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Quasi-Optimality of Adaptive Mixed FEMs for Non-selfadjoint Indefinite Second-Order Linear Elliptic Problems

Carstensen, Carsten and Dond, Asha K and Rabus, Hella (2019) Quasi-Optimality of Adaptive Mixed FEMs for Non-selfadjoint Indefinite Second-Order Linear Elliptic Problems. In: COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, 19 (2, SI). pp. 233-250.

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Official URL: https://dx.doi.org/10.1515/cmam-2019-0034

Abstract

The well-posedness and the a priori and a posteriori error analysis of the lowest-order Raviart-Thomas mixed finite element method (MFEM) has been established for non-selfadjoint indefinite second-order linear elliptic problems recently in an article by Carstensen, Dond, Nataraj and Pani (Numer. Math., 2016). The associated adaptive mesh-refinement strategy faces the difficulty of the flux error control in H(div, Omega) and so involves a data-approximation error parallel to f - Pi(0)f parallel to in the L-2 norm of the right-hand side f and its piecewise constant approximation Pi(0)f. The separate marking strategy has recently been suggested with a split of a Dorfler marking for the remaining error estimator and an optimal data approximation strategy for the appropriate treatment of parallel to f -Pi(0)f parallel to(L)2((Omega)). The resulting strategy presented in this paper utilizes the abstract algorithm and convergence analysis of Carstensen and Rabus (SINUM, 2017) and generalizes it to general second-order elliptic linear PDEs. The argument for the treatment of the piecewise constant displacement approximation u(RT) is its supercloseness to the piecewise constant approximation Pi(0u) of the exact displacement u. The overall convergence analysis then indeed follows the axioms of adaptivity for separate marking. Some results on mixed and nonconforming finite element approximations on the multiply connected polygonal 2D Lipschitz domain are of general interest.

Item Type: Journal Article
Additional Information: Copyright of this article belongs to WALTER DE GRUYTER GMBH
Keywords: Adaptivity; Separate Marking; Finite Element Method; Mixed Finite Element Method; Optimal Convergence; Axioms of Adaptivity; General Second-order Linear Elliptic Problems
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Depositing User: Id for Latest eprints
Date Deposited: 17 May 2019 11:13
Last Modified: 17 May 2019 11:13
URI: http://eprints.iisc.ac.in/id/eprint/62332

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