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A comparison of duality and energy a posteriori estimates for $ \mathrm{L}_{\infty}(0,T;\mathrm{L}_2(\varOmega))$ in parabolic problems


Authors: Omar Lakkis, Charalambos Makridakis and Tristan Pryer
Journal: Math. Comp. 84 (2015), 1537-1569
MSC (2010): Primary 65N30, 65M15, 80M10; Secondary 65M50, 93C40
DOI: https://doi.org/10.1090/S0025-5718-2014-02912-8
Published electronically: December 17, 2014
MathSciNet review: 3335883
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Abstract: We use the elliptic reconstruction technique in combination with a duality approach to prove a posteriori error estimates for fully discrete backward Euler scheme for linear parabolic equations. As an application, we combine our result with the residual based estimators from the a posteriori estimation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the $ \mathrm {L}_{\infty }(0,T;\mathrm {L}_2(\varOmega ))$ norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson in 1991 by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estimators. For comparison with previous results we derive also an energy-based a posteriori estimate for the $ \mathrm {L}_{\infty }(0,T;\mathrm {L}_2(\varOmega ))$-error which simplifies a previous one given by Lakkis and Makridakis in 2006. We then compare both estimators (duality vs. energy) in practical situations and draw conclusions.


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Additional Information

Omar Lakkis
Affiliation: Department of Mathematics, University of Sussex, Brighton, GB-BN1 9QH, England United Kingdom
Email: o.lakkis@sussex.ac.uk

Charalambos Makridakis
Affiliation: Department of Applied Mathematics, University of Crete, GR-71409 Heraklion, Greece — and — Institute for Applied and Computational Mathematics, Foundation for Research and Technology-Hellas, Vasilika Vouton P.O.Box 1527, GR-71110 Heraklion, Greece
Email: makr@tem.uoc.gr

Tristan Pryer
Affiliation: School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, GB-CT2 7NF, England United Kingdom
Email: T.Pryer@kent.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-2014-02912-8
Keywords: Finite element methods, heat equation, \aposteriori error estimate, linear parabolic PDE, reaction-diffusion, elliptic reconstruction, convergence, superconvergence, optimality, duality, dissipation
Received by editor(s): September 14, 2012
Received by editor(s) in revised form: July 6, 2013, and November 16, 2013
Published electronically: December 17, 2014
Additional Notes: This work was partially supported by the E.U. RTN Hyke HPRN-CT-2002-00282 and the Marie Curie Fellowship Foundation. The first author wishes to thank the Hausdorff Institute for Mathematics, Bonn.
The second author was supported at Sussex by a EPSRC D. Phil. postgraduate research fellowship.
Article copyright: © Copyright 2014 by the authors