On the Crank-Nicolson anisotropic a posteriori error analysis for parabolic integro-differential equations

Reddy, G. Murali; Sinha, Rajen

doi:10.1090/mcom/3067

On the Crank-Nicolson anisotropic a posteriori error analysis for parabolic integro-differential equations

Authors: G. Murali Mohan Reddy and Rajen K. Sinha
Journal: Math. Comp. 85 (2016), 2365-2390
MSC (2010): Primary 65M15, 65M60
DOI: https://doi.org/10.1090/mcom/3067
Published electronically: December 30, 2015
MathSciNet review: 3511285
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Abstract: In this exposition, we derive two anisotropic error estimators for parabolic integro-differential equations in a two-dimensional convex polygonal domain. A continuous, piecewise linear finite element space is employed for the space discretization and the time discretization is based on the Crank-Nicolson method. The a posteriori contributions corresponding to space discretization is derived using the anisotropic interpolation estimates together with the Zienkiewicz-Zhu error estimator to approximate the error gradient. Two different continuous, piecewise quadratic reconstructions are used to obtain the error due to time discretization. Moreover, linear approximations of the Volterra integral term are used in a crucial way to estimate the quadrature error in the approximation of the Volterra integral term.

[1] Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308
[2] Georgios Akrivis, Charalambos Makridakis, and Ricardo H. Nochetto, A posteriori error estimates for the Crank-Nicolson method for parabolic equations, Math. Comp. 75 (2006), no. 254, 511–531. MR 2196979, https://doi.org/10.1090/S0025-5718-05-01800-4
[3] E. Bänsch, F. Karakatsani, and Ch. Makridakis, A posteriori error control for fully discrete Crank-Nicolson schemes, SIAM J. Numer. Anal. 50 (2012), no. 6, 2845–2872. MR 3022245, https://doi.org/10.1137/110839424
[4] V. Capasso, Asymptotic stability for an integro-differential reaction-diffusion system, J. Math. Anal. Appl. 103 (1984), no. 2, 575–588. MR 762575, https://doi.org/10.1016/0022-247X(84)90147-1
[5] L. Formaggia and S. Perotto, New anisotropic a priori error estimates, Numer. Math. 89 (2001), no. 4, 641–667. MR 1865506, https://doi.org/10.1007/s002110100273
[6] L. Formaggia and S. Perotto, Anisotropic error estimates for elliptic problems, Numer. Math. 94 (2003), no. 1, 67–92. MR 1971213, https://doi.org/10.1007/s00211-002-0415-z
[7] Morton E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), no. 2, 113–126. MR 1553521, https://doi.org/10.1007/BF00281373
[8] G. J. Habetler and R. L. Schiffman, A finite difference method for analyzing the compression of poro-viscoelastic media, Computing (Arch. Elektron. Rechnen) 6 (1970), 342–348 (English, with German summary). MR 286371, https://doi.org/10.1007/bf02238819
[9] W. E. Kastenberg and P. L. Chambre, On the stability of nonlinear space dependent reactor kinetics, Nucl. Sci. Eng., 31 (1968), pp. 67-79.
[10] Alexei Lozinski, Marco Picasso, and Virabouth Prachittham, An anisotropic error estimator for the Crank-Nicolson method: application to a parabolic problem, SIAM J. Sci. Comput. 31 (2009), no. 4, 2757–2783. MR 2520298, https://doi.org/10.1137/080715135
[11] Stefano Micheletti and Simona Perotto, Reliability and efficiency of an anisotropic Zienkiewicz-Zhu error estimator, Comput. Methods Appl. Mech. Engrg. 195 (2006), no. 9-12, 799–835. MR 2195291, https://doi.org/10.1016/j.cma.2005.02.009
[12] M. Picasso, An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: application to elliptic and parabolic problems, SIAM J. Sci. Comput. 24 (2003), no. 4, 1328–1355. MR 1976219, https://doi.org/10.1137/S1064827501398578
[13] M. Picasso, Numerical study of the effectivity index for an anisotropic error indicator based on Zienkiewicz-Zhu error estimator, Comm. Numer. Methods Engrg. 19 (2003), no. 1, 13–23. MR 1952014, https://doi.org/10.1002/cnm.546
[14] M. Picasso, Adaptive finite elements with large aspect ratio based on an anisotropic error estimator involving first order derivatives, Comput. Methods Appl. Mech. Engrg. 196 (2006), no. 1-3, 14–23. MR 2270123, https://doi.org/10.1016/j.cma.2005.11.018
[15] Marco Picasso and Virabouth Prachittham, An adaptive algorithm for the Crank-Nicolson scheme applied to a time-dependent convection-diffusion problem, J. Comput. Appl. Math. 233 (2009), no. 4, 1139–1154. MR 2557303, https://doi.org/10.1016/j.cam.2009.09.004
[16] G. Murali Mohan Reddy and Rajen K. Sinha, Ritz-Volterra reconstructions and a posteriori error analysis of finite element method for parabolic integro-differential equations, IMA J. Numer. Anal. 35 (2015), no. 1, 341–371. MR 3335208, https://doi.org/10.1093/imanum/drt059
[17] Simon Shaw and J. R. Whiteman, Numerical solution of linear quasistatic hereditary viscoelasticity problems, SIAM J. Numer. Anal. 38 (2000), no. 1, 80–97. MR 1770343, https://doi.org/10.1137/S0036142998337855
[18] Simon Shaw and J. R. Whiteman, A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems, Comput. Methods Appl. Mech. Engrg. 193 (2004), no. 52, 5551–5572. MR 2103153, https://doi.org/10.1016/j.cma.2004.04.005
[19] Vidar Thomée and Nai Ying Zhang, Error estimates for semidiscrete finite element methods for parabolic integro-differential equations, Math. Comp. 53 (1989), no. 187, 121–139. MR 969493, https://doi.org/10.1090/S0025-5718-1989-0969493-9
[20] R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation, Calcolo 40 (2003), no. 3, 195–212. MR 2025602, https://doi.org/10.1007/s10092-003-0073-2
[21] Elizabeth G. Yanik and Graeme Fairweather, Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Anal. 12 (1988), no. 8, 785–809. MR 954953, https://doi.org/10.1016/0362-546X(88)90039-9
[22] Nai Ying Zhang, On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type, Math. Comp. 60 (1993), no. 201, 133–166. MR 1149295, https://doi.org/10.1090/S0025-5718-1993-1149295-4
[23] O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (1987), no. 2, 337–357. MR 875306, https://doi.org/10.1002/nme.1620240206
[24] O. C. Zienkiewicz and J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique, Internat. J. Numer. Methods Engrg. 33 (1992), no. 7, 1331–1364. MR 1161557, https://doi.org/10.1002/nme.1620330702