We present a posteriori error analysis of diffusion problems where the diffusion tensor is not necessarily symmetric and positive definite and can in particular change its sign. We first identify the correct intrinsic error norm for such problems, covering both conforming and nonconforming approximations. It combines a dual (residual) norm together with the distance to the correct functional space. Importantly, we show the equivalence of both these quantities defined globally over the entire computational domain with the Hilbertian sums of their localizations over patches of elements. In this framework, we then design a posteriori estimators which deliver simultaneously guaranteed error upper bound, global and local error lower bounds, and robustness with respect to the (sign-changing) diffusion tensor. Robustness with respect to the approximation polynomial degree is achieved as well. The estimators are given in a unified setting covering at once conforming, nonconforming, mixed, and discontinuous Galerkin finite element discretizations in two or three space dimensions. Numerical results illustrate the theoretical developments.
Keywords: Noncoercive problem, sign change, metamaterial, a posteriori error estimate, dual norm, distance to energy space, localization, equivalence local–global, minimization, best approximation, equilibrated flux, unified framework, robustness, finite element methods
Ciarlet, Patrick Jr. 1 ; Vohralík, Martin 1
@article{M2AN_2018__52_5_2037_0,
author = {Ciarlet, Patrick Jr. and Vohral{\'\i}k, Martin},
title = {Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2037--2064},
year = {2018},
publisher = {EDP Sciences},
volume = {52},
number = {5},
doi = {10.1051/m2an/2018034},
mrnumber = {3891753},
zbl = {1417.65187},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2018034/}
}
TY - JOUR AU - Ciarlet, Patrick Jr. AU - Vohralík, Martin TI - Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 2037 EP - 2064 VL - 52 IS - 5 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2018034/ DO - 10.1051/m2an/2018034 LA - en ID - M2AN_2018__52_5_2037_0 ER -
%0 Journal Article %A Ciarlet, Patrick Jr. %A Vohralík, Martin %T Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 2037-2064 %V 52 %N 5 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2018034/ %R 10.1051/m2an/2018034 %G en %F M2AN_2018__52_5_2037_0
Ciarlet, Patrick Jr.; Vohralík, Martin. Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 2037-2064. doi: 10.1051/m2an/2018034
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