Volume 52, Number 5, September–October 2018
|Page(s)||2037 - 2064|
|Published online||14 December 2018|
Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients★
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 828 Boulevard des Maréchaux,
2 INRIA, 2 rue Simone Iff, 75589 Paris, France.
3 Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 9 May 2018
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.
Mathematics Subject Classification: 65N15 / 65N30 / 65N50 / 78A48
Key words: 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
© EDP Sciences, SMAI 2018
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