmission problems; second order elliptic equations; Laplace equation; unbounded region; error estimate; convergence; numerical estimate; performance
UR - http://eudml.org/doc/194103
ER -

References

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