A posteriori error estimates for the Crank–Nicolson method for parabolic equations
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- by Georgios Akrivis, Charalambos Makridakis and Ricardo H. Nochetto PDF
- Math. Comp. 75 (2006), 511-531 Request permission
Abstract:
We derive optimal order a posteriori error estimates for time discretizations by both the Crank–Nicolson and the Crank–Nicolson–Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second-order Crank–Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank–Nicolson method.References
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Additional Information
- Georgios Akrivis
- Affiliation: Computer Science Department, University of Ioannina, 451 10 Ioannina, Greece
- MR Author ID: 24080
- Email: akrivis@cs.uoi.gr
- Charalambos Makridakis
- Affiliation: Department of Applied Mathematics, University of Crete, 71409 Heraklion-Crete, Greece – and – Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion-Crete, Greece
- MR Author ID: 289627
- Email: makr@math.uoc.gr, makr@tem.uoc.gr
- Ricardo H. Nochetto
- Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
- MR Author ID: 131850
- Email: rhn@math.umd.edu
- Received by editor(s): June 10, 2004
- Received by editor(s) in revised form: February 23, 2005
- Published electronically: November 30, 2005
- Additional Notes: The first author was partially supported by a “Pythagoras” grant funded by the Greek Ministry of National Education and the European Commission.
The second author was partially supported by the European Union RTN-network HYKE, HPRN-CT-2002-00282, the EU Marie Curie Development Host Site, HPMD-CT-2001-00121 and the program Pythagoras of EPEAEK II
The third author was partially supported by NSF Grants DMS-9971450 and DMS-0204670. - © Copyright 2005 American Mathematical Society
- Journal: Math. Comp. 75 (2006), 511-531
- MSC (2000): Primary 65M15, 65M50
- DOI: https://doi.org/10.1090/S0025-5718-05-01800-4
- MathSciNet review: 2196979