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Mathematics of Computation

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A posteriori error estimates for the Crank–Nicolson method for parabolic equations

Authors: Georgios Akrivis, Charalambos Makridakis and Ricardo H. Nochetto
Journal: Math. Comp. 75 (2006), 511-531
MSC (2000): Primary 65M15, 65M50
Published electronically: November 30, 2005
MathSciNet review: 2196979
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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.

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Additional Information

Georgios Akrivis
Affiliation: Computer Science Department, University of Ioannina, 451 10 Ioannina, Greece
MR Author ID: 24080

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

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

Keywords: Parabolic equations, Crank–Nicolson method, Crank–Nicolson–Galerkin method, Crank–Nicolson reconstruction, Crank–Nicolson–Galerkin reconstruction, a posteriori error analysis
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.
Article copyright: © Copyright 2005 American Mathematical Society