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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
DOI: https://doi.org/10.1090/S0025-5718-05-01800-4
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.


References [Enhancements On Off] (What's this?)

  • 1. G. Akrivis, M. Crouzeix, and Ch. Makridakis,
    Implicit-explicit multistep finite element methods for nonlinear parabolic problems,
    Math. Comp. 67 (1998) 457-477. MR 1458216 (98g:65088)
  • 2. G. Akrivis, M. Crouzeix, and Ch. Makridakis,
    Implicit-explicit multistep methods for quasilinear parabolic equations,
    Numer. Math. 82 (1999) 521-541. MR 1701828 (2000e:65075)
  • 3. G. Akrivis and Ch. Makridakis,
    Galerkin time-stepping methods for nonlinear parabolic equations,
    M$ ^2$AN Math. Mod. Numer. Anal. 38 (2004) 261-289. MR 2069147 (2005f:65124)
  • 4. A. K. Aziz and P. Monk,
    Continuous finite elements in space and time for the heat equation,
    Math. Comp. 52 (1989) 255-274. MR 0983310 (90d:65189)
  • 5. M. Crouzeix,
    Parabolic Evolution Problems.
    Unpublished manuscript, 2003.
  • 6. W. Dörfler,
    A time- and space-adaptive algorithm for the linear time-dependent Schrödinger equation,
    Numer. Math. 73 (1996) 419-448. MR 1393174 (97g:65183)
  • 7. K. Eriksson and C. Johnson,
    Adaptive finite element methods for parabolic problems. I. A linear model problem,
    SIAM J. Numer. Anal. 28 (1991) 43-77. MR 1083324 (91m:65274)
  • 8. K. Eriksson and C. Johnson,
    Adaptive finite element methods for parabolic problems. IV. Nonlinear problems,
    SIAM J. Numer. Anal. 32 (1995) 1729-1749. MR 1360457 (96i:65081)
  • 9. K. Eriksson, C. Johnson, and S. Larsson,
    Adaptive finite element methods for parabolic problems. VI. Analytic semigroups,
    SIAM J. Numer. Anal. 35 (1998) 1315-1325. MR 1620144 (99d:65281)
  • 10. D. Estep and D. French,
    Global error control for the continuous Galerkin finite element method for ordinary differential equations,
    RAIRO Math. Mod. Numer. Anal. 28 (1994) 815-852. MR 1309416 (95k:65079)
  • 11. C. Johnson,
    Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations,
    SIAM J. Numer. Anal. 25 (1988) 908-926. MR 0954791 (90a:65160)
  • 12. C. Johnson, Y.-Y. Nie, and V. Thomée,
    An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem,
    SIAM J. Numer. Anal. 27 (1990) 277-291. MR 1043607 (91g:65199)
  • 13. C. Johnson and A. Szepessy,
    Adaptive finite element methods for conservation laws based on a posteriori error estimates,
    Comm. Pure Appl. Math. 48 (1995) 199-234. MR 1322810 (97b:76084)
  • 14. O. Karakashian and Ch. Makridakis,
    A space-time finite element method for the nonlinear Schrödinger equation: the continuous Galerkin method,
    SIAM J. Numer. Anal. 36 (1999) 1779-1807. MR 1712169 (2000h:65139)
  • 15. O. Karakashian and Ch. Makridakis,
    Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations,
    Math. Comp. 74 (2005) 85-102. MR 2085403 (2005g:65147)
  • 16. O. Lakkis and R. H. Nochetto,
    A posteriori error analysis for the mean curvature flow of graphs,
    SIAM J. Numer. Anal. 42 (2005) 1875-1898. MR 2139228
  • 17. Ch. Makridakis and R. H. Nochetto,
    Elliptic reconstruction and a posteriori error estimates for parabolic problems,
    SIAM J. Numer. Anal. 41 (2003) 1585-1594. MR 2034895 (2004k:65157)
  • 18. Ch. Makridakis and R. H. Nochetto,
    A posteriori error analysis for higher order dissipative methods for evolution problems.
    (Submitted for publication).
  • 19. R. H. Nochetto, G. Savaré, and C. Verdi,
    A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations,
    Comm. Pure Appl. Math. 53 (2000) 525-589. MR 1737503 (2000k:65142)
  • 20. R. H. Nochetto, A. Schmidt, and C. Verdi,
    A posteriori error estimation and adaptivity for degenerate parabolic problems,
    Math. Comp. 69 (2000) 1-24. MR 1648399 (2000i:65136)
  • 21. V. Thomée,
    Galerkin Finite Element Methods for Parabolic Problems.
    Springer-Verlag, Berlin, 1997. MR 1479170 (98m:65007)
  • 22. R. Verfürth,
    A posteriori error estimates for finite element discretizations of the heat equation,
    Calcolo 40 (2003) 195-212. MR 2025602 (2005f:65131)

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Additional Information

Georgios Akrivis
Affiliation: Computer Science Department, University of Ioannina, 451 10 Ioannina, Greece
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
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
Email: rhn@math.umd.edu

DOI: https://doi.org/10.1090/S0025-5718-05-01800-4
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

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